FFOLIATED CORONA DECOMPOSITIONS
ASSAF NAOR AND ROBERT YOUNG
Dedicated to the memory of Louis Nirenberg A BSTRACT . We prove that the L norm of the vertical perimeter of any measurable subsetof the 3–dimensional Heisenberg group (cid:72) is at most a universal constant multiple of the(Heisenberg) perimeter of the subset. We show that this isoperimetric-type inequality isoptimal in the sense that there are sets for which it fails to hold with the L norm replacedby the L q norm for any q <
4. This is in contrast to the 5–dimensional setting, where theabove result holds with the L norm replaced by the L norm.The proof of the aforementioned isoperimetric inequality introduces a new structuralmethodology for understanding the geometry of surfaces in (cid:72) . In previous work (2017)we showed how to obtain a hierarchical decomposition of Ahlfors-regular surfaces intopieces that are approximately intrinsic Lipschitz graphs. Here we prove that any suchgraph admits a foliated corona decomposition , which is a family of nested partitions intopieces that are close to ruled surfaces.Apart from the intrinsic geometric and analytic significance of these results, whichsettle questions posed by Cheeger–Kleiner–Naor (2009) and Lafforgue–Naor (2012), theyhave several noteworthy implications. We deduce that the L distortion of a word-ballof radius n (cid:202) (cid:112) log n ; this is in contrast to higher dimensionalHeisenberg groups, where our previous work (2017) showed that the distortion of a word-ball of radius n (cid:202) (cid:112) log n . We also show that for any p > (cid:96) and (cid:96) p , yet not into a Hilbert space. This answers theclassical question of whether there is a metric analogue of the Kadec–Pełczy´nski theorem(1962), which implies that a normed space that embeds into both L p and L q for p < < q is isomorphic to a Hilbert space. Another consequence is that for any p > f : (cid:96) p → (cid:96) that cannot be factored through a subset of a Hilbertspace using Lipschitz functions, i.e., there are no Lipschitz functions g : (cid:96) p → (cid:96) and h : g ( (cid:96) p ) → (cid:96) such that f = h ◦ g ; this answers the question, first broached by Johnson–Lindenstrauss (1983), whether there is an analogue of Maurey’s theorem (1974) that sucha factorization exists if f is linear. Finally, we obtain conceptually new examples thatdemonstrate the failure of the Johnson–Lindenstrauss dimension reduction lemma (1983)for subsets of (cid:96) ; these are markedly different from the previously available examples(Brinkman–Charikar, 2003) which do not embed into any uniformly convex normedspace, while for any p > (cid:96) for which the Johnson–Lindenstrausslemma fails, yet they embed into (cid:96) p .(A.N.) P RINCETON U NIVERSITY , D
EPARTMENT OF M ATHEMATICS , F
INE H ALL , W
ASHINGTON R OAD , P
RINCE - TON , NJ 08544-1000, USA. E-
MAIL ADDRESS : [email protected]. (R.Y.) N EW Y ORK U NIVERSITY , C
OURANT I NSTITUTE OF M ATHEMATICAL S CIENCES , 251 M
ERCER S TREET ,N EW Y ORK , NY 10012, USA. E-
MAIL ADDRESS : [email protected]. A.N. was supported by the Packard Foundation and the Simons Foundation. R.Y. was supported by NSFgrant 1612061 and the Sloan Foundation. The research that is presented here was conducted under theauspices of the Simons Algorithms and Geometry (A&G) Think Tank. a r X i v : . [ m a t h . M G ] A p r C ONTENTS
1. Introduction 31.1. Geometric implications 61.1.1. Embeddings 61.1.2. Aspects of the Ribe program 81.1.3. Dimension reduction 131.1.4. Permanence of compression rates of groups 151.2. Decomposing surfaces into approximately ruled pieces 171.2.1. Fractal Venetian blinds abound 181.2.2. A maximally bumpy surface 231.3. Roadmap 252. Preliminaries 252.1. The Heisenberg group 262.2. Intrinsic graphs and intrinsic Lipschitz graphs 282.3. Characteristic curves 302.4. Automorphisms and characteristic curves 322.5. Measures on lines and the kinematic formula 342.6. Vertical perimeter and parametric vertical perimeter 343. Constructing surfaces and embeddings 363.1. Obtaining an embedding from an intrinsic graph 393.2. Constructing a bumpy intrinsic graph 433.2.1. The horizontal perimeter of Γ ψ i Γ ψ i (cid:82) (cid:72) L bounds and characteristic curves on monotone intrinsic graphs 9312.1. Bounding the tails of f (cid:107) f − F (cid:107) L (10 Q ) p in [LN14b] 108 OLIATED CORONA DECOMPOSITIONS 3
1. I
NTRODUCTION
Since our main theorem (Theorem 1.1 below) can be stated without the need to recallany specialized background, we will start by formulating it. After doing so, we will explainits significance and context, as well as geometric applications that answer longstandingopen questions. We will then describe our main conceptual contribution, called a foliatedcorona decomposition , which is a new structural methodology that we introduce in the proof of this theorem; see Remark 1.2 and mainly Section 1.2 for an overview.For a smooth function f : (cid:82) → (cid:82) define X f , Y f : (cid:82) → (cid:82) by setting for h = ( x , y , z ) ∈ (cid:82) , X f ( h ) def = ∂ f ∂ x ( h ) + y ∂ f ∂ z ( h ) and Y f ( h ) def = ∂ f ∂ y ( h ) − x ∂ f ∂ z ( h ). (1)Also, for t ∈ (0, ∞ ) define D t v f : (cid:82) → (cid:82) by setting for h = ( x , y , z ) ∈ (cid:82) , D t v ( h ) def = f ( x , y , z + t ) − f ( h ) (cid:112) t . (2) Theorem 1.1.
Every compactly supported smooth function f : (cid:82) → (cid:82) satisfies (cid:181) ˆ ∞ (cid:181) ˆ (cid:82) | D t v f ( h ) | d h (cid:182) d tt (cid:182) (cid:46) ˆ (cid:82) (cid:161) | X f ( h ) | + | Y f ( h ) | (cid:162) d h . (3) Moreover, one cannot replace the L ( d tt ) norm above by an L q ( d tt ) norm for any < q < . The second assertion (sharpness) of Theorem 1.1 resolves negatively the conjectureof [LN14b] that (3) holds with the L ( d tt ) norm in the left hand side replaced by the L ( d tt )norm. Notwithstanding the optimality of (3), it should be noted that it was previouslyunknown whether such a bound holds true merely for some finite exponent, namely thatthere exists 0 < p < ∞ such that in the setting of Theorem 1.1 we have (cid:181) ˆ ∞ (cid:181) ˆ (cid:82) | D t v f ( h ) | d h (cid:182) p d tt (cid:182) p (cid:46) ˆ (cid:82) (cid:161) | X f ( h ) | + | Y f ( h ) | (cid:162) d h . (4)It is simple to justify (see [NY18, Remark 4]) that if (4) holds, then the analogous boundholds for any larger exponent P > p . Remark . To briefly indicate what goes into Theorem 1.1, we first note that the func-tional inequality (3) is equivalent to a certain isoperimetric-type inequality (see (30)) forsufficiently smooth surfaces in (cid:82) . By [NY18], it turns out that it suffices to prove thisisoperimetric-type inequality for a more restricted class of surfaces (intrinsic Lipschitzgraphs; see Section 2.2). Such surfaces can still be very complicated, as one can see inFigure 1. However, notice that the example in Figure 1 has an anisotropic texture, withfeatures of many different scales that line up along a one-dimensional foliation. We will use throughout the following (standard) asymptotic notation. For a , b ∈ (0, ∞ ), the notations a (cid:46) b and b (cid:38) a mean that a (cid:201) Cb for some universal constant C ∈ (0, ∞ ). The notation a (cid:179) b stands for( a (cid:46) b ) ∧ ( b (cid:46) a ). If we need to allow for dependence on parameters, we indicate this by subscripts. Forexample, in the presence of an auxiliary parameter q , the notation a (cid:46) q b means that a (cid:201) C ( q ) b , where C ( q ) ∈ (0, ∞ ) is allowed to depend only on q , and analogously for the notations a (cid:38) q b and a (cid:179) q b . ASSAF NAOR AND ROBERT YOUNG F IGURE An example of an intrinsic Lipschitz graph.
We prove the desired isoperimetric-type inequality by showing that the texture of any intrinsic Lipschitz graph can be encoded as a foliated corona decomposition, which isa multi-scale hierarchical partition of the surface. The pieces of this decomposition areroughly rectangular regions that mimic the dimensions and orientation of the features ofthe surface. Crucially, we can control the number and size of these pieces. The desired in-equality holds locally on each piece up to suitably controlled error, and the full inequalityis obtained by summing the resulting estimates. This process is illustrated in Figure 2 andFigure 3, and a more detailed overview can be found in Section 1.2.In contrast to Theorem 1.1, we have the following theorem, the case p = p ∈ (1, 2] of which is due (via a different proof ) to [LN14b]. OLIATED CORONA DECOMPOSITIONS 5
Theorem 1.3.
If f : (cid:82) → (cid:82) is smooth and compactly supported, then for every p ∈ (1, 2] , (cid:181) ˆ ∞ (cid:181) ˆ (cid:82) | D t v f ( h ) | p d h (cid:182) p d tt (cid:182) (cid:46) (cid:112) p − (cid:181) ˆ (cid:82) (cid:161) | X f ( h ) | p + | Y f ( h ) | p (cid:162) d h (cid:182) p . (5)See [LN14b] for a variant of Theorem 1.3 when p ∈ (2, ∞ ). The pertinent point ofcomparison to (3) is as p → + , namely there is a jump discontinuity at the endpoint p = p in the right hand side of (5) is not specifiedin [LN14b], but one obtains (5) in the form stated above by tracking the dependence on p in the proof of [LN14b]; we explain how to do so in Appendix A below. We conjecture thatthe following bound holds, which is better than (5) only in terms of the dependence on p ; its geometric ramifications will be derived later (see Remark 1.15), at which point itwill become clear why we need to record an explicit (power-type) dependence as p → + in (5) rather than using the implicit (cid:46) p notation as done in [LN14b]. Conjecture . In the setting of Theorem 1.3 we have (cid:181) ˆ ∞ (cid:181) ˆ (cid:82) | D t v f ( h ) | p d h (cid:182) p d tt (cid:182) (cid:46) (cid:112) p − (cid:181) ˆ (cid:82) (cid:161) | X f ( h ) | p + | Y f ( h ) | p (cid:162) d h (cid:182) p . (6)Another key point of comparison between Theorem 1.1 and the literature is with itshigher-dimensional counterpart due to [NY18]. For a smooth function f : (cid:82) → (cid:82) , denotein analogy to (1) and (2) for every h = ( x , y , x , y , z ) ∈ (cid:82) and t ∈ (0, ∞ ), X f ( h ) def = ∂ f ∂ x ( h ) − y ∂ f ∂ z ( h ), X f ( h ) def = ∂ f ∂ x ( h ) − y ∂ f ∂ z ( h ), Y f ( h ) def = ∂ f ∂ y ( h ) + x ∂ f ∂ z ( h ), Y f ( h ) def = ∂ f ∂ y ( h ) + x ∂ f ∂ z ( h ),and D t v ( h ) def = f ( x , y , x , y , z + t ) − f ( h ) (cid:112) t .We then have the following theorem (it holds with (cid:82) replaced mutatis mutandis by (cid:82) k + for every k (cid:202)
2; we are focusing only on (cid:82) because the crucial qualitative difference thatwe establish here is between dimension 3 and all the larger odd dimensions). Theorem 1.5.
If f : (cid:82) → (cid:82) is smooth and compactly supported, then for every p ∈ [1, 2] , (cid:181) ˆ ∞ (cid:181) ˆ (cid:82) | D t v f ( h ) | p d h (cid:182) p d tt (cid:182) (cid:46) (cid:181) ˆ (cid:82) (cid:161) | X f ( h ) | p + | Y f ( h ) | p + | X f ( h ) | q + | Y f ( h ) | p (cid:162) d h (cid:182) p . (7)The case p = p ∈ (1, 2] the bound (7)but with (cid:46) replaced by (cid:46) p is from [LN14b]. The case p = p as in [LN14b], follows by interpolatingbetween the cases p = p = ASSAF NAOR AND ROBERT YOUNG the boundedness of a linear operator, the L ( L p ) norms in the left hand side of (7) are aninterpolation family by classical interpolation theory [BL76], and the Sobolev W p normsin the right hand side of (7) are an interpolation family by [Bad09, Theorem 8.8].1.1. Geometric implications.
Let (cid:72) be the 3–dimensional Heisenberg group with realcoefficients. As a set, (cid:72) is identified with (cid:82) , and the group structure on (cid:72) is given by ∀ g = ( x , y , z ), h = ( χ , υ , ζ ) ∈ (cid:82) , g h def = (cid:179) x + χ , y + υ , z + ζ +
12 ( x υ − y χ ) (cid:180) . (8)The identity element is = (0, 0, 0) and the inverse of g = ( x , y , z ) is g − = ( − x , − y , − z ). Thecenter of (cid:72) is {0} × {0} × (cid:82) and if we let (cid:72) (cid:90) be the discrete subgroup of (cid:72) that is generatedby (1, 0, 0) and (0, 1, 0), then we have (cid:72) (cid:90) = (cid:189)(cid:179) x , y , z + x y (cid:180) : x , y , z ∈ (cid:90) (cid:190) ⊆ (cid:90) × (cid:90) × (cid:90) d W : (cid:72) (cid:90) × (cid:72) (cid:90) → (cid:78) ∪ {0} be the left-invariant word metric on (cid:72) (cid:90) that is induced bythe symmetric set of generators {( −
1, 0, 0), (1, 0, 0), (0, −
1, 0), (0, 1, 0)}. It is well-known (andelementary to verify) that for every g = ( x , y , z ), h = ( χ , υ , ζ ) ∈ (cid:72) (cid:90) we have d W ( g , h ) (cid:179) | x − χ | + | y − υ | + (cid:113) | z − ζ − x υ + y χ | (9)In fact, an exact formula for d W ( g , h ), which directly implies (9), is derived in [Bla03]. Forevery n ∈ (cid:78) , denote the word-ball of radius n centered at the identity element by B n def = (cid:169) g ∈ (cid:72) (cid:90) : d W ( g , ) (cid:201) n (cid:170) . (10)1.1.1. Embeddings.
Recall that a metric space ( M , d M ) is said to admit a bi-Lipschitzembedding into a Banach space ( X , (cid:107)·(cid:107) X ) if there exist D ∈ [1, ∞ ) and φ : M → X such that ∀ x , y ∈ M , d M ( x , y ) (cid:201) (cid:107) φ ( x ) − φ ( y ) (cid:107) X (cid:201) Dd M ( x , y ). (11)The infimum over those D ∈ [1, ∞ ) for which this holds is called the X –distortion of M and is denoted c X ( M ). If no such D exists, then one writes c X ( M ) = ∞ .Theorem 1.6 below is a sharp asymptotic evaluation of c (cid:96) ( B n ). It answers a questionposed in [LN06, CK10a, CK10b, CKN09, CKN11, Nao10, Pan13, LN14b]; these referencesask for the asymptotic evaluation of c (cid:96) ( B n ), but most of them also conjecture that c (cid:96) ( B n ) (cid:179) (cid:112) log n , so Theorem 1.6 constitutes both a resolution of an open problem,and an unexpected answer. The fact that lim n →∞ c (cid:96) ( B n ) = ∞ is due to [CK10a], thepreviously best known upper bound [Ass83] was c (cid:96) ( B n ) (cid:46) (cid:112) log n and the previouslybest-known lower bound [CKN11] was c (cid:96) ( B n ) (cid:202) (log n ) δ for some positive but very smalluniversal constant δ ; thus both the upper and the lower bounds of Theorem 1.6 are new. Theorem 1.6. c (cid:96) ( B n ) (cid:179) (cid:112) log n for every integer n (cid:202) . In contrast, the word-ball of radius n (cid:202) (cid:96) –distortion of order (cid:112) log n ; this was proved in [NY18] using Theorem 1.5.The statement of Theorem 1.6 has two parts. While the lower bound c (cid:96) ( B n ) (cid:38) (cid:112) log n is framed above as a “negative result” (impossibility of embedding), it encapsulates a“positive result,” namely the aforementioned new structural information on surfacesin (cid:72) , to which most of this article is devoted. The upper bound c (cid:96) ( B n ) (cid:46) (cid:112) log n is a OLIATED CORONA DECOMPOSITIONS 7 “positive result,” namely a new geometric realization of B n , but we will soon see that ithas ramifications for counterexamples to natural geometric questions.The estimate (3) of Theorem 1.1 implies the lower bound c (cid:96) ( B n ) (cid:38) (cid:112) log n . In fact,such vertical-versus-horizontal Poincaré inequalities were originally envisaged as obstruc-tions to embeddings of B n into various spaces; see [ANT13, NN12, LN14b, NY17], andmost pertinently Section 3 of [NY18], where we treated such matters in greater general-ity than what is needed here; in particular, for any p (cid:202)
1, if every compactly supportedsmooth function f : (cid:82) → (cid:82) satisfies the inequality (cid:181) ˆ ∞ (cid:181) ˆ (cid:82) | D t v f ( h ) | d h (cid:182) p d tt (cid:182) p (cid:46) ˆ (cid:82) (cid:161) | X f ( h ) | + | Y f ( h ) | (cid:162) d h , (12)then by [NY18, § 3] and the reasoning in [NY18, § 1.3] we have c (cid:96) ( B n ) (cid:38) (log n ) p .Thus, c (cid:96) ( B n ) (cid:38) (cid:112) log n , since Theorem 1.1 asserts that (12) holds for p =
4. This alsodemonstrates that the matching upper bound c (cid:96) ( B n ) (cid:46) (cid:112) log n of Theorem 1.6 impliesthe second assertion of Theorem 1.1, namely the optimality of the L ( d tt ) norm in the lefthand side of (3). Here we prove the following more refined embedding statement whichwe formulate as a separate theorem because it has further noteworthy applications. Theorem 1.7.
For every ϑ (cid:202) and every integer n (cid:202) there exists φ = φ n , ϑ : (cid:72) (cid:90) → (cid:96) withrespect to which every two points g = ( x , y , z ), h = ( χ , υ , ζ ) ∈ (cid:72) (cid:90) with d W ( g , h ) (cid:201) n satisfy (cid:107) φ ( g ) − φ ( h ) (cid:107) (cid:96) (cid:179) | x − χ | + | y − υ | + (cid:112) | z − ζ − x υ + y χ | (log n ) ϑ . (13)By (9) and the case ϑ = of Theorem 1.7, the following weakening of (13) holds. ∀ g , h ∈ B n , d W ( g , h ) (cid:112) log n (cid:46) (cid:107) φ ( g ) − φ ( h ) (cid:107) (cid:96) (cid:46) d W ( g , h ).So, the upper bound c (cid:96) ( B n ) (cid:46) (cid:112) log n of Theorem 1.6 follows from Theorem 1.7. How-ever, Theorem 1.7 is of further use thanks to the following embedding result of [LN14a]. Atpresent, the fact that both our embedding and that of [LN14a] yield the same expression(up to universal constant factors) for the metric in the image seems to be a fortunate andconsequential coincidence; it would be valuable, if possible, to explain conceptually whythose formulas coincided (e.g. is this inevitable due to underlying symmetries?). Theorem 1.8.
For any p > , any ϑ (cid:202) p and any integer n (cid:202) there is ψ = ψ n , p , ϑ : (cid:72) (cid:90) → (cid:96) p such that every g = ( x , y , z ), h = ( χ , υ , ζ ) ∈ (cid:72) (cid:90) with d W ( g , h ) (cid:201) n satisfy (cid:107) ψ ( g ) − ψ ( h ) (cid:107) (cid:96) p (cid:179) | x − χ | + | y − υ | + (cid:112) | z − ζ − x υ + y χ | (log n ) ϑ . (14)Theorem 1.8 is not formulated explicitly in [LN14a], but it is a direct consequence ofLemma 3.1 in [LN14a] combined with the finite-determinacy theorem of [Ost12], whichtogether imply that for every ε ∈ (0, ] there exists an embedding σ = σ ε , p : (cid:72) → (cid:96) p forwhich every g = ( x , y , z ), h = ( χ , υ , ζ ) ∈ (cid:72) (cid:90) satisfy (cid:107) σ ( g ) − σ ( h ) (cid:107) (cid:96) p (cid:179) | x − χ | − ε + | y − υ | − ε + ε p | z − ζ − x υ + y χ | − ε . (15) ASSAF NAOR AND ROBERT YOUNG (Without reference to [Ost12], Lemma 3.1 in [LN14a] asserts the existence of such anembedding into L p rather than into (cid:96) p .) To derive Theorem 1.8 from (15), let π : (cid:72) → (cid:82) bethe map that is given by setting π ( x , y , z ) = ( x , y ) for ( x , y , z ) ∈ (cid:72) and choose ε = n and ψ = σ (log n ) ϑ − p ⊕ π : (cid:72) (cid:90) → (cid:96) p ⊕ (cid:82) ∼= (cid:96) p . (16)1.1.2. Aspects of the Ribe program.
Inspired by a fundamental rigidity theorem of [Rib76]and first put forth in [Bou86], the Ribe program is a web of conjectures and analogieswhose goal is to transfer linear phenomena in the geometry of Banach spaces to questionsabout metric spaces, where Lipschitz mappings take the role of bounded linear operators;see e.g. the surveys [Kal08, Nao12, Bal13, Ost13, Nao18]. We will next explain how theabove results answer natural questions in this area.Theorem 1.9 below follows from Theorem 1.7, Theorem 1.8 and [ANT13, LN14b]. Itanswers a longstanding question in metric embedding theory; even though (to the bestof our knowledge) this question never appeared in published texts, it was a folkloreopen problem. To briefly explain the context, the classical work [KP62] (together with adifferentiation argument of [Man72]) implies that for 1 (cid:201) p < r < q < ∞ , if a Banach space X admits a bi-Lipschitz embedding into both L p and L q , then X also admits a bi-Lipschitzembedding into L r . The case r = X embeds into L p for twofinite values of p that lie on both sides of 2, then X must embed into (hence, by [Enf70],be linearly isomorphic to) a Hilbert space; a different proof of the latter statement, as aspecial case of a much more general phenomenon, follows from [Kwa72]. In light of thesefacts about the geometry of Banach spaces, one is naturally led to ask if a metric space M that embeds bi-Lipschitzly into L p for two finite values of p that lie on both sides of 2must admit a bi-Lipschitz embedding into a Hilbert space. Theorem 1.9.
For any < p (cid:201) there is a metric space M that admits a bi-Lipschitzembedding into (cid:96) as well as into (cid:96) r for all r (cid:202) p, yet M does not admit a bi-Lipschitzembedding into L q for any < q < p. More generally, M does not admit a bi-Lipschitzembedding into a Banach space whose modulus of uniform convexity has power type q for (cid:201) q < p. For the statement of Theorem 1.9, recall that a Banach space ( X , (cid:107) · (cid:107) ) has modulusof uniform convexity of power type q if there is C > (cid:107) x + y (cid:107) (cid:201) − C (cid:107) x − y (cid:107) q holds for any unit vectors x , y ∈ X . By [Cla36, Han56],for 1 < q < ∞ any L q ( µ ) space has modulus of uniform convexity of power type max{ q , 2}. Proof of Theorem 1.9 assuming Theorem 1.7 and Theorem 1.8.
For every n ∈ (cid:78) , we define M n = φ n , ϑ ( B n ) ⊆ (cid:96) , where φ n , ϑ is as in Theorem 1.7 applied with ϑ = p ,4} (cid:202) .By considering the union of sufficiently widely-spaced translations in (cid:96) of the finitesets { M n } ∞ n = , we see that there is M ⊆ (cid:96) such that sup n ∈ (cid:78) c M ( M n ) < ∞ .For every r (cid:202) p , consider ψ n , r , ϑ ( B n ) ⊆ (cid:96) r , where ψ n , r , ϑ is as in Theorem 1.8. The-orem 1.7 and Theorem 1.8 show that ψ n , r , ϑ ( B n ) is bi-Lipschitz equivalent with O (1)distortion to M n . Hence, by considering a suitable union of translations in (cid:96) r of the finite We have seen it appear in writing only in grant proposals, and it was posed verbally among experts. Inparticular, we are indebted to Gideon Schechtman for valuable discussions on this matter over the years.
OLIATED CORONA DECOMPOSITIONS 9 sets { ψ n , r , ϑ ( B n )} ∞ n = , we see that c (cid:96) r ( M ) < ∞ . Let X be a Banach space whose modulus ofuniform convexity has power type q for 2 (cid:201) q < p . By [LN14b] we have(log n ) q (cid:46) X c X ( B n ) (cid:46) (log n ) ϑ c X ( M n ) = (log n ) p ,4} c X ( M n ),where the penultimate step holds because, due to (13), M n and B n are bi-Lipschitzequivalent with distortion O ((log n ) ϑ ). Therefore, since q < p , c X ( M n ) (cid:38) X (log n ) q − p ,4} −−−−→ n →∞ ∞ .Hence, c X ( M ) = ∞ , as required. For future reference we record in passing that we obtainedthe following bound when X = L q and 1 < q < p . c L q ( M n ) (cid:38) q (log n ) q ,2} − p ,4} . (17)Note that the bound in [ANT13], which is asymptotically weaker than that of [LN14b],suffices for the qualitative conclusion c X ( M ) = ∞ of Theorem 1.9. The above estimatesseem to be the best that one could achieve using available methods; it would be veryinteresting to determine the optimal behavior, e.g. if an n –point metric space W embedswith O (1) distortion into (cid:96) and also into (cid:96) p for some p >
2, how large can c (cid:96) ( W ) be? (cid:3) Remark . With more care it is possible to ensure that the metric space M of Theo-rem 1.9 is a left-invariant metric δ = δ p on (cid:72) (cid:90) ; see Theorem 3.2. Concretely, for p = δ can be taken to satisfy the following bounds for any ( a , b , c ) ∈ (cid:72) (cid:90) with | c | (cid:202) δ (cid:161) , ( a , b , c ) (cid:162) (cid:179) | a | + | b | + (cid:112)| c | (cid:112) log | c | · (log log | c | ) .By the reasoning in [NP11, Section 9], since (cid:72) (cid:90) is amenable, it follows that ( (cid:72) (cid:90) , δ ) admits abi-Lipschitz embedding into L and L r for all r (cid:202) p which is also equivariant (with respectto an action of (cid:72) (cid:90) on, respectively, L and L p by affine isometries); we did not investigateif this holds for equivariant embeddings into the sequence spaces (cid:96) and (cid:96) r .The natural question how the embeddability of a group into L p depends on p was alsostudied in the literature; see [CDH10, Czu17], and especially the recent solution of thisquestion in [MdlS20], where it is proved that the phenomenon of Theorem 1.9 does nothold for equivariant coarse embeddings (namely, for such embeddings the correspondingset of p is always an interval). Note that for coarse embeddings that need not be equivari-ant, the statement of [MdlS20] was previously known as a direct consequence of [MN04,Remark 5.10] (from here, using [NP11], one gets the full equivariant statement of [MdlS20]for amenable groups). Theorem 3.2 shows that the situation is markedly different if oneconsiders bi-Lipschitz embeddings rather than coarse embeddings.The following question arises naturally from Theorem 1.9 and seems quite difficult. Question 1.11.
For a metric space M , how complicated can the following set be? (cid:169) (cid:201) p < ∞ : c L p ( M ) < ∞ (cid:170) .Theorem 1.9 leaves the possibility that there is better behavior in the reflexive range, i.e.,that if a metric space M embeds bi-Lipschitzly into (cid:96) p and (cid:96) q for 1 < q < < p < ∞ , then M embeds bi-Lipschitzly into a Hilbert space. If true, this would be an excellent theorem, but due to Theorem 1.9 we speculate that the answer is negative. A substantial new ideaseems to be needed here. Less ambitiously, does the above assumption (even allowing q =
1) imply that M embeds into a Hilbert space with finite average distortion (see [Nao19]for the relevant definition)? Does this imply that every n –point subset of M embeds intoa Hilbert space with bi-Lipschitz distortion o (log n ), i.e., asymptotically better than thedistortion that is guaranteed by the general embedding theorem of [Bou85]?The above reasoning also leads to Theorem 1.12 below, which answers another naturalquestion arising in the Ribe program, on the factorization of Lipschitz functions.We first briefly make preparatory observations that will be also useful elsewhere. Recallthat for K ∈ (cid:78) a metric space X is said to be K - doubling if for every r >
0, any ball B ⊆ X of radius r can be covered by K balls of radius r /2. X is doubling if it is K –doubling forsome K ∈ (cid:78) . The metric space M of Theorem 1.9 can be taken to be doubling. Indeed,fix p > n ∈ (cid:78) . As in the proof of Theorem 1.9, write ϑ =
1/ min{ p , 4}. It was shownin [LN14a] that ψ n , p , ϑ ( (cid:72) (cid:90) ) is a O (1)–doubling subset of (cid:96) p . Let S ⊆ (cid:96) p be the disjointunion of translates in (cid:96) p of the finite sets { ψ n , p , ϑ ( B n )} ∞ n = that are sufficiently widely-spaced so as to ensure that S is a doubling subset of (cid:96) p , and sup n ∈ (cid:78) c S ( M n ) < ∞ . As inthe proof of Theorem 1.9, using Theorem 1.7 we get an embedding ϕ : S → (cid:96) satisfying (cid:107) ϕ ( x ) − ϕ ( y ) (cid:107) (cid:96) (cid:179) (cid:107) x − y (cid:107) (cid:96) p for all x , y ∈ S . Thus ϕ ( S ) = M is a doubling subset of (cid:96) .Since S is doubling, by [LN05] we can extend ϕ to a Lipschitz function f : (cid:96) p → (cid:96) . Ifthere were Lipschitz mappings g : (cid:96) p → (cid:96) and h : g ( (cid:96) p ) → (cid:96) such that f = h ◦ g , then itwould follow that for all x , y ∈ S we have (cid:107) x − y (cid:107) (cid:96) p (cid:179) (cid:107) ϕ ( x ) − ϕ ( y ) (cid:107) (cid:96) = (cid:176)(cid:176) h (cid:161) g ( x ) (cid:162) − h (cid:161) g ( y ) (cid:162)(cid:176)(cid:176) (cid:96) (cid:46) (cid:107) g ( x ) − g ( y ) (cid:107) (cid:96) (cid:46) (cid:107) x − y (cid:107) (cid:96) p .Therefore, g ◦ ϕ − would be a bi-Lipschitz embedding of M into (cid:96) , which we provedabove was impossible. We thus arrive at the following statement. Theorem 1.12.
For any < p < ∞ there is a Lipschitz mapping f : (cid:96) p → (cid:96) that cannotbe factored through a subset of a Hilbert space using Lipschitz mappings. Namely, theredo not exist Lipschitz mappings g : (cid:96) p → (cid:96) and h : g ( (cid:96) p ) → (cid:96) such that f = h ◦ g . Moregenerally, f cannot be factored using Lipschitz mappings through a subset of a Banachspace whose modulus of uniform convexity has power type q for (cid:201) q < min{4, p } . By [LP68, Theorem 5.2], for p (cid:202) (cid:96) p to (cid:96) factors through (cid:96) (the factorization is via linear operators, though by [JMS09] this is equivalent to factor-ization using Lipschitz functions as above). Theorem 1.12 demonstrates that there is noanalogue of this factorization phenomenon for Lipschitz mappings.Such investigations arose in the Ribe program in the seminal work [JL84] which had amajor influence on the subsequent fruitful efforts by many mathematicians in search ofmetric analogues of the extension and factorization paradigm of [Mau74]. This search isitself intimately intertwined with the search for metric theories of type and cotype.We refer to the survey [Mau03] for an exposition of the powerful and deep theoryof type and cotype of Banach spaces; it suffices to say here that one can define linearinvariants of Banach spaces that are called type cotype
2, such that L p has type2 if 2 (cid:201) p < ∞ and cotype 2 if 1 (cid:201) p (cid:201)
2, and such that the following extension andfactorization phenomenon [Mau74] holds.
OLIATED CORONA DECOMPOSITIONS 11
Suppose that Y is a Banach space of type 2 and that Z is a Banach space of cotype 2.Let X be a linear subspace of Y and let τ : X → Z be a bounded linear operator. Thenthere exist a bounded linear operator T : Y → Z that extends τ , a Hilbert space H andbounded linear operators A : Y → H , B : A ( Y ) → Z with T = B A .[JL84] raised the question of when the analogous statement holds in the metric setting.Namely, now Y , Z are metric spaces, X is an arbitrary subset of Y , f : X → Z is a Lipschitzmapping, and we ask for the same extension and factorization through a Hilbert space H ,i.e., to establish the existence of Lipschitz mappings F : Y → Z , α : Y → H and β : α ( Y ) → Z , such that the following diagram commutes. Y α (cid:47) (cid:47) F (cid:33) (cid:33) α ( Y ) (cid:31) (cid:127) ⊆ (cid:47) (cid:47) β (cid:15) (cid:15) HX (cid:63)(cid:31) ⊆ (cid:79) (cid:79) f (cid:47) (cid:47) Z (18)An implicit but central part of this endeavor encompasses the important issue of how todefine useful notions of type 2 and cotype 2 for metric spaces so that, at the very least, (cid:96) p has type 2 for 2 (cid:201) p < ∞ and cotype 2 for 1 (cid:201) p (cid:201)
2. Clearly (18) has two components. Thefirst is if f admits the Lipschitz extension F . The second is if F can be factored through asubset of a Hilbert space. While these questions come hand-in-hand in the linear theoryof [Mau74] (see also [Pis86a]), they are different issues in the metric setting.The main focus of [JL84] was the Lipschitz extension problem, so it highlighted thefirst component above. At the time, the metric version of the extension problem was abold and speculative question, but [Bal92] introduced metric notions of type 2 and cotype2 and obtained a powerful extension result for maps from spaces of Markov type 2 tospaces of Markov cotype 2. Combined with [NPSS06], this provides a quite satisfactoryunderstanding of the extension component of (18) when the target space is (cid:96) p , 1 < p < (cid:96) (see [Kal12, MN13b] for apartial negative answer, and [MM16] for an intriguing algorithmic reformulation).In contrast to the achievement of [Bal92], Theorem 1.12 demonstrates that there is no way to define notions of type 2 and cotype 2 for metric spaces so that any map from a spaceof type 2 to a space of cotype 2 factors through Hilbert space and such that (cid:96) p has type 2when 2 < p < ∞ and cotype 2 when p =
1. Though this resolves the factorization questionwhen the target is (cid:96) , it remains a fascinating open problem to see if a factorization theoryanalogous to [Bal92] can be developed when the target is (cid:96) q for 1 < q < does not shed light on Lipschitz factorization, but Quoting what [Bal92] says about this crucial duality step: “
This lemma is a variant of one used byMaurey. A related lemma was found earlier by Johnson, Lindenstrauss and Schechtman: their result actuallycharacterises extensions which factor through subsets of Hilbert space, a problem much closer to Maurey’sargument. Their lemma provided much of the stimulus for the present work. ” Unfortunately, it seems that thework of Johnson, Lindenstrauss and Schechtman that is mentioned in [Bal92] was never published. the factorization issue was broached in [FJ09, CD14]. One can deduce from [CD14]the following factorization criterion. Given Φ >
0, metric spaces ( X , d X ), ( Z , d Z ) and f : X → Z , there exists a Hilbert space H and a factorization f = β ◦ α for some Lipschitzmappings α : X → H and β : α ( X ) → Z with (cid:107) α (cid:107) Lip (cid:107) β (cid:107) Lip (cid:201) Φ if and only if for all n ∈ (cid:78) and x , . . . , x n ∈ X , any two symmetric stochastic matrices A = ( a i j ), B = ( b i j ) ∈ M n ( (cid:82) )such that A − B is positive semidefinite satisfy the following quadratic inequality. n (cid:88) i = n (cid:88) j = a i j d Z (cid:161) f ( x i ), f ( x j ) (cid:162) (cid:201) Φ n (cid:88) i = n (cid:88) j = b i j d X ( x i , x j ) . (19)Theorem 1.12 yields the first example of a Lipschitz mapping f : (cid:96) p → (cid:96) for 2 < p < ∞ that fails to satisfy (19) for any Φ >
0, despite the fact that if f were a linear operator, thenby [Mau74] it would automatically satisfy (19) with Φ (cid:46) p (cid:107) f (cid:107) Lip . Remark . Another counterexample to the nonlinear version of [Mau74] arises froman embedding of the Laakso graphs into a non-classical Banach space. Let { Λ n } ∞ n = be theLaakso graphs [Laa00, Laa02], indexed so that | Λ n | = n ; these are series-parallel (henceplanar) graphs that are O (1)–doubling when equipped with their shortest-path metric.On one hand, the Laakso graphs do not admit a bi-Lipschitz embedding into a Hilbertspace. In fact, by [Laa00, LP01], we have c (cid:96) ( Λ n ) (cid:38) (cid:112) log n (this is sharp by the general em-bedding theorem of [Rao99]). Moreover, by [MN08], for every uniformly convex Banachspace X we have lim n →∞ c X ( Λ n ) = ∞ .On the other hand, by [GNRS04], we have sup n ∈ (cid:78) c (cid:96) ( Λ n ) < ∞ , and by [JS09], we havesup n ∈ (cid:78) c Y ( Λ n ) < ∞ when Y is a Banach space that is not reflexive. By considering trans-lates of the images of the embeddings in (cid:96) that are sufficiently widely spaced, we obtaina doubling subset Λ ⊆ (cid:96) such that c Y ( Λ ) < ∞ for any nonreflexive Banach space Y and c X ( Λ ) = ∞ for any uniformly convex Banach space X .By [Jam78], there exists a Banach space (cid:74) that has type 2, yet (cid:74) is not reflexive; a differentconstruction of such a Banach space was found in [PX87]. So, Λ embeds bi-Lipschitzlyinto both the cotype 2 space (cid:96) and the type 2 space (cid:74) , yet not into a Hilbert space. Thisis impossible in the linear setting; by [Kwa72] a Banach space of type 2 and cotype 2 isisomorphic to a Hilbert space (this is a far reaching generalization of the aforementionedconsequence of [KP62] that motivates Theorem 1.9). This reasoning also produces astronger asymptotic estimate than (17), since c (cid:96) ( Λ n ) (cid:38) (cid:112) log n , but it cannot shed lighton the (cid:96) p setting of (17) because it relies precisely on the non-reflexivity of (cid:74) (through theuse of [JS09]) to deduce that sup n ∈ (cid:78) c (cid:74) ( Λ n ) < ∞ .The Laakso graphs also lead to a counterexample to the metric version of [Mau74].Let ϕ : Λ → (cid:74) be a bilipschitz embedding. Since Λ is a doubling subset of (cid:96) , one canuse [LN05] to construct a Lipschitz map f : (cid:96) → (cid:74) that extends ϕ . As above, f cannotfactor through a Hilbert space (or even through any uniformly convex Banach space X )by Lipschitz maps, because such a factorization would produce a bilipschitz embeddingof Λ into a Hilbert space (respectively, into X ).This discussion shows that if one is allowed to replace (cid:96) p in Theorem 1.9 and Theo-rem 1.12 by non-classical (indeed, “exotic” and hard to come by) Banach spaces such as (cid:74) ,then it is possible to demonstrate the failure of the metric space version of [Mau74] andits important precursor [Kwa72] using well-known examples. OLIATED CORONA DECOMPOSITIONS 13
Part of the impetus for the search for definitions of metric space notions of type 2 andcotype 2 was the hope of obtaining a metric version of the theorem of [Kwa72], but it waswell-known to experts that the metric definitions of type 2 and cotype 2 found over thepast decades are not suitable for this purpose (see e.g. the discussion in [DLP13]). Theabove discussion demonstrates conclusively that it is impossible to define metric spacenotions of type 2 and cotype 2 that are bi-Lipschitz invariant, pass to subsets, coincide forBanach spaces with type 2 and cotype 2, and for which [Kwa72] holds for doubling metricspaces, i.e., any doubling space that has both type 2 and cotype 2 admits a bi-Lipschitzembedding into a Hilbert space (the corresponding statement with Λ replaced by a metricspace that is not doubling follows by using [Bou86] instead of the Laakso graphs in theabove reasoning; in fact, using the improvement [Bau07] of [Bou86], the infinite binarytree embeds bilipschitzly into both (cid:96) and (cid:74) , but not into a Hilbert space). Theorem 1.9shows that this is so even if one restricts attention to subsets of (cid:96) p for p > Dimension reduction.
By a highly influential lemma of [JL84], any finite subset S ofa Hilbert space embeds with bi-Lipschitz distortion O (1) into a k –dimensional Hilbertspace for k (cid:46) log | S | ; see [Nao18] for an indication of the significance of this statement.The question whether this phenomenon holds with Hilbert space replaced by (cid:96) was aprominent open problem until it was resolved negatively in [BC05], where it was shownthat for arbitrarily large n ∈ (cid:78) there is an n –point subset D n of (cid:96) such that if D n embedswith bi-Lipschitz distortion O (1) into (cid:96) k , then necessarily k (cid:202) n c for some universalconstant c >
0. In [LMN05] it was shown that D n can be taken to be O (1)–doubling, andin [NPS18] it was shown that (cid:96) k can be replaced by an arbitrary k –dimensional subspaceof the Schatten–von Neumann trace class S ; both of these enhancements hold withoutchanging the conclusion (other than perhaps values of universal constants).The examples { D n } ∞ n = of [BC05] are the diamond graphs [NR03], while their aforemen-tioned doubling counterparts in [LMN05] are the Laakso graphs { Λ n } ∞ n = that we discussedin Remark 1.13. By [MN08, JS09] we have sup n ∈ (cid:78) c X ( D n ) = sup n ∈ (cid:78) c X ( Λ n ) = ∞ for everyuniformly convex Banach space X . In fact, by [JS09] the converse of this statement holdstrue (though we do not need it below), namely X admits an equivalent uniformly convexnorm if and only if sup n ∈ (cid:78) c X ( D n ) = ∞ or sup n ∈ (cid:78) c X ( Λ n ) = ∞ . Theorem 1.14 below ob-tains new examples that demonstrate the failure of dimension reduction in (cid:96) à la [JL84],which are qualitatively different than the previously known examples, since our examplesdo admit a bi-Lipschitz embedding into a uniformly convex Banach space (specifically,into (cid:96) p for any p > Theorem 1.14.
There is a universal constant c > with the following property. For every n ∈ (cid:78) and < p (cid:201) there exists a O (1) –doubling subset H n = H n ( p ) of (cid:96) with | H n | (cid:201) n suchthat c (cid:96) q ( H n ) (cid:46) for all q (cid:202) p, and for every D (cid:202) , if X is a finite-dimensional subspace ofthe Schatten–von Neumann trace class S for which c X ( H n ) (cid:201) D, then necessarily dim( X ) (cid:202) exp (cid:181) cD (log n ) − p (cid:182) . (20) In the statement of Theorem 1.14, recall that for p (cid:202) S p is the Banach space of all the compact operators T : (cid:96) → (cid:96) that satisfy (cid:107) T (cid:107) S p def = (cid:179) Trace (cid:163) ( T ∗ T ) p (cid:164)(cid:180) p < ∞ .Note that (cid:96) p is the subspace of S p consisting of the diagonal operators. Thus, the dimen-sion reduction lower bound (20) holds in particular for any subspace X of (cid:96) .The proof of Theorem 1.14 is short (modulo previously stated results and the availableliterature), so we present the quick derivation now instead of postponing it to a latersection; it mimics the reasoning of [LN04] while combining it with [LN14b], Theorem 1.7and Theorem 1.8, as well as structural information on subspaces of S from [NPS18]. Proof of Theorem 1.14.
By (9) we have | B m | (cid:179) m for all m ∈ (cid:78) . So, fix m ∈ (cid:78) with m (cid:179) (cid:112) n such that n (cid:46) | B m | (cid:201) n . Using the mapping φ m , p : (cid:72) (cid:90) → (cid:96) of Theorem 1.7, define H n def = φ m , p ( B m ).By combining Theorem 1.7 and Theorem 1.8, we indeed have c (cid:96) q ( H n ) (cid:46) q (cid:202) p .Let X be a finite-dimensional subspace of S . Fix 1 < r (cid:201) c S r ( X ) (cid:201) dim( X ) − r .Hence, if c X ( H n ) (cid:201) D , then, since c H n ( B m ) (cid:46) (log n ) p by Theorem 1.7, we have c S r ( B m ) (cid:46) (log n ) p c S r ( H n ) (cid:201) (log n ) p D c S r ( X ) (cid:201) (log n ) p D dim( X ) − r .At the same time, by [LN14b] we have c S r ( B m ) (cid:38) (cid:112) ( r −
1) log n , so we conclude thatinf < r (cid:201) dim( X ) − r (cid:112) r − (cid:38) (log n ) − p D .This gives the desired bound (20) by choosing r − (cid:179)
1/ log(dim( X )). (cid:3) Remark . By substituting (6) into the reasoning of [LN14b], a positive resolution ofConjecture 1.4 would imply that for every r ∈ (1, 2] and n ∈ (cid:78) we have c (cid:96) r ( B n ) (cid:38) (cid:112) r − · (cid:113) log n . (21)An incorporation of this improved distortion lower bound into the above proof of Theo-rem 1.14 (while using [LTJ80] in place of [NPS18] since we are in the simpler (cid:96) p setting)would imply that for any finite-dimensional subspace X of (cid:96) , if c X ( H n ( p )) (cid:201) D , then thefollowing improvement over (20) holds true.dim( X ) (cid:202) exp (cid:181) cD (log n ) − p (cid:182) . (22) If one only wishes to rule out embeddings into low-dimensional subspaces of (cid:96) rather than of S , thenit suffices to use here [LTJ80, Theorem 1.2], which yields an embedding into (cid:96) r rather than S r . As in the discussion before Conjecture 1.4, the dependence on r in this estimate is not stated in [LN14b],while it is crucial for us here; a justification why the reasoning in [LN14b] implies this appears in Appendix A. OLIATED CORONA DECOMPOSITIONS 15
Notably, for p = X ) (cid:202) exp (cid:161) cD (cid:112) log n (cid:162) todim( X ) (cid:202) n cD , (23)namely a power-type dimension reduction lower bound as in [BC05]. Understanding whatis the correct behavior as p → + remains an intriguing open question; some deteriorationof the lower bound as in (20) or (22) must occur because by [JL84] logarithmic dimensionreduction is possible for finite subsets of a Hilbert space.Another question that this discussion obviously raises is if (21) could be enhanced to c S r ( B n ) (cid:38) (cid:112) r − · (cid:113) log n . (24)If so, then (23) would hold when X is a subspace of S rather than (cid:96) . More substantially,this would resolve a difficult open question (see the discussion following Question 13in [NY18]) by showing that (cid:72) (cid:90) does not admit a bi-Lipschitz embedding into S . In fact,for the latter conclusion it would suffice to establish the weaker propertylim n →∞ c S + n ( B n ) = ∞ . (25)Indeed, by [NPS18] we have c S ( B n ) (cid:38) c S r ( B n ) when r = +
1/ log n . Due to its significantconsequences, we expect that proving (25), and all the more so its stronger version (24),would require a major and conceptually new idea.We end this discussion on dimension reduction by noting that [Tao19] shows that onecould embed B n with optimal distortion (up to universal constant factors) into Euclideanspace of dimension O (1). Theorem 1.14 shows that this fails badly if one aims for optimal (cid:96) –distortion embedding of B n into a bounded dimensional subspace of (cid:96) .1.1.4. Permanence of compression rates of groups.
Suppose that ( M , d M ) is a metric and( X , (cid:107) · (cid:107) X ) is a Banach space. The compression rate of a Lipschitz mapping f : M → X isthe non-decreasing function ω f : [0, ∞ ) → [0, ∞ ) that is defined [Gro93] by ∀ s (cid:202) ω f ( s ) def = inf x , y ∈ Md M ( x , y ) (cid:202) s (cid:107) f ( x ) − f ( y ) (cid:107) X .Equivalently, ω f is the largest non-decreasing function from [0, ∞ ) to [0, ∞ ) such that ∀ x , y ∈ M , (cid:107) f ( x ) − f ( y ) (cid:107) X (cid:202) ω f (cid:161) d M ( x , y ) (cid:162) .There is a great deal of interest in determining the largest possible compression rate of1–Lipschitz mappings from a finitely generated group G (equipped with a word metricthat is induced by some finite generating set) to certain Banach spaces, notable and usefulexamples of which are Hilbert space and L . The literature on this topic is too extensive todiscuss here, and we only mention that a substantial part of it is devoted to understandingthe extent to which compression rates are preserved under various group operations(e.g. various semidirect products). Theorem 1.16 below provides a new example of thelack of such permanence which does not seem to be accessible using previously availablemethods. It leverages the fact that we establish here a marked difference between the L embeddability of Heisenberg groups of dimension 3 and dimension 5. Theorem 1.16.
There exists a finitely group G that has two finitely generated normalsubgroups H , K (cid:47) G such that the following properties hold true. (1)
Any h ∈ H and k ∈ K commute. (2) H ∩ K is the center of G. (3)
H and K are isomorphic. (4)
H and K are undistorted in G; in fact, they admit generating sets S H and S K such that S H ∪ S K generates G and the word metric on G that is induced by S H ∪ S K restricts to the word metrics on H and K that are induced by S H and S K , respectively. (5) The L compression of G is asymptotically smaller than that of H (hence also ofK ∼= H ). Concretely, there exists a Lipschitz mapping f : H → (cid:96) that satisfies ∀ s (cid:202) ω f ( s ) (cid:38) s (cid:112) log s · (log log s ) , (26) yet for any Lipschitz mapping F : G → L there are arbitrarily large s (cid:202) for which ω F ( s ) (cid:201) s (cid:112) (log s ) log log s . (27) Proof.
Let G (cid:82) be the 5–dimensional Heisenberg group, i.e., (cid:82) with the group operation( x , y , x , y , z )( x (cid:48) , y (cid:48) , x (cid:48) , y (cid:48) , z (cid:48) ) = (cid:179) x + x (cid:48) , y + y (cid:48) , x + x (cid:48) , y + y (cid:48) , z + z (cid:48) +
12 ( x y (cid:48) + x y (cid:48) − y x (cid:48) − y x (cid:48) ) (cid:180) for ( x , y , x , y , z ), ( x (cid:48) , y (cid:48) , x (cid:48) , y (cid:48) , z (cid:48) ) ∈ (cid:82) . Let G be the 5–dimensional integer Heisenberggroup, which is the subgroup G = (cid:169) ( x , y , x , y , z + ( x y + x y )/2) : x , x , y , y , z ∈ (cid:90) (cid:170) .The subgroups H , K are natural copies of (cid:72) (cid:90) in G , namely H = {( x , y , x , y , z ) ∈ G : x = y =
0} and K = {( x , y , x , y , z ) ∈ G : x = y = f : (cid:72) (cid:90) → (cid:96) ( (cid:96) ) ∼= (cid:96) that is given by f def = ∞ (cid:77) n = n φ n , ,where the mappings that are being concatenated are those of Theorem 1.7. The finalassertion (27) of Theorem 1.16 follows from [NY18, Theorem 9]. (cid:3) Obviously, Theorem 1.16 raises the question if a similar phenomenon could occurfor embeddings into a Hilbert space rather than into L . Also, in Theorem 1.16 thecompression rate of the subgroups H , K grows roughly (suppressing lower-order factors)like s / (cid:112) log s as s → ∞ , while the compression rate of G grows slower than s / (cid:112) log s .What are the possible asymptotic profiles of the compression rates that exhibit suchphenomena? OLIATED CORONA DECOMPOSITIONS 17
Decomposing surfaces into approximately ruled pieces.
In the previous sections,we discussed consequences of Theorem 1.1 (and the refined version of its second partin Theorem 1.7). In this section, we will give an overview of the concepts involved in theproof of Theorem 1.1, especially our main contribution, which is a new way to describethe structure of surfaces in (cid:72) .The statement of Theorem 1.1 is in terms of smooth functions f : (cid:72) → (cid:82) , but the mainbound (3) has an equivalent formulation in terms of surfaces in (cid:72) ; see (31) below. We willprove it by showing that surfaces in (cid:72) admit a multi-scale hierarchical decompositioninto pieces that are close to ruled surfaces (unions of horizontal lines) and that most ofthese pieces (in a quantitative sense) are long and narrow, giving the decomposition theappearance of a Venetian blind with many narrow slats; see Figure 2 and Figure 3 forexamples. For reasons that will be clarified soon, we call the above structure a foliatedcorona decomposition . This decomposition is conceptually central to this work, and themost involved part of this paper is to formulate this decomposition, prove its existence,and demonstrate its utility for the aforementioned applications (more are forthcoming).The defining feature of this decomposition is that its pieces, which we call pseudoquads ,have widely varying aspect ratios. Each pseudoquad is roughly rectangular, and we definethe aspect ratio of a pseudoquad to be its width divided by its height; long, narrowrectangles have large aspect ratios, while tall, skinny rectangles have small aspect ratios.The fact that the pieces of the decomposition (the slats of the Venetian blind) can haveunbounded aspect ratios allows the decomposition to have additional symmetries andultimately leads to the exponent 4 in Theorem 1.1.Specifically, in order to work with long, narrow pieces, we must prove results on thegeometry of (cid:72) that are invariant not only under the usual scaling automorphisms, but alsounder automorphisms that stretch and shear (cid:72) . The resulting automorphism-invariantbounds allow us to produce a decomposition that is likewise invariant under rescaling,stretching, and shearing. Furthermore, the overlap of the pieces of our decompositionis controlled by a coercive quantity that scales like the fourth power of the aspect ratiounder automorphisms. This leads to a new weighted Carleson packing condition in whichoverlaps are normalized by the fourth power of the aspect ratio; this condition leadsdirectly to the exponent 4 in the bound (3) of Theorem 1.1.Proving the optimality of Theorem 1.1 entails finding a surface for which (31) is sharp.Part of the construction of such a surface can be seen in Figure 3. The surface in the figurecan be viewed as a surface with a foliated corona decomposition for which the weightedCarleson packing condition is sharp. For this reason, it is pedagogically beneficial todescribe that construction after describing foliated corona decompositions. In truth, thegeneral decomposition methodology and the construction that demonstrates its optimal-ity are intertwined: limitations of such a construction indicate what decomposition tolook for. We therefore suggest to also consider the alternative route of first examining theconstruction of the specific (sharp) example prior to considering the task of decomposinggeneral surfaces; the proofs in the rest of this article follow the latter (“reverse”) route asthis leads to a more gradual introduction of notations and concepts. The ensuing considerations belong firmly to the setting of the continuous Heisenberggroup and its Carnot–Carathéodory geometry. They therefore assume some familiaritywith notions from that setting; the pertinent background appears in Section 2 below.1.2.1.
Fractal Venetian blinds abound.
In what follows, for any s > H s on (cid:72) will be with respect to the Carnot–Carathéodory metric d on (cid:72) . We denotethe standard generators of (cid:72) by X = (1, 0, 0), Y = (0, 1, 0), and Z = (0, 0, 1).For Ω ⊆ (cid:72) and a ∈ (cid:82) , consider the symmetric difference D a Ω def = Ω (cid:52) Ω Z − a = (cid:161) Ω (cid:224) Ω Z − a (cid:162) ∪ (cid:161) Ω Z − a (cid:224) Ω (cid:162) (28)If Ω , U ⊆ (cid:72) are measurable, then, following [LN14b, NY18], we define v U ( Ω ) : (cid:82) → (cid:82) by ∀ a ∈ (cid:82) , v U ( Ω )( a ) def = a H ( U ∩ D a Ω ) = a ˆ U (cid:175)(cid:175) Ω ( h ) − Ω (cid:161) hZ − − a (cid:162)(cid:175)(cid:175) d H ( h ). (29)Thus, v U ( Ω )( a ) is a (normalized) measurement of the amount that Ω changes within U when translated up and down by the specified (Carnot–Carathéodory) distance 2 − a .By [NY18, Lemma 38], in order to prove the first part of Theorem 1.1, namely inequal-ity (3) for any compactly supported smooth function f : (cid:72) → (cid:82) , it suffices to prove thatevery measurable subset Ω ⊆ (cid:72) satisfies the following isoperimetric-type inequality. (cid:176)(cid:176) v (cid:72) ( Ω ) (cid:176)(cid:176) L ( (cid:82) ) (cid:46) H ( ∂ Ω ). (30)This amounts in essence to an application of the coarea formula (e.g. [Amb01]).A central step of [NY18] is a further reduction of (30) to the special case that Ω is (a pieceof) an intrinsic Lipschitz epigraph . An intrinsic Lipschitz epigraph Γ + is a region of (cid:72) thatis bounded by an intrinsic Lipschitz graph Γ . The notion of an intrinsic Lipschitz graphwas introduced in [FSSC06] and all of the relevant background is explained in Section 2.2below. The intrinsic Lipschitz condition is parametrized by a intrinsic Lipschitz constant λ ∈ (0, 1). By combining Proposition 55, Theorem 57 and Lemma 58 of [NY18] (see thededuction on page 232 of [NY18]) it follows that to prove (30) it suffices to show that forevery 0 < λ < λ –Lipschitz epigraph Γ + ⊆ (cid:72) satisfiesthe growth bound ∀ r > (cid:176)(cid:176) v B r ( ) (cid:161) Γ + (cid:162)(cid:176)(cid:176) L ( (cid:82) ) (cid:46) λ r , (31)where B r ( ) denotes the (Carnot–Carathéodory) ball of radius r centered at = (0, 0, 0).The structural information that underlies the reduction of (30) to (31) is that for any0 < λ <
1, any (sufficiently nice; see [NY18] for precise assumptions) surface in (cid:72) has amulti-scale hierarchical decomposition into pieces that are close to intrinsic λ –Lipschitzgraphs, and moreover that decomposition has controlled overlap in the sense that itsatisfies a O (1)–Carleson packing condition. As such, this decomposition is an intrinsicHeisenberg analog of the corona decompositions that were introduced and developedfor subsets of Euclidean space in [DS91] and have since led to a variety of powerfulapplications in harmonic analysis (see also the monograph [DS93]).The corona decomposition of [NY18] is in some respects a Heisenberg variant of a“vanilla” corona decomposition. Like corona decompositions in (cid:82) n , it is a hierarchicalpartition of a surface into pieces of bounded aspect ratio, and the Carleson packing OLIATED CORONA DECOMPOSITIONS 19 condition governing overlaps of pieces depends only on the diameter of the pieces. Nev-ertheless, there are key differences, including the fact that the proof in [NY18] relies on anew “stopping rule” (based on the quantitative nonmonotonicity of [CKN11]) that yields,in fact, a different proof of the existence of corona decompositions even in Euclideanspace (though, for less general sets than those that [DS91] treats). In addition, while“vanilla” Euclidean corona decompositions cover a surface in (cid:82) n by pieces that are approx-imately graphs of Lipschitz functions, the approximating graphs in [NY18] are intrinsicLipschitz. Such graphs need not be graphs of functions that are Lipschitz with respect tothe Carnot–Carathéodory metric, and can even have Hausdorff dimension 3 as subsetsof (cid:82) (i.e., with respect to the Euclidean metric); see examples in [FSSC11]. Thus, thesurfaces resulting from this decomposition can be unwieldy and difficult to understand,such as the intrinsic Lipschitz graph depicted in Figure 1. Even after the decompositionstep of [NY18], the challenge of establishing estimates such as (31) remains.In [NY18], we addressed this challenge for the 5–dimensional Heisenberg group (cid:72) ,but our techniques do not shed light on the 3–dimensional setting of Theorem 1.1. Anintrinsic Lipschitz graph in (cid:72) is the intrinsic graph of a function ψ that is defined on a4–dimensional vertical hyperplane V . An inspection of the intrinsic Lipschitz conditionshows that the restriction of ψ to any coset of (cid:72) that is contained in V is Lipschitz withrespect to the Carnot–Carathéodory metric on (cid:72) . In [NY18], we applied a representation-theoretic functional inequality of [ANT13] to each of these restrictions, yielding a boundon the vertical variation of ψ . The desired control on the vertical perimeter of intrinsicLipschitz graphs in (cid:72) followed by integrating this bound over the cosets of (cid:72) in V .In the 3–dimensional setting of the present work, the intrinsic graph Γ in (31) corre-sponds to an intrinsic Lipschitz function ψ : V → (cid:82) , where V is a 2–dimensional verticalplane in (cid:72) . For concreteness, assume in what follows that V = {( x , 0, z ) : x , z ∈ (cid:82) } is the xz –plane. The reasoning of [NY18] is irrelevant to proving (31): one cannot restrict ψ tocosets of a lower-dimensional Heisenberg group, as there is no such group!Our strategy here is therefore entirely different from that of [NY18]. We will prove (31) byfinding a new structural description of intrinsic Lipschitz graphs in (cid:72) . Specifically, we willprove that they admit a hierarchical family of partitions into pieces that are approximatelyruled surfaces and bound the total error of these approximations.We call this description of Γ a foliated corona decomposition . It is a sequence ofnested partitions of Γ into approximately rectangular regions, called pseudoquads , ofvarying heights and widths. On each pseudoquad, Γ is close to a vertical plane, and thesevertical planes can be glued together to form a collection of ruled surfaces such that atmost locations and scales, Γ is approximated by one of the ruled surfaces. Furthermore,the decomposition satisfies a new weighted variant of the classical Carleson packingcondition . Namely, we bound the weighted sum of the measures of the pseudoquads inthe decomposition, where the measure of each pseudoquad is normalized by the fourthpower of its aspect ratio. We will see that the occurrence of the fourth power here is dictated by the requirement that this decomposition should be invariant under certainautomorphisms of (cid:72) (scaling, stretch, and shear automorphisms).
Theorem 1.17.
Any intrinsic Lipschitz graph in (cid:72) has a foliated corona decomposition.
The above description of foliated corona decompositions and the statement of Theo-rem 1.17 clearly lack rigorous definitions, but they convey the essence of what is achievedhere. The necessary technical matters are treated in Section 5 below, where a preciseformulation of Theorem 1.17 appears as Theorem 5.2. The justification that Theorem 1.17can be used to achieve our goal (31) is carried out in Section 6 below; the groundwork ofconstructing a foliated corona decomposition makes this deduction quite mechanical.We will next cover a few technical details necessary to describe foliated corona decom-positions and the subdivision mechanism that produces them. Recall that V ⊆ (cid:72) is the xz –plane. Fix 0 < λ < Γ be an intrinsic λ –Lipschitz graph that is the intrinsicgraph of ψ : V → (cid:82) . That is, Γ = Ψ ( V ), where Ψ ( v ) = vY ψ ( v ) for all v ∈ V . The function ψ satisfies the intrinsic Lipschitz condition (Definition 2.2); the nonlinear nature of thiscondition is the source of subtleties that ensue (and the reason why basic questions onthe rectifiability properties of intrinsic Lipschitz graphs remain open; see e.g. [Orp19]).For any p ∈ Γ , there is a horizontal curve γ contained in Γ that passes through p , so Γ is the union of all such curves. It is often convenient to work in V instead of Γ . To thisend, let Π : (cid:72) → V be the projection to V , so Π ( Ψ ( v )) = v for v ∈ V . The projected curve Π ◦ γ is a curve in V which we call a characteristic curve ; see Section 2.3 for a detaileddiscussion. Parametrize γ so that Π ( γ ( t )) = ( t , 0, g ( t )) for some continuous function g .This function is a solution of the differential equation g (cid:48) ( t ) = − ψ ( t , 0, g ( t )), and conversely,each solution gives a characteristic curve. If Γ is a vertical plane, then ψ ( x , 0, z ) = ax + b for some a , b ∈ (cid:82) , in which case the characteristic curves are parallel parabolas.Since horizontal curves pass through every point of Γ , there is a characteristic curvethrough every point of V , so one can reconstruct Γ from its set of characteristic curves.Note that the characteristic curve through p is not necessarily unique: when ψ is notsmooth, these curves can split and rejoin [BCSC15]. When ψ is smooth, the characteristiccurves foliate V , so there is a coordinate system on V such that the foliation forms oneset of coordinate lines. However, it is difficult to use this coordinate system to study thegeometry of Γ because the distance between two characteristic curves can vary wildly.Foliated corona decompositions provide a way to overcome this difficulty.A pseudoquad for Γ is a region in V that is bounded by characteristic curves above andbelow and by vertical line segments on either side. We call a pseudoquad Q rectilinear if its top and bottom boundaries approximate two parallel parabolas; if the top andbottom boundaries of Q are exactly two parallel parabolas, we call Q a parabolic rectangle .Parabolic rectangles are the projections to V of rectangles in (cid:72) bounded by two horizontalline segments and two vertical line segments. The width δ x ( Q ) and height δ z ( Q ) of sucha pseudoquad are defined to be, respectively, the width and height of its approximatingparabolic rectangle; see Section 4. The aspect ratio of Q is α ( Q ) = δ x ( Q )/ (cid:112) δ z ( Q ).Let Q ⊆ V be a rectilinear pseudoquad. A foliated corona decomposition for Γ withroot at Q is a sequence of nested partitions of Q into rectilinear pseudoquads. We con-struct such a decomposition using the following subdivision algorithm which, importantly,outputs pseudoquads that can be divided into two sets V V and V H , called, respectively, the vertically cut pseudoquads and horizontally cut pseudoquads. The algorithm repeatedlycuts pseudoquads into halves. Let Q be a pseudoquad in the decomposition. If Ψ ( Q )is a region in Γ that is sufficiently close to a vertical plane V Q and if the characteristic OLIATED CORONA DECOMPOSITIONS 21 curves through Q are close to characteristic curves for V Q , then cut Q in half along one ofthe characteristic curves of Γ . In this case, say that Q is horizontally cut and add it to V H .Otherwise, cut Q in half along a vertical line through its center, say that Q is vertically cut,and add it to V V . By applying this procedure iteratively, we obtain a sequence of nestedpartitions of Q ; see Figure 2.A crucial part of the algorithm is the mechanism determining whether to cut thepseudoquad horizontally or vertically. We stated qualitatively how this step dependson the geometry of Ψ ( Q ), but we implement it quantitatively by introducing a coercivequantity called R–extended nonmonotonicity . This is a family of measures Ω P Γ + , R onthe vertical plane V , parametrized by R >
0; see Section 8. These are inspired by thequantitative nonmonotonicity of [CKN11], but there are key differences. For instance,while the nonmonotonicity of Γ on a subset U ⊆ (cid:72) measures how lines intersect Γ inside U , the R –extended nonmonotonicity of Γ on a subset W ⊆ V measures how lines intersect Γ inside an R –neighborhood of Ψ ( W ). We refer to Section 8 for the details, in particularto Lemma 9.2 which shows that for any measurable U ⊆ V , (cid:88) i ∈ (cid:90) Ω P Γ + ,2 − i ( U ) (cid:46) λ | U | , (32)where | U | is the area of U and λ is the intrinsic Lipschitz constant of ψ .Analogously to [CKN11], extended nonmonotonicity is coercive in the following sense.Let U = [0, 1] × {0} × [0, 1] ⊆ V and for r >
0, let rU be the square of side r concentric with U . There is a universal constant r > δ is sufficiently small, R is sufficientlylarge, ψ (0) is bounded, and Ω Γ + , R ( rU ) < δ , then Ψ ( U ) is close to a vertical plane and thecharacteristic curves that pass through U are close to characteristic curves of that verticalplane (i.e., parabolas). The proof of this geometric statement (whose precise formulationappears as Proposition 7.2) is the most technically involved part of this work; it is outlinedin Section 10 and carried out in Section 11 and Section 12.By translation, rescaling, and applying a shear automorphism, a similar coercive prop-erty applies to any pseudoquad of aspect ratio 1, but for the subdivision algorithm, weneed a coercive property for pseudoquads of arbitrary aspect ratio. If Q is a pseudoquadof aspect ratio α ( Q ), the stretch automorphism s ( x , y , z ) = ( α ( Q ) − x , α ( Q ) y , z ) sends Q to apseudoquad of aspect ratio 1. The extended nonmonotonicity of s ( Q ) scales like α ( Q ) , soif the extended nonmonotonicity of Γ + on Q is at most δ | Q | / α ( Q ) , then Ψ ( Q ) is close to avertical plane and the characteristic curves that pass through Q are close to characteristiccurves of that vertical plane.Therefore, in the subdivision algorithm above, there is δ > Q hori-zontally if and only if the extended nonmonotonicity of Γ + on Q is at most δ | Q | / α ( Q ) .This criterion, combined with (32), leads to a crucial bound on the total pseudoquadsthat have been vertically cut by the subdivision algorithm. Specifically, if Q is a pseudo-quad of the decomposition and D V ( Q ) is the set of vertically cut pseudoquads Q (cid:48) in thedecomposition that are contained in Q , then (cid:88) Q (cid:48) ∈ D V ( Q ) | Q (cid:48) | α ( Q (cid:48) ) (cid:46) λ | Q | . (33) F IGURE Stages in the construction of a foliated corona decomposition for abump function as in the top row of Figure 3. The aspect ratio of the regions in thedecomposition varies widely. On the sides, where the surface is close to a verticalplane, the aspect ratio is large and the regions are short and wide; near the topand bottom, where it is further from a plane, the regions are tall and narrow.
OLIATED CORONA DECOMPOSITIONS 23
The condition (33) is the aforementioned weighted Carleson packing condition, and the L norm that appears in Theorem 1.1 arises directly from the exponent 4 in (33).Thus, the L norm in Theorem 1.1 is ultimately dictated by having to prove a coerciveproperty for intrinsic Lipschitz graphs that is invariant under stretch automorphisms.This stretch-invariance has multiple effects. On one hand, stretch-invariance means thatit suffices to prove the coercive property for pseudoquads of aspect ratio 1; indeed, it isenough to consider pseudoquads that approximate the unit square. On the other hand, itinduces a substantial complication in the proofs: since the intrinsic Lipschitz constant isnot invariant under stretch automorphisms, the coercivity must be independent of theintrinsic Lipschitz constant.1.2.2. A maximally bumpy surface.
The optimality part of Theorem 1.1 corresponds toconstructing (in Section 3) an intrinsic Lipschitz graph for which the L ( (cid:82) ) norm in (31)cannot be replaced by the L q ( (cid:82) ) norm for any 0 < q <
4. Theorem 1.7 is deduced inSection 3.1 by analyzing this construction; the level sets of the resulting embeddinginto L are a superposition of certain random rotations, scalings and translations of thissurface.We will show that for any sufficiently small ε >
0, there are intrinsic Lipschitz surfacesin (cid:72) of bounded (Heisenberg) perimeter that are ε –far from planes at ε − different scales,many more than the ε − different scales that are possible (by [NY18]) for such surfaces inthe 5–dimensional Heisenberg group (cid:72) (or, for that matter, in (cid:82) n , by the Jones travellingsalesman theorem [Jon90] and the higher-dimensional analogues thereof [DS91]).We construct these surfaces by adding bumps to a vertical plane. While surfaces thatdemonstrate that the bound of [NY18] for (cid:72) is optimal can be constructed by addinground bumps with equal width and height, it is more natural in (cid:72) to add oblong bumpswith width (horizontal size) w , depth d (size perpendicular to the surface), and height h (vertical size). The automorphisms of the Heisenberg group preserve the ratio d w / h , sowe can construct a family of bump functions by applying automorphisms to a prototypebump with d = w = h =
1. The resulting bumps have h = d w , and we define the aspectratio α of such a bump to be α = w (cid:112) h = w (cid:112) d w = (cid:114) wd .A horizontal curve connecting one side of the bump to its other side has slope roughly d / w = α − , so adding a layer of bumps with aspect ratio α (cid:202) + α − . Thus, we can start with a unit square, then add ε − layers ofbumps of width ε − r i , depth ε r i , and height r i , for r (cid:192) · · · (cid:192) r ε − . These bumps all haveaspect ratio ε − , so the resulting surface Σ has bounded perimeter, and for any x ∈ Σ , theintersections B r i ( x ) ∩ Σ are each ε r i –far away from any plane. So, Σ is ε –far from planesat ε − different scales. The implementation of this strategy in Section 3 is in essencean example of a foliated corona decomposition. At each stage we use the characteristiccurves of the surface that was obtained in the previous stage to guide us where to glue thenext layer of bumps. Figure 3 shows a sketch of the construction.It is highly informative to examine why this construction does not work in (cid:72) . Bumpson a surface in (cid:72) have five dimensions, which we denote w , w , d , d , and h , so that F IGURE The first three steps of the construction of a maximally rough surfacein (cid:72) . The left and right column show the same surface from two different angles.The center column shows a projection of the surface to the plane, with charac-teristic curves marked. Since the second derivatives of these curves are small, theHeisenberg area of the surface is bounded, but the surface can be made ε –far froma plane at ε − different scales — much more than what is possible in (cid:72) . h is vertical, the other four dimensions are horizontal, and d is normal to the surface.The automorphisms of (cid:72) preserve the ratios d w /( d w ), d w / h , and d w / h . If β isa bump with d w = d w = h and d (cid:201) w , then the slopes of β in the three horizontal OLIATED CORONA DECOMPOSITIONS 25 directions are roughly d / w , d / w , and d / d . So, adding β to a vertical rectangle withdimensions w × w × d × h increases the volume of the rectangle by a factor of roughly ν ( w , w , d , d , h ) def = + max (cid:40) d w , d w , d d (cid:41) ,and the resulting bump is roughly d / (cid:112) h –far from a 4–dimensional hyperplane at scale (cid:112) h . If d / (cid:112) h = ε , then d w = h = ε − d , and ν ( w , w , d , d , h ) (cid:202) + d max{ d , w } (cid:202) + d d w = + ε .Hence, this construction results, at best, in a surface that is ε –far from planes at ε − differ-ent scales. One may also consider bumps where d w , d w , and h are not proportional,such as bumps with d = w = w = d − = r (cid:192) = h . This is more subtle than it mightinitially seem. Indeed, because the d – and w –directions do not commute, there are no r × r × r × r − × (cid:72) that stay close to horizontal. Consequently, a bump of thesedimensions behaves similarly to a collection of smaller bumps with d w = d w = h ,which are governed by the previous reasoning.1.3. Roadmap.
In Section 2, we present notation for working with the Heisenberg groupand some definitions and results related to intrinsic graphs and characteristic curves.In Section 3, we construct an intrinsic graph with large vertical perimeter and use it toconstruct the embeddings used in Theorem 1.7 and its consequences.The rest of the paper is devoted to defining and constructing foliated corona decom-positions and using them to prove equation (31) bounding the vertical perimeter of anintrinsic Lipschitz graph. In Sections 4, we define a rectilinear foliated patchwork, whichdecomposes an intrinsic Lipschitz graph into rectilinear pseudoquads, and in Section 5,we define the weighted Carleson packing condition required for such a patchwork to be afoliated corona decomposition. Then, in Section 6, we show that an intrinsic Lipschitzgraph that admits a foliated corona decomposition satisfies equation (31).It remains to show that every intrinsic Lipschitz graph admits a foliated corona de-composition. We produce foliated corona decompositions by the subdivision algorithmdescribed in Section 7. The fact that the patchworks produced by this algorithm satisfythe weighted Carleson packing condition relies on careful analysis of a coercive quantity,the extended parametric nonmonotonicity, defined in Section 8. When this coercivequantity is small, the graph satisfies strong geometric bounds, detailed in Proposition 7.2.Assuming Proposition 7.2, we prove the weighted Carleson condition in Section 9. InSection 10, we outline the proof of Proposition 7.2, and in Sections 11–12, we prove it.2. P
RELIMINARIES
Most of this section presents initial facts about the Heisenberg group that will be usedthroughout what follows. However, we will start by briefly setting notation for measuretheoretical boundaries and interiors that are best described in greater generality (thoughthey will be applied below only to either the Heisenberg group or the real line).
Let ( (cid:77) , d (cid:77) , µ ) be a non-degenerate metric measure space, i.e., ( (cid:77) , d (cid:77) ) is a metric spaceand µ is a Borel measure on (cid:77) such that µ ( B (cid:77) ( x , r )) > x ∈ (cid:77) and r >
0, where B (cid:77) ( x , r ) = { y ∈ (cid:77) : d (cid:77) ( x , y ) (cid:201) r } is the closed d (cid:77) –ball of radius r centered at x .Given a subset S ⊆ (cid:77) , we define the measure-theoretic support supp µ ( S ) of S to be theusual measure-theoretic support of the indicator function S : (cid:77) → {0, 1}, namelysupp µ ( S ) def = (cid:92) r > (cid:169) x ∈ (cid:77) : µ ( B (cid:77) ( x , r ) ∩ S ) > (cid:170) . (34)The measure-theoretic boundary of S is defined as ∂ µ S def = supp µ ( S ) ∩ supp µ ( (cid:77) (cid:224) S ) = (cid:92) r > (cid:110) x ∈ (cid:77) : 0 < µ ( B (cid:77) ( x , r ) ∩ S ) µ ( B (cid:77) ( x , r )) < (cid:111) . (35)The measure-theoretic interior of S is defined asint µ ( F ) def = (cid:77) (cid:224) supp µ ( (cid:77) (cid:224) S ) = (cid:91) r > (cid:169) x ∈ (cid:77) : µ ( B (cid:77) ( x , r ) (cid:224) S ) = (cid:170) . (36)These definitions are nonstandard; other works define the measure-theoretic boundaryas the set of points where the density of S is not 0 or 1. The advantage of our definition isthat one may check that int µ ( S ) is open in (cid:77) and its (topological) boundary ∂ int µ ( S ) iscontained in ∂ µ S . The sets int µ ( S ), int µ ( (cid:77) (cid:224) S ), ∂ µ S are disjoint and their union is (cid:77) , i.e., (cid:77) = int µ ( S ) (cid:71) (cid:161) int µ ( (cid:77) (cid:224) S ) (cid:162) (cid:71) ∂ µ S . (37)2.1. The Heisenberg group.
Here we summarize basic notation and terminology relatedto the Heisenberg group.Throughout what follows, (cid:107) · (cid:107) : (cid:82) → (cid:82) will denote the Euclidean norm on (cid:82) , namely (cid:107) ( a , b , c ) (cid:107) = (cid:112) a + b + c for all a , b , c ∈ (cid:82) . Let X def = (1, 0, 0), Y def = (0, 1, 0), Z def = (0, 0, 1)be the standard basis of (cid:82) , and let x , y , z : (cid:82) → (cid:82) be the coordinate functions. Namely, for u = ( a , b , c ) ∈ (cid:82) we set x ( u ) = a , y ( u ) = b and z ( u ) = c . With this notation, the Heisenberggroup operation (8) can be written as ∀ u , v ∈ (cid:72) = (cid:82) , uv = u + v + x ( u ) y ( v ) − y ( u ) x ( v )2 Z . (38)The linear span of a set of vectors S ⊆ (cid:82) will be denoted 〈 S 〉 . The plane H def = 〈 X , Y 〉 is called the space of horizontal vectors . Let π : (cid:82) → H be the orthogonal projection. A horizontal line in (cid:72) is a coset of the form w 〈 h 〉 ⊆ (cid:72) for some w ∈ (cid:72) and h ∈ H .The union of the horizontal lines passing through a point u ∈ (cid:72) is the plane u H , whichwe denote H u and call the horizontal plane centered at u . Every plane P ⊆ (cid:82) eithercontains a coset of 〈 Z 〉 (a vertical line ), in which case we call P a vertical plane , or can bewritten P = H u for some unique u ∈ (cid:72) .If I ⊆ (cid:82) is an interval and γ : I → (cid:72) is a curve such that x ◦ γ , y ◦ γ , z ◦ γ : I → (cid:82) areLipschitz, then γ (cid:48) ( t ) is defined for almost all t ∈ I . One then says that γ is a horizontalcurve if γ is tangent to H γ ( t ) at γ ( t ) for almost all t ∈ I , i.e., for almost all t ∈ I we havedd s (cid:161) γ ( t ) − γ ( s ) (cid:162)(cid:175)(cid:175)(cid:175)(cid:175) s = t ∈ H . OLIATED CORONA DECOMPOSITIONS 27
Note that horizontality is left-invariant; if γ is a horizontal curve and g ∈ (cid:72) , then g · γ isalso a horizontal curve. If γ ( t ) = ( γ x ( t ), γ y ( t ), γ z ( t )), then this requirement is equivalentto the differential equation 2 γ (cid:48) z ( t ) = γ x ( t ) γ (cid:48) y ( t ) − γ y ( t ) γ (cid:48) x ( t ).Define (cid:96) ( γ ) def = ˆ I (cid:107) π ( γ (cid:48) ( t )) (cid:107) d t .The sub-Riemannian or Carnot–Carathéodory metric d : (cid:72) × (cid:72) → [0, ∞ ) is defined byletting d ( v , w ) be the infimum of (cid:96) ( γ ) over all horizontal curves γ connecting v ∈ (cid:72) to w ∈ (cid:72) . This metric is left-invariant, i.e., d ( g a , g b ) = d ( a , b ) for all a , b , g ∈ (cid:72) .If γ is a horizontal curve connecting v to w , then π ◦ γ is a curve in (cid:82) of the samelength connecting π ( v ) to π ( w ), so d ( v , w ) (cid:202) (cid:107) π ( v ) − π ( w ) (cid:107) . Consequently, any horizontalline in (cid:72) is a geodesic. Also, d satisfies (e.g. [BR96, Gro96, Mon02]) the ball-box inequality ∀ h = ( x , y , z ) ∈ (cid:72) , d ( , h ) (cid:201) | x | + | y | + (cid:112) | z | (cid:201) d ( , h ) + · d ( , h ) (cid:112) π (cid:201) d ( , h ). (39)For h ∈ (cid:72) and r (cid:202) B r ( h ) = { g ∈ (cid:72) : d ( g , h ) (cid:201) r } = hB r ( ) denote the closed ballof radius r centered at h with respect to the sub-Riemannian metric d on (cid:72) ; throughoutwhat follows we will not use this notation for balls with respect to any other metric.For σ > H σ the σ –dimensional Hausdorff measure that d induces on (cid:72) .Thus H is the Lebesgue measure on (cid:82) , which is also the Haar measure on (cid:72) . Givena measurable subset E ⊆ (cid:72) , the associated perimeter measure that is induced by d willbe denoted by Per E ( · ); we refer to [FSSC01] for background on this fundamental notion,noting only that there exists η > E ⊆ (cid:72) has a piecewise smooth boundary,then Per E ( U ) = η H ( U ∩ ∂ E ) for every open subset U ⊆ (cid:72) .It is also beneficial to describe the group operation on (cid:72) in terms of a symplectic form.Let ω (cid:82) : (cid:82) × (cid:82) → (cid:82) be the standard symplectic form, i.e., ∀ ( a , b ), ( α , β ) ∈ (cid:82) , ω (cid:82) (cid:161) ( a , b ), ( α , β ) (cid:162) def = a β − b α = det (cid:181) a b α β (cid:182) .Under this notation, (38) can be written as follows. ∀ u , v ∈ (cid:72) , uv = u + v + ω (cid:82) ( π ( u ), π ( v ))2 Z . (40)This lets us define automorphisms of (cid:72) . Let A : (cid:82) → (cid:82) be an invertible linear mapwith determinant J ∈ (cid:82) (cid:224) {0}, so that ω (cid:82) ( A ( v ), A ( w )) = J ω (cid:82) ( v , w ) for any v , w ∈ (cid:82) . Itfollows from (40) that the map ˜ A : (cid:72) → (cid:72) that is defined by ∀ ( x , y , z ) ∈ (cid:72) , ˜ A ( x , y , z ) def = ( A ( x , y ), J z ) (41)is an automorphism of (cid:72) which, since ˜ A ( H ) = H , sends horizontal curves to horizontalcurves and is thus Lipschitz with respect to the sub-Riemannian metric on (cid:72) . If A is anorthogonal matrix, then ˜ A is an isometry. As a notable special case, for a , b >
0, we define ∀ ( x , y , z ) ∈ (cid:72) , s a , b ( x , y , z ) def = ( ax , b y , abz ), (42)which we call a stretch map . When a = b = t , s t , t is the usual scaling automorphism of (cid:72) , which scales the sub-Riemannian metric on (cid:72) by a factor of t . For simplicity, in whatfollows we will sometimes write s t , t = s t . Intrinsic graphs and intrinsic Lipschitz graphs.
Throughout what follows, we de-note the xz –plane by V , namely V = {( x , y , z ) ∈ (cid:72) : y = = (cid:82) × {0} × (cid:82) ⊆ (cid:72) .Note that the restriction of H to V is proportional to the Lebesgue measure on V .Fix U ⊆ V . The intrinsic graph of a function ψ : U → (cid:82) is defined in [FSSC06] to be Γ ψ def = (cid:169) vY ψ ( v ) : v ∈ U (cid:170) = (cid:110)(cid:161) x ( v ), ψ ( v ), z ( v ) + x ( v ) ψ ( v ) (cid:162) : v ∈ U (cid:111) ⊆ (cid:72) , (43)where in (43), as well as throughout what follows, it is convenient to use the exponentialnotation u t = t u = ( t x ( u ), t y ( u ), t z ( u )) for u ∈ (cid:72) and t ∈ (cid:82) . Observe that any coset of 〈 Y 〉 that passes through U intersects Γ ψ in exactly one point. We will also use the followingnotation for the intrinsic epigraph of ψ . Γ + ψ def = (cid:169) vY t : ( v , t ) ∈ U × [ ψ ( v ), ∞ ) (cid:170) .Suppose that U ⊆ V is an open subset of V and that g : U → (cid:82) is smooth. For every ψ : U → (cid:82) define a function ∂ ψ g : U → (cid:82) by ∂ ψ g def = ∂ g ∂ x − ψ ∂ g ∂ z . (44)If ψ is smooth, then we define the horizontal derivative of ψ to be the function ∂ ψ ψ = ∂ψ∂ x − ψ ∂ψ∂ z . (45)Let v ∈ U and let p = vY ψ ( t ) ∈ Γ ψ . Then the horizontal plane H p locally intersects Γ ψ in acurve, and the slope of this curve at p is given by ∂ ψ ψ ( v ). This also determines the slopeof the intrinsic tangent plane to Γ ψ ; as r →
0, rescalings of the intersections B r ( p ) ∩ Γ ψ converge to a vertical tangent plane with slope ∂ ψ ψ ( v ).The following proposition is part of Theorem 1.2 of [ASCV06]. It expresses the area H ( Γ ψ ) of Γ ψ , namely the 3-dimensional Hausdorff measure (with respect to the sub-Riemannian metric) of Γ ψ , in terms of ∂ ψ ψ . Proposition 2.1 ([ASCV06]) . There exists a constant c > such that if U ⊆ V is a measur-able set and ψ : U → (cid:82) is a smooth function, then H ( Γ ψ ) = c ˆ U (cid:113) + ( ∂ ψ ψ ) d w (cid:179) H ( U ) + (cid:107) ∂ ψ ψ (cid:107) L ( U ) . (46)For λ ∈ (0, 1), define the double coneCone λ def = (cid:169) h ∈ (cid:72) : | y ( h ) | > λ d ( , h ) (cid:170) .This is a cone centered on the horizontal line 〈 Y 〉 which is scale-invariant, i.e., ∀ t > s t , t (Cone λ ) = Cone λ . OLIATED CORONA DECOMPOSITIONS 29
The intersection H ∩ Cone λ is a double cone in H with angle depending on λ . Specifically, H ∩ Cone λ = (cid:189) ( x , y , 0) ∈ (cid:72) : | y | > λ (cid:113) x + y (cid:190) = (cid:189) ( x , y , 0) ∈ (cid:72) : | y | > λ (cid:112) − λ | x | (cid:190) . (47) Definition 2.2.
Let U ⊆ V and let Γ ⊆ (cid:72) be an intrinsic graph over U . For any λ ∈ (0, 1) ,we say that Γ is an intrinsic λ –Lipschitz graph if ( h Cone λ ( V )) ∩ Γ = ∅ for every h ∈ Γ .Equivalently, for every p , q ∈ Γ , | y ( q ) − y ( p ) | (cid:201) λ d ( p , q ). We say that Γ is an intrinsic Lipschitz graph if it is intrinsic λ –Lipschitz for some λ ∈ (0, 1) .If Γ = Γ ψ for some ψ : U → (cid:82) , then we say that ψ is an intrinsic Lipschitz function . Definition 2.2 gives the same class of intrinsic Lipschitz graphs as the definition in-troduced in [FSSC06], but it gives different classes of intrinsic λ –Lipschitz graphs; seeSection 3.2 of [Rig19] for a proof that the definitions are equivalent.The following simple bound will be convenient later. Lemma 2.3.
Let (cid:201) λ (cid:201) and let Γ = Γ ψ be an intrinsic λ –Lipschitz graph of a function ψ : U ⊆ V → (cid:82) . Let v , w ∈ U and write p = vY ψ ( v ) ∈ Γ and q = wY ψ ( w ) ∈ Γ . Then | y ( p ) − y ( q ) | = | ψ ( v ) − ψ ( w ) | (cid:201) − λ d ( p , q 〈 Y 〉 ). Proof.
Denote m = d ( p , w 〈 Y 〉 ). Let c ∈ w 〈 Y 〉 be a point such that d ( p , c ) = m . By theintrinsic Lipschitz condition, | y ( c ) − y ( q ) | (cid:201) m + | y ( p ) − y ( q ) | (cid:201) m + λ d ( p , q ) (cid:201) m + λ ( m + | y ( c ) − y ( q ) | ).This simplifies to give | y ( c ) − y ( q ) | (cid:201) + λ − λ m .Hence, | y ( p ) − y ( q ) | (cid:201) | y ( p ) − y ( c ) | + | y ( c ) − y ( q ) | (cid:201) m − λ . (cid:3) Intrinsic Lipschitz graphs satisfy the following version of Rademacher’s differentiationtheorem due to [FSSC11, Theorem 4.29].
Theorem 2.4 ([FSSC11]) . Let < λ < , let U ⊆ V be an open set and let f : U → (cid:82) be afunction such that Γ ψ ⊆ (cid:72) is an intrinsic λ –Lipschitz graph. Then for almost every p ∈ U , Γ ψ has an intrinsic tangent plane at pY ψ ( p ) whose slope satisfies | ∂ ψ ψ ( p ) | (cid:201) λ (cid:112) − λ . (48)We note that [FSSC11, Theorem 4.29] is concerned with the (almost everywhere) ex-istential statement of horizontal derivatives, while the upper bound in (48) is a straight-forward consequence of the above definitions, using (47). This bound on the horizon-tal derivatives of an intrinsic Lipschitz graph leads to a bound on the perimeter mea-sure. The following result follows from Theorem 4.1 of [FSC07], which proves a similar bound on the Hausdorff measure of Γ , and the results of [FSSC01], which imply thatthe Hausdorff measure of Γ and the perimeter measure of Γ + differ by a multiplicativeconstant. Let Π : (cid:72) → V be the natural (nonlinear) projection to V along cosets of 〈 Y 〉 ,i.e., Π ( v ) = vY − y ( v ) for every v ∈ (cid:72) . Equivalently, ∀ ( x , y , z ) ∈ (cid:72) , Π ( x , y , z ) def = (cid:161) x , 0, z − x y (cid:162) . (49) Lemma 2.5 ([FSC07]) . Fix λ ∈ (0, 1) . Let ψ : V → (cid:82) be λ –intrinsic Lipschitz. The perimetermeasure Per Γ + ψ satisfies the following equivalence for measurable subsets A ⊆ Γ ψ . Per Γ + ψ ( A ) (cid:179) λ | Π ( A ) | , where here, and henceforth, | · | denotes the Haar measure on V , normalized to coincidewith the usual –dimensional area measure in (cid:82) . Characteristic curves.
Let U ⊆ V be an open set and let ψ : U → (cid:82) be an intrinsicLipschitz function. The differential operator ∂ ψ given in (44) defines a vector field on V that is continuous and has x –coordinate 1, so by the Peano existence theorem, there isat least one flow line of ∂ ψ through every point of U ; these flow lines are the graphs offunctions g : (cid:82) → (cid:82) satisfying ∀ t ∈ (cid:82) , g (cid:48) ( t ) + ψ (cid:161) t , 0, g ( t ) (cid:162) =
0. (50)We call these flow lines characteristic curves of Γ ψ .In this section, we will show that the characteristic curves of Γ ψ are the projections ofhorizontal curves in Γ ψ and use them to describe Γ ψ . In the next section, we will describehow characteristic curves transform under automorphisms of (cid:72) ; later, we will use thesecurves to describe how horizontal lines intersect an intrinsic Lipschitz graph. Lemma 2.6.
Let Γ = Γ ψ . The characteristic curves of Γ are exactly the projections (under Π )of horizontal curves φ : I → Γ such that x ( φ ( t )) = t for every t ∈ I .
Because characteristic curves can branch and rejoin (see [BCSC15] for such examples),there are intrinsic Lipschitz graphs with horizontal curves whose x –coordinate is notmonotone. Thus the condition x ( φ ( t )) = t of Lemma 2.6 cannot be dropped. Proof of Lemma 2.6.
First, we claim that if φ is a horizontal curve in Γ with x ( φ ( t )) = t ,then Π ◦ φ is a characteristic curve of Γ . Write Γ = Γ ψ and let φ : I → Γ be a horizontalcurve of the form φ ( t ) = X t Y f ( t ) Z g ( t ) . Then Π ( φ ( t )) = ( t , 0, g ( t )), and, since φ ( t ) ∈ Γ , wehave f ( t ) = ψ ( t , 0, g ( t )). Since φ is horizontal,dd u φ ( t ) − φ ( t + u ) (cid:175)(cid:175)(cid:175)(cid:175) u = ∈ H for almost every t ∈ I . Observe that φ ( t ) − φ ( t + u ) = (cid:161) X t Y f ( t ) Z g ( t ) (cid:162) − (cid:161) X t + u Y f ( t + u ) Z g ( t + u ) (cid:162) = X u Y f ( t + u ) − f ( t ) Z g ( t + u ) − g ( t ) + u f ( t ) .Since ψ is intrinsic Lipschitz, the following identity holds almost everywhere.dd u φ ( t ) − φ ( t + u ) (cid:175)(cid:175)(cid:175)(cid:175) u = = X + f (cid:48) ( t ) Y + ( g (cid:48) ( t ) + f ( t )) Z . (51) OLIATED CORONA DECOMPOSITIONS 31
That is, g satisfies (50).Conversely, suppose that g is a solution of (50) and let f ( t ) = ψ ( t , 0, g ( t )). By Theo-rem 1.1 and Theorem 1.2 of [BCSC15], f is Lipschitz. Therefore, φ ( t ) = X t Y f ( t ) Z g ( t ) is aLipschitz curve in Γ such that Π ( φ ( t )) = ( t , 0, g ( t )) and such that φ satisfies (51) almosteverywhere. In combination with (50), this implies that φ is horizontal. (cid:3) If ψ is smooth, the characteristic curves of Γ ψ foliate U . If ψ is merely intrinsic Lipschitz,characteristic curves can branch and rejoin, but if two characteristic curves pass throughthe same point, then they are tangent at that point; see Figure 1 of [BCSC15] for anexample of this phenomenon.Characteristic curves satisfy bounds based on the intrinsic Lipschitz constant of Γ . Lemma 2.7.
Fix λ ∈ (0, 1) and denote L def = λ (cid:112) − λ . Let Γ = Γ ψ be an intrinsic λ –Lipschitz graph over an open set and let γ : I → V be acharacteristic curve for Γ parametrized so that x ( γ ( t )) = t for all t ∈ I . Then, ∀ s , t ∈ I , | ψ ( γ ( s )) − ψ ( γ ( t )) | (cid:201) L | s − t | . (52) Also, if we denote g ( t ) = z ( γ ( t )) , then ∀ s , t ∈ I , (cid:175)(cid:175) g ( t ) − g ( s ) − g (cid:48) ( s ) · ( t − s ) (cid:175)(cid:175) (cid:201) L ( t − s ) Proof.
Since γ is characteristic, the curve φ ( t ) = γ ( t ) · Y ψ ( γ ( t )) is horizontal. The intrinsicLipschitz condition implies that ∀ δ ∈ (cid:82) (cid:224) {0}, | y ( φ ( t + δ )) − y ( φ ( t )) | d ( φ ( t ), φ ( t + δ )) (cid:201) λ . (54)By Pansu’s theorem [Pan89], for almost every t ∈ I , there is a vector h t ∈ H such thatlim δ → d (cid:161) φ ( t ) h δ t , φ ( t + δ ) (cid:162) δ = h t = (1, m , 0), where m = ( ψ ◦ γ ) (cid:48) ( t ). Thenlim inf δ → | y ( φ ( t + δ )) − y ( φ ( t )) | d ( φ ( t ), φ ( t + δ )) (cid:202) lim inf δ → | δ m | − d ( φ ( t ) h δ t , φ ( t + δ )) δ (cid:107) h t (cid:107) + d ( φ ( t ) h δ t , φ ( t + δ )) = | m |(cid:112) + m .By (54) it follows that | m |(cid:112) + m (cid:201) λ , so for almost every t ∈ I , | ( ψ ◦ γ ) (cid:48) ( t ) | = | m | (cid:201) L . (55)This implies (52). By (50), g (cid:48) ( t ) = − ψ ( γ ( t )), so it follows from (55) that | g (cid:48)(cid:48) ( t ) | (cid:201) L foralmost every t ∈ I . The remaining bound (53) is therefore justified as follows. (cid:175)(cid:175) g ( t ) − (cid:161) g ( s ) + g (cid:48) ( s ) · ( t − s ) (cid:162)(cid:175)(cid:175) = (cid:175)(cid:175)(cid:175)(cid:175) ˆ t s ( t − u ) g (cid:48)(cid:48) ( u ) d u (cid:175)(cid:175)(cid:175)(cid:175) (cid:201) L (cid:175)(cid:175)(cid:175)(cid:175) ˆ t s ( t − u ) d u (cid:175)(cid:175)(cid:175)(cid:175) = L ( t − s ) (cid:3) Since there is a characteristic curve through every point p ∈ U and the derivative of sucha curve at p is − ψ ( p ), an intrinsic graph Γ can be reconstructed from its characteristiccurves. Indeed, one way to construct intrinsic Lipschitz graphs is to construct a foliationof V by C curves { z = g α ( x )}, α ∈ A such that Lip( g (cid:48) α ) (cid:46) α ∈ A . Each such curvelifts to a horizontal curve, and one can show that the union of these lifts is an intrinsicLipschitz graph. (This is how the graphs in Figure 3 were constructed.)For illustration, we consider planes in (cid:72) . A vertical plane V that is not orthogonal to V is an intrinsic graph over V . The horizontal curves in V are parallel lines; let L be onesuch line. The image Π ( L ) is a parabola in V , and the characteristic curves of V are theparabolas parallel to Π ( L ). The second derivative of these parabolas depends on the anglebetween V and V .Let v ∈ (cid:72) . The horizontal plane H v centered at v is not an intrinsic graph, but thehorizontal line v 〈 Y 〉 divides H v into two intrinsic graphs. The horizontal lines in H v all pass through v , and their projections to V are parabolas through Π ( v ). Since theyall intersect at v , their projections are all tangent at Π ( v ). These parabolas foliate thecomplement in V of the vertical line through Π ( v ). They have unboundedly large secondderivatives, so the two halves of H v are locally intrinsic Lipschitz graphs, but not globally.2.4. Automorphisms and characteristic curves.
Recall that any invertible linear map A : (cid:82) → (cid:82) induces an automorphism ˜ A of (cid:72) as in (41). We are particularly interested inthe case that Y is an eigenvector of A . In this case, ˜ A ( 〈 Y 〉 ) = 〈 Y 〉 , so ˜ A sends cosets of 〈 Y 〉 to cosets of 〈 Y 〉 . A set Γ is an intrinsic graph if and only if it intersects each coset of 〈 Y 〉 atmost once, so ˜ A sends intrinsic graphs to intrinsic graphs.One family of maps with this property are the stretch maps s a , b ( x , y , z ) = ( ax , b y , abz )defined in (42). To construct a second family of maps with the above property, let b ∈ (cid:82) and consider the linear map A b ( x , y ) = ( x , y + bx ), which is a shear of the plane (cid:82) . Theinduced map ˜ A b , is an automorphism of (cid:72) given by the formula ∀ ( x , y , z ) ∈ (cid:72) , ˜ A b ( x , y , z ) = ( x , y + bx , z ),and we call such maps shear maps . (Note that these are different from the shear mapsconsidered in [Xie16].)Let Π : (cid:72) → V be as in (49), i.e., the projection to V along cosets of 〈 Y 〉 . The mapsabove preserve cosets of 〈 Y 〉 , so composed with Π they induce maps from V to V . Lemma 2.8.
Fix h = ( x , y , z ) ∈ (cid:72) and v = ( x , 0, z ) ∈ V . For any a , b , t ∈ (cid:82) we have Π (cid:161) s a , b ( vY t ) (cid:162) = s a , b ( v ) = ( ax , 0, abz ), Π (cid:161) ˜ A b ( vY t ) (cid:162) = (cid:161) x , 0, z − bx (cid:162) , and Π ( hvY t ) = (cid:161) x + x , 0, z + z − x y − x y (cid:162) . Proof. Π ( g Y t ) = Π ( g ) for all g ∈ (cid:72) and t ∈ (cid:82) . Since s a , b and ˜ A b are homomorphisms, Π (cid:161) s a , b ( vY t ) (cid:162) = Π (cid:161) s a , b ( v ) Y bt (cid:162) = s a , b ( v ) = ( ax , 0, abz ), OLIATED CORONA DECOMPOSITIONS 33 and Π (cid:161) ˜ A b ( vY t ) (cid:162) = Π (cid:161) ˜ A b ( v ) Y t (cid:162) = ( x , bx , z ) Y − bx = (cid:161) x , 0, z − bx (cid:162) .Finally, Π ( hvY t ) = Π ( hv ) = (cid:161) x + x , y , z + z − x y (cid:162) Y − y = (cid:161) x + x , 0, z + z − x y −
12 ( x + x ) y (cid:162) . (cid:3) We next describe how these maps affect characteristic curves and intrinsic graphs.
Lemma 2.9.
Fix U ⊆ V and ψ : U → (cid:82) . Write Γ = Γ ψ . Let C = {( x , 0, z ) ∈ V : z = g ( x )} be acharacteristic curve of Γ . Let q : (cid:72) → (cid:72) be a stretch map, shear map, or left translation, andlet ˆ q : V → V , ˆ q ( v ) = Π ( q ( v )) be the map that q induces on V . Then q ( Γ ) is the intrinsicgraph of a function ˆ ψ : ˆ q ( U ) → (cid:82) and ˆ q ( C ) is a characteristic curve of q ( Γ ) . Also, • If a , b ∈ (cid:82) (cid:224) {0} and q = s a , b , then ˆ ψ ( ˆ q ( v )) = b ψ ( v ) for all v ∈ U . • If b ∈ (cid:82) and q = ˜ A b , then ˆ ψ ( ˆ q ( v )) = ψ ( v ) + bx ( v ) for all v ∈ U . • If h ∈ (cid:72) and q ( p ) = hp for all h ∈ (cid:72) , then ˆ ψ ( ˆ q ( v )) = ψ ( v ) + y ( h ) for all v ∈ U .Proof.
Any coset of 〈 Y 〉 intersects q ( Γ ) at most once, so q ( Γ ) is an intrinsic graph withdomain Π ( q ( Γ )) = ˆ q ( Γ ).Let γ ⊆ Γ be the horizontal curve such that Π ( γ ) = C . Then q ( γ ) is a horizontal curvein q ( Γ ). For all g ∈ (cid:72) and t ∈ (cid:82) we have Π ( g Y t ) = Π ( g ). Consequently, we have Π ( q ( γ )) = Π ( q ( C )) = ˆ q ( C ), and ˆ q ( C ) is characteristic for q ( Γ ).For any v ∈ U , we have q ( vY ψ ( v ) ) ∈ q ( Γ ), and since q ( Γ ) is an intrinsic graph, we musthave q ( vY ψ ( v ) ) = ˆ q ( v ) Y ˆ ψ ( v ) . The claimed expressions for ˆ ψ follow directly. (cid:3) One application of Lemma 2.9 (used in Remark 4.3) is that if a , b , c ∈ (cid:82) and q ( v ) = Y b Z − c ˜ A a ( v ) for all v ∈ (cid:72) , then ˆ q ( x , 0, z ) = ( x , 0, z − ax − bx − c ). That is, for any quadraticfunction f , there is a map q : (cid:72) → (cid:72) so that the characteristic curves of q ( Γ ) are thecharacteristic curves of Γ translated by f .Finally, stretch maps and shear maps send intrinsic Lipschitz graphs to intrinsic Lips-chitz graphs (with a possible change in the Lipschitz constant). Lemma 2.10.
Let Γ be an intrinsic Lipschitz graph, and let a , b ∈ (cid:82) (cid:224) {0} . Then s a , b ( Γ ) and ˜ A b ( Γ ) are intrinsic Lipschitz graphs, with an intrinsic Lipschitz constant depending on a , b,and the intrinsic Lipschitz constant of Γ .Proof. Let q = s a , b or q = ˜ A b . As Γ is an intrinsic Lipschitz graph, there is a scale-invariantdouble cone C ⊆ (cid:72) containing a neighborhood of Y such that pC ∩ Γ = ∅ for all p ∈ Γ .The image q ( C ) is a scale-invariant double cone containing a neighborhood of Y , so thereis a 0 < λ < λ ⊆ q ( C ) (see [Rig19]). For all p ∈ Γ , q ( p )Cone λ ∩ q ( Γ ) ⊆ q ( p ) q ( C ) ∩ q ( Γ ) = q ( pC ∩ Γ ) = ∅ ,so q ( Γ ) is intrinsic λ –Lipschitz. (cid:3) Measures on lines and the kinematic formula.
Let L be the space of horizontallines in (cid:72) . For U ⊆ (cid:72) , denote the set of horizontal lines that intersect U by L ( U ) def = { L ∈ L : L ∩ U (cid:54)= ∅ }.Let N be the unique (up to constants) measure on L that is invariant under the actionof the isometry group of (cid:72) . Scalings of horizontal lines are horizontal lines, so scalingautomorphisms of (cid:72) act on L , and L ( s t , t ( M )) = t L ( M ) for all t >
0. Henceforth N willbe normalized so that N ( L ( B r ( x ))) = r for every r > x ∈ (cid:72) .The Heisenberg group satisfies the following kinematic formula, which we record herefor ease of later use (see [Mon05] or equation (6.1) in [CKN11]). There exists a constant c > E ⊆ (cid:72) and any open subset U ⊆ (cid:72) ,Per E ( U ) = c ˆ L Per E ∩ L ( U ∩ L ) d N ( L ). (56)Consider also the set L = {( L , p ) : L ∈ L ∧ p ∈ L } of pointed horizontal lines . Asso-ciate to each measurable subset K ⊆ L the following two quantities. ˆ L H (cid:161) { p ∈ L : ( L , p ) ∈ K } (cid:162) d N ( L ), (57)and ˆ π ˆ (cid:72) K (cid:161) p 〈 cos( θ ) X + sin( θ ) Y 〉 , p (cid:162) d H ( p ) d θ . (58)Both of the expressions in (57) and (58) define measures on L that are invariant underthe isometry group of (cid:72) , which acts transitively on L . Therefore, they are proportional,and there is a constant C > K ⊆ L , ˆ L H ( K L ) d N ( L ) = C ˆ π ˆ (cid:72) K ( L p , θ , p ) d H ( p ) d θ , (59)where we use the following notations for every L ∈ L , p ∈ (cid:72) and θ ∈ [0, 2 π ]. K L def = { p ∈ L : ( L , p ) ∈ K } ⊆ L and L p , θ def = p 〈 cos( θ ) X + sin( θ ) Y 〉 ∈ L . (60)2.6. Vertical perimeter and parametric vertical perimeter.
Given a measurable subset E ⊆ V , a measurable function ψ : V → (cid:82) and (a scale) a ∈ (cid:82) , we define the (normalized)parametric vertical perimeter at scale a of ψ on E by v PE , ψ ( a ) def = ´ E (cid:175)(cid:175) ψ ( v ) − ψ (cid:161) v Z − − a (cid:162)(cid:175)(cid:175) d H ( v )2 − a . (61)This notion relates to the usual vertical perimeter (29) of the epigraph of ψ as follows. Lemma 2.11 (parametric vertical perimeter versus vertical perimeter of epigraph) . Forany measurable subset E ⊆ V , any measurable function ψ : V → (cid:82) , and any a ∈ (cid:82) , v PE , ψ ( a ) = v Π − ( E ) (cid:161) Γ + ψ (cid:162) ( a ). OLIATED CORONA DECOMPOSITIONS 35
Proof.
Recalling (28), for Ω ⊆ (cid:72) and a ∈ (cid:82) we denote D a Ω = Ω (cid:52) Ω Z − a . Then D a Γ + ψ = (cid:169) vY t : v ∈ V ∧ ψ ( v ) < t (cid:201) ψ (cid:161) v Z − − a (cid:162)(cid:170) (cid:91) (cid:169) vY t : v ∈ V ∧ ψ (cid:161) v Z − − a (cid:162) < t (cid:201) ψ ( v ) (cid:170) ,since, by definition, Γ + ψ Z − a = (cid:169) vY t : v ∈ V ∧ ψ (cid:161) v Z − − a (cid:162) < t (cid:170) . Therefore, v Π − ( E ) (cid:161) Γ + ψ (cid:162) ( a ) = H (cid:161) Π − ( E ) ∩ D a Γ + ψ (cid:162) − a = ´ E (cid:175)(cid:175) ψ ( v ) − ψ (cid:161) v Z − − a (cid:162)(cid:175)(cid:175) d H ( v )2 − a = v PE , ψ ( a ). (cid:3) An advantage of the parametric vertical perimeter is that it increases or decreases by aconstant factor under a stretch map or a shear map, as computed in the following lemma.
Lemma 2.12.
Let ψ : V → (cid:82) and E ⊆ V be measurable. Let q : (cid:72) → (cid:72) , ˆ q : V → V , and ˆ ψ : V → (cid:82) be as in Lemma 2.9, i.e., q is a stretch map or a shear map, ˆ q is the map inducedon V , and ˆ ψ is the function such that q ( Γ ψ ) = Γ ˆ ψ . Then for all t ∈ (cid:82) we have • If a , b ∈ (cid:82) (cid:224) {0} and q = s a , b , then v P ˆ q ( E ), ˆ ψ ( t ) = | ab | · v PE , ψ (cid:179) t + log (cid:112) | ab | (cid:180) . • If b ∈ (cid:82) (cid:224) {0} and q = ˜ A b , then v P ˆ q ( E ), ˆ ψ ( t ) = v PE , ψ ( t ). Proof. If q = s a , b for some a , b ∈ (cid:82) (cid:224) {0}, then ˆ q ( x , 0, z ) = ( ax , 0, abz ) and ˆ ψ ( ˆ q ( v )) = b ψ ( v )for every v = ( x , 0, z ) ∈ V . So, v P ˆ q ( E ), ˆ ψ ( t ) = t ˆ ˆ q ( E ) (cid:175)(cid:175) ˆ ψ ( v ) − ˆ ψ (cid:161) v Z − − t (cid:162)(cid:175)(cid:175) d H ( v ) = t | b | ˆ ˆ q ( E ) (cid:175)(cid:175) ψ (cid:161) ˆ q − ( v ) (cid:162) − ψ (cid:161) ˆ q − ( v ) Z − ( ab ) − − t (cid:162)(cid:175)(cid:175) d H ( v ) = t a b ˆ E (cid:175)(cid:175) ψ ( v ) − ψ (cid:161) v Z − ( ab ) − − t (cid:162)(cid:175)(cid:175) d H ( v ) = | ab | · v PE , ψ (cid:179) t + log (cid:112) | ab | (cid:180) .Next, if q = ˜ A b for some b ∈ (cid:82) (cid:224) {0}, then ˆ ψ ( ˆ q ( v )) = ψ ( v ) + bx ( v ) for all v = ( x , 0, z ) ∈ E ,and by Lemma 2.8 we haveˆ q ( v ) = Π (cid:161) q ( x , 0, z ) (cid:162) = (cid:161) x , 0, z − bx (cid:162) .So, ˆ ψ ( v Z − t ) = ψ ( ˆ q − ( v Z − − t )) + bx ( ˆ q − ( v Z − − t )) = ψ ( ˆ q − ( v ) Z − − t ) + bx ( v ), and hence v P ˆ q ( E ), ˆ ψ ( t ) = t ˆ ˆ q ( E ) (cid:175)(cid:175) ψ (cid:161) ˆ q − ( v ) (cid:162) − ψ (cid:161) ˆ q − ( v ) Z − − t (cid:162)(cid:175)(cid:175) d H ( v ) = t ˆ E (cid:175)(cid:175) ψ ( v ) − ψ (cid:161) v Z − − t (cid:162)(cid:175)(cid:175) d H ( v ) = v P U , ψ ( t ). (cid:3) We end this section by recording a straightforward a priori upper bound on v P E , ψ ( a ). Lemma 2.13.
Suppose that E ⊆ V is measurable and ψ : V → (cid:82) is smooth. Then ∀ a ∈ (cid:82) , v PE , ψ ( a ) (cid:201) min (cid:189) a + (cid:107) ψ (cid:107) L ∞ ( V ) , 2 − a (cid:176)(cid:176)(cid:176)(cid:176) ∂ψ∂ z (cid:176)(cid:176)(cid:176)(cid:176) L ∞ ( V ) (cid:190) H ( E ). Proof.
For all v = ( x , 0, z ) ∈ E , we (trivially) have (cid:175)(cid:175) ψ ( v ) − ψ (cid:161) v Z − − a (cid:162)(cid:175)(cid:175) = | ψ ( x , 0, z ) − ψ ( x , 0, z − − a ) | (cid:201) (cid:107) ψ (cid:107) L ∞ ( V ) ,and (cid:175)(cid:175) ψ ( v ) − ψ (cid:161) v Z − − a (cid:162)(cid:175)(cid:175) = | ψ ( x , 0, z ) − ψ ( x , 0, z − − a ) | (cid:201) − a (cid:176)(cid:176)(cid:176)(cid:176) ∂ψ∂ z (cid:176)(cid:176)(cid:176)(cid:176) L ∞ ( V ) .Recalling the definition (61), we obtain the desired inequality by integrating over E . (cid:3)
3. C
ONSTRUCTING SURFACES AND EMBEDDINGS
In this section, we will prove Proposition 3.3, following the reasoning sketched inSection 1.2.2, to construct surfaces that are α –far from planes at α − different scales. Weuse these surfaces to prove the following theorem. Theorem 3.1.
For any k > , there is a left-invariant metric ∆ = ∆ k : (cid:72) × (cid:72) → [0, ∞ ) on (cid:72) and a measure space ( S , µ ) such that ( (cid:72) , ∆ ) embeds isometrically in L ( µ ) and such that forany h = ( a , b , c ) ∈ (cid:72) we have ∆ ( , h ) (cid:46) | a | + | b | + min (cid:169) (cid:112)| c | , k (cid:170) (cid:112) log k . (62) If moreover (cid:201) | c | (cid:201) k , then, in fact ∆ ( , h ) (cid:179) | a | + | b | + (cid:112)| c | (cid:112) log k . (63)We will prove Theorem 3.1 in Section 3.1 after deriving two of its applications, andstating Proposition 3.3. The first application of Theorem 3.1 is the proof of Theorem 1.7. Proof of Theorem 1.7 assuming Theorem 3.1.
Letting ∆ and ( S , µ ) be as in Theorem 3.1, fix ξ : (cid:72) → L ( µ ) such that (cid:107) ξ ( g ) − ξ ( h ) (cid:107) L ( µ ) = ∆ ( g , h ) for all g , h ∈ (cid:72) . Also, using [Ass83], fix m ∈ (cid:78) and ϕ : (cid:72) → (cid:82) m such that (cid:107) ϕ ( g ) − ϕ ( h ) (cid:107) (cid:96) m (cid:179) (cid:112) d ( g , h ) for all g , h ∈ (cid:72) .Suppose that ϑ (cid:202) . Consider the function τ : (cid:72) → L ( µ ) ⊕ (cid:82) ⊕ (cid:82) m ∼= L ( ν ) given by τ def = ξ (log k ) ϑ − ⊕ π ⊕ ϕ (log k ) ϑ . (64)Since ∆ is left-invariant, every g = ( x , y , z ), h = ( χ , υ , ζ ) ∈ (cid:72) with 1 (cid:201) d ( g , h ) (cid:201) k satisfy (cid:107) τ ( g ) − τ ( h ) (cid:107) L ( ν ) (cid:179) | x − χ | + | y − υ | + (cid:112) | z − ζ − x υ + y χ | (log k ) ϑ , (65)using (39) and Theorem 3.1. While (65) would hold even without the third componentof τ in (64), thanks to that component τ ( (cid:72) (cid:90) ) is a locally-finite subset of L ( ν ). By [Ost12],it follows that τ ( (cid:72) (cid:90) ) admits a bi-Lipschitz embedding into (cid:96) of distortion O (1). As theword metric d W on (cid:72) (cid:90) is bounded above and below by universal constant multiples of d ,this gives Theorem 1.7 provided k is a large enough universal constant multiple of n . (cid:3) OLIATED CORONA DECOMPOSITIONS 37
A second application of Theorem 3.1 is to construct a left-invariant metric on (cid:72) (cid:90) withthe properties of Theorem 1.9, at the cost of losing an iterated logarithm in the quantitativebound. We suspect that with more care one could remove this lower order factor, but wedid not attempt to do so (it has no impact on the main qualitative conclusion).
Theorem 3.2.
For any < p (cid:201) there is a left-invariant metric δ = δ p on (cid:72) (cid:90) that admits abi-Lipschitz embedding into both (cid:96) and (cid:96) q for all q (cid:202) p, yet not into any Banach spacewhose modulus of uniform convexity has power-type r for (cid:201) r < p (in particular, ( (cid:72) (cid:90) , δ ) does not admit a bi-Lipschitz embedding into a Hilbert space or (cid:96) s for < s < p). Moreover,if we denote ϑ = ϑ p = max{1/4, 1/ p } , then for every h = ( a , b , c ) ∈ (cid:72) (cid:90) with | c | (cid:202) we have δ ( , h ) (cid:179) | a | + | b | + (cid:112)| c | (log | c | ) ϑ (log log | c | ) . (66) Proof.
Define a left-invariant metric δ : (cid:72) (cid:90) × (cid:72) (cid:90) → [0, ∞ ) as a superposition of the metrics{ ∆ k } k > of Theorem 3.1, by setting for every h = ( a , b , c ) ∈ (cid:72) (cid:90) , δ ( , h ) def = ∞ (cid:88) n = n e (4 ϑ − n ∆ e e n ( , h ). (67)We will first verify (66), which in particular implies that the sum defining δ converges, andhence by Theorem 3.1 we would know that δ is indeed a left-invariant metric on (cid:72) (cid:90) , andthat ( (cid:72) (cid:90) , δ ) admits an isometric embedding into (cid:96) ( L ( µ )). By [Ost12], it follows from thisthat ( (cid:72) (cid:90) , δ ) also admits a bi-Lipschitz embedding into the sequence space (cid:96) .Fix h = ( a , b , c ) ∈ (cid:72) (cid:90) with | c | (cid:202) e e and choose m = m ( c ) ∈ (cid:78) such that e e m (cid:201) (cid:112) | c | < e e m + . (68)Then, δ ( , h ) (62) (cid:46) | a | + | b | + ∞ (cid:88) n = min (cid:110) (cid:112)| c | , e e n (cid:111) n e ϑ n (cid:46) | a | + | b | + m (cid:88) n = e e n n e ϑ n + ∞ (cid:88) n = m + (cid:112)| c | n e ϑ n (cid:179) | a | + | b | + e e m m e ϑ m + (cid:112)| c | m e ϑ m (68) (cid:46) | a | + | b | + (cid:112)| c | (log | c | ) ϑ (log log | c | ) .Conversely, since the sum in (67) is at least its summands for n = n = m + δ ( , h ) (cid:38) | a | + | b | + (cid:112)| c | ( m + e ϑ ( m + (68) (cid:38) | a | + | b | + (cid:112)| c | (log | c | ) ϑ (log log | c | ) .This is (66) if | c | (cid:202) e e , but then (66) follows formally in the remaining range 3 (cid:201) | c | < e e (simply use the triangle inequality to reduce the upper bound to the case of large enough | c | that we just proved, and take only the n = n (cid:202) c ( B n , d W ) ( B n , δ ) (cid:46) (log n ) ϑ (log log n ) . (69)At the same time, if 2 (cid:201) r < p and X is a Banach space whose modulus of uniform convexityhas power-type r , then by [LN14b] we have c X ( B n , d W ) (cid:38) X (log n ) r . (70) By combining (69) and (70) we deduce that c X ( B n , d W ) (cid:38) X (log n ) r − ϑ (log log n ) (cid:202) (log n ) r − q (log log n ) −−−−→ n →∞ ∞ .Consequently, ( (cid:72) (cid:90) , δ ) does not admit a bi-Lipschitz embedding into X .It remains to show that ( (cid:72) (cid:90) , δ ) admits a bi-Lipschitz embedding into (cid:96) q for any q (cid:202) p .For this, due to [Ost12], since ( (cid:72) (cid:90) , δ ) is locally finite it suffices to show that ( (cid:72) (cid:90) , δ ) admitsa bi-Lipschitz embedding into L q . By [LN14a, Lemma 3.1], for any 0 < ε < , there exists aleft-invariant metric ρ ε on (cid:72) (cid:90) such that ( (cid:72) (cid:90) , ρ ε ) embeds isometrically into L q , and ∀ h = ( a , b , c ) ∈ (cid:72) (cid:90) , ρ ε ( , h ) (cid:179) | a | − ε + | b | − ε + ε q | c | − ε . (71)Define a left-invariant metric ρ : (cid:72) (cid:90) × (cid:72) (cid:90) → [0, ∞ ) by setting for every h = ( a , b , c ) ∈ (cid:72) (cid:90) , ρ ( , h ) def = (cid:181) | a | q + | b | q + ∞ (cid:88) n = n q e n ( q ϑ − ρ e − n ( , h ) q (cid:182) q .By design, ( (cid:72) (cid:90) , ρ ) embeds isometrically into (cid:96) q ( L q ). So, the proof of Theorem 3.2 will becomplete if we show that δ ( , h ) (cid:179) ρ ( , h ) for all h = ( a , b , c ) ∈ (cid:72) (cid:90) with, say, | c | (cid:202) (cid:181) ∞ (cid:88) n = n q e nq ϑ | c | qe − n (cid:182) q (cid:179) | c | ) ϑ (log log | c | ) . (72)Fix s = s ( c ) ∈ (cid:78) such that 2 e s (cid:201) log | c | < e s + (this is possible because | c | (cid:202) (cid:181) ∞ (cid:88) n = n q e nq ϑ | c | qe − n (cid:182) q (cid:46) (cid:181) s − (cid:88) k = s − k ) q e ( s − k ) q ϑ | c | qe − ( s − k ) (cid:182) q + (cid:181) ∞ (cid:88) n = s + n q e nq ϑ (cid:182) q (cid:179) e s ϑ (cid:181) s − (cid:88) k = e qk ϑ ( s − k ) q ( | c | e − s ) qe k (cid:182) q + s e s ϑ (cid:179) s e s ϑ (cid:179) | c | ) ϑ (log log | c | ) ,where the final step holds by our choice of s , and the penultimate step holds as | c | e − s (cid:202) e by our choice of s , and therefore the sum in question is dominated by its k = n = s summand. (cid:3) The main ingredient in the proof of Theorem 3.1 is the following proposition, which isproved in Section 3.2. It constructs a function ψ : V → (cid:82) whose intrinsic graph has smallhorizontal perimeter but large vertical perimeter due to bumps at many different scales.Here and throughout the rest of this section, we denote the unit square in V by U , i.e., U def = [0, 1] × {0} × [0, 1] ⊆ V . Proposition 3.3.
There are universal constants ρ , R , r ∈ (cid:82) with R > r and ρ > R − r ) suchthat for any α ∈ (cid:78) , there is a smooth function ψ : V → (cid:82) that has the following properties. (1) ψ is periodic with respect to the integer lattice (cid:90) × {0} × (cid:90) of V . (2) (cid:107) ∂ ψ ψ (cid:107) L ( U ) (cid:46) . (3) (cid:107) ψ (cid:107) L ∞ ( V ) (cid:201) α . OLIATED CORONA DECOMPOSITIONS 39 (4) v PU , ψ ( a ) (cid:38) α for any integer (cid:201) n < α and any a ∈ I + log ( αρ n ) , where I = [ r , R ] .Hence, (cid:176)(cid:176)(cid:176) v PU , ψ (cid:176)(cid:176)(cid:176) L ([log ( αρ n ) + r ,log ( αρ n ) + R ]) (cid:38) α ,(5) For any q > , we have (cid:176)(cid:176)(cid:176) v PU , ψ (cid:176)(cid:176)(cid:176) L q ( (cid:82) ) (cid:38) α q − .(6) v PU , ψ ( a ) (cid:46) min (cid:110) α , a α (cid:111) for any a ∈ (cid:82) . By Proposition 2.1, the second assertion of Proposition 3.3 implies that H ( ∂ E ) (cid:46) E is the epigraph of the restriction of ψ to the unit square U ⊆ V . In combinationwith Proposition 3.3.(5), since α can be arbitrarily large, this shows that the L ( (cid:82) ) normin (30) cannot be replaced by L q ( (cid:82) ) for any q ∈ (0, 4); as explained in the introduction, thisalso implies the optimality of Theorem 1.1.Proposition 3.3.(5) follows directly from Proposition 3.3.(4). Indeed, since ρ > R − r ) ,the intervals {[log ( αρ n ) + r , log ( αρ n ) + R ]} n ∈ (cid:90) are disjoint. Consequently, (cid:176)(cid:176)(cid:176) v PU , ψ (cid:176)(cid:176)(cid:176) qL q ( (cid:82) ) (cid:202) α − (cid:88) n = (cid:176)(cid:176)(cid:176) v PU , ψ (cid:176)(cid:176)(cid:176) qL q ([log ( αρ n ) + r ,log ( αρ n ) + R ]) (cid:202) α − (cid:88) n = R − r ) q − (cid:176)(cid:176)(cid:176) v PU , ψ (cid:176)(cid:176)(cid:176) qL ([log ( αρ n ) + r ,log ( αρ n ) + R ]) (cid:38) α − q . (73)where the penultimate step is an application of Jensen’s inequality and the final step holdsbecause R − r > α − q .3.1. Obtaining an embedding from an intrinsic graph.
Here we show how Theorem 3.1follows from Proposition 3.3. Let ρ , r , R > k >
8. Let α ∈ (cid:78) be the unique integer satisfying (cid:115) log ρ (cid:181) k (cid:182) (cid:201) α < + (cid:115) log ρ (cid:181) k (cid:182) . (74)Let ψ = ψ α be the function produced by Proposition 3.3. Write Γ = Γ ψ and Γ + = Γ + ψ .Denote by A ⊆ V ∩ (cid:72) (cid:90) the discrete subgroup that is generated by X and Z , so that as asubset of (cid:82) we have A = (cid:90) × {0} × (cid:90) . For every p ∈ (cid:72) define ∀ h , h ∈ (cid:72) , λ p ( h , h ) def = (cid:175)(cid:175) p − Γ + ( h ) − p − Γ + ( h ) (cid:175)(cid:175) = (cid:189) | { ph , ph } ∩ Γ + | = A –periodicity of ψ we have a Γ = Γ and λ ap ( h , h ) = λ p ( h , h ) for all a ∈ A and p , h , h ∈ (cid:72) . We can therefore define λ p also when p is an equivalence class in thequotient A \ (cid:72) . Consider the following fundamental domain for A . P def = (cid:169) X a Z c Y b : a , c ∈ [0, 1) and b ∈ (cid:82) (cid:170) = (cid:169)(cid:161) a , b , c + ab (cid:162) : ( a , b , c ) ∈ [0, 1) × (cid:82) × [0, 1) (cid:170) . We may define l : (cid:72) × (cid:72) → [0, ∞ ) by l ( h , h ) def = ˆ A \ (cid:72) λ p ( h , h ) d H ( p ) = ˆ P λ p ( h , h ) d H ( p ).Since (cid:72) is a unimodular group (namely, one directly checks that the Lebesgue measure H is a bi-invariant Haar measure on (cid:72) ), and λ p ( g h , g h ) = λ pg ( h , h ), we have ∀ g , h , h ∈ (cid:72) , l ( g h , g h ) = l ( h , h ),i.e., l is a left-invariant semi-metric on (cid:72) . Lemma 3.4.
For every a ∈ (cid:82) we have l (cid:161) , Z − a (cid:162) = − a · v PU , ψ ( a ). Proof.
For v ∈ V and b ∈ (cid:82) , we have vY b ∈ Γ + if and only if b > ψ ( v ). So, for any c > λ vY b ( , Z c ) = λ ( vY b , v Z c Y b ) = (cid:40) ψ ( v ) < b (cid:201) ψ ( v Z c ) or ψ ( v Z c ) < b (cid:201) ψ ( v ).0 otherwise.Consequently, ˆ (cid:82) λ vY b ( , Z c ) d b = | ψ ( v Z c ) − ψ ( v ) | .Therefore, fixing a ∈ (cid:82) and denoting c = − a , we see that l (cid:161) , Z − a (cid:162) = ˆ P λ p ( , Z c ) d p = ˆ (cid:82) ˆ U λ vY b ( , Z c ) d v d b = ˆ U | ψ ( v Z c ) − ψ ( v ) | d v = − a · v PU , ψ ( t ). (cid:3) For every θ ∈ [0, 2 π ) let R θ : (cid:72) → (cid:72) be rotation around the z –axis by angle θ . Define thefollowing left-invariant semi-metric on (cid:72) , which is also (by design) invariant under thefamily of { R θ : θ ∈ [0, 2 π )} automorphisms of (cid:72) . ∀ h , h ∈ (cid:72) , M ( h , h ) def = ˆ π l (cid:161) R θ ( h ), R θ ( h ) (cid:162) d θ . Lemma 3.5.
For every w ∈ H we have M ( , w ) (cid:46) (cid:107) w (cid:107) .Proof. By the rotation-invariance of M , it suffices to show that M ( , X t ) (cid:46) | t | for all t .In fact, by the left-invariance of M and the triangle inequality, it suffices to prove that M ( X t , X − t ) (cid:46) t for 0 < t < .Let L = 〈 X 〉 ⊆ (cid:72) be the x –axis. Recall that L p , θ = pR θ ( L ) for p ∈ (cid:72) and θ ∈ [0, 2 π ).The map ( p , θ ) (cid:55)→ ( L p , θ , p ) is a bijection between (cid:72) × [0, π ) and the set of pointed lines L = {( L , p ) : L ∈ L ∧ p ∈ L }.By the above definitions, we have M ( X − t , X t ) = ˆ π ˆ P λ p (cid:161) R θ ( X − t ), R θ ( X t ) (cid:162) d H ( p ) d θ .Let K ⊆ P × [0, 2 π ) be the set such that pR θ ( X ± t ) (cid:54)∈ Γ and L p , θ intersects Γ transversally.Then L p , θ ∩ Γ is a discrete set of points and L p , θ crosses from one side of Γ to the other ateach point. The complement of K has measure zero. OLIATED CORONA DECOMPOSITIONS 41
Choose a point and angle ( p , θ ) ∈ K such that λ p ( R θ ( X − t ), R θ ( X t )) is nonzero. Then pR θ ( X − t ) and pR θ ( X t ) lie on opposite sides of Γ , so there is a point g ∈ L p , θ ∩ Γ such that d ( p , g ) < t . Let v = Π ( g ) ∈ V , let u = Π ( p ) ∈ U , and let b = y ( p ), so that g = vY ψ ( v ) and p = uY b . Then | x ( v ) − x ( u ) | = | x ( g ) − x ( p ) | (cid:201) d ( g , p ) (cid:201) t . By Proposition 3.3.(3), we have | ψ ( v ) | (cid:201)
1, so | b | (cid:201) | ψ ( v ) | + d ( p , g ) (cid:201) t + (cid:201)
2, and d (cid:161) Y − b uY b , Y − b vY b (cid:162) (cid:201) d (cid:161) Y − b uY b , Y − b vY ψ ( v ) (cid:162) + | ψ ( v ) − b | < d ( p , g ) + t < t . (75)Since Y − b ( a , 0, c ) Y b = ( a , 0, c + ab ) for every a , b ∈ (cid:82) , (75) implies that | z ( v ) − z ( u ) − b ( x ( v ) − x ( u )) | < (2 t ) (cid:201) t .Because | b | (cid:201)
2, it follows that x ( v ) ∈ [ − t , 1 + t ] and z ( v ) ∈ [ − t , 1 + t ]. If we denote U (cid:48) = [ −
1, 2] × {0} × [ −
1, 2] ⊆ V , then by our restrictions on t , we have v ∈ U (cid:48) and therefore d ( p , L p , θ ∩ Γ ( U (cid:48) )) < t , where for W ⊆ V we let Γ ( W ) def = (cid:169) wY ψ ( w ) : w ∈ W (cid:170) = Γ ∩ Π − ( W )be the intrinsic graph of ψ restricted to W .For L ∈ L , let I L = (cid:169) p ∈ L : d (cid:161) p , L ∩ Γ ( U (cid:48) ) (cid:162) < t (cid:170) .If L intersects Γ transversally, then H ( I L ) (cid:201) t | L ∩ Γ ( U (cid:48) ) | = t Per Γ + ∩ L (cid:161) Π − ( U (cid:48) ) (cid:162) .We have seen above that if ( p , θ ) ∈ K and λ p ( R θ ( X − t ), R θ ( X t )) (cid:54)=
0, then p ∈ I L p , θ . Hence, M ( X − t , X t ) = ˆ π ˆ P λ p (cid:161) R θ ( X − t ), R θ ( X t ) (cid:162) d H ( p ) d θ (59) (cid:46) ˆ L H ( I L ) d N ( L ) (cid:46) t ˆ L Per Γ + ∩ L (cid:161) Π − ( U (cid:48) ) (cid:162) d N ( L ) (56) (cid:179) t Per Γ + (cid:161) Π − ( U (cid:48) ) (cid:162) (cid:46) t .where Per Γ + (cid:161) Π − ( U (cid:48) ) (cid:162) (cid:46) (cid:3) Next, define a left-invariant semi-metric Λ on (cid:72) by ∀ h , h ∈ (cid:72) , Λ ( h , h ) def = ˆ R + log ρ r − log ρ a M (cid:161) s − a ( h ), s − a ( h ) (cid:162) d a . Lemma 3.6. Λ ( , Z c ) (cid:38) (cid:112) c α for all α ρ α (cid:201) c (cid:201) α and Λ ( , Z c ) (cid:46) min (cid:110) (cid:112) c α , α (cid:111) for all c > .Proof. Write c = − t for some t ∈ (cid:82) . By Lemma 3.4 we have the following identity. Λ ( , Z c ) = π ˆ R + log ρ r − log ρ a l (cid:179) , Z − t + a ) (cid:180) d a = π − t ˆ R + log ρ r − log ρ v P U , ψ ( t + a ) d a . (76)So, Λ ( , Z c ) (cid:46) min (cid:110) (cid:112) c α , α (cid:111) for c ∈ (0, ∞ ) by (76) and the final assertion of Proposition 3.3. If α ρ α (cid:201) c (cid:201) α , then t ∈ [log ( αρ n ), log ( αρ n + )] for some integer 0 (cid:201) n < α . Hence,[ t − log ρ + r , t + log ρ + R ] ⊇ [log ( αρ n ) + r , log ( αρ n ) + R ],so (76) implies that Λ ( , Z c ) (cid:202) π − t (cid:176)(cid:176)(cid:176) v PU , ψ (cid:176)(cid:176)(cid:176) L ([log ( αρ n ) + r ,log ( αρ n ) + R ]) (cid:38) (cid:112) c α ,where the final step is the third assertion of Proposition 3.3 (and the definition of t ). (cid:3) Lemma 3.7. Λ ( h , h ) (cid:46) d ( h , h ) for all h , h ∈ (cid:72) .Proof. By Lemma 3.5 we have M ( , X t ) (cid:46) | t | for any t ∈ (cid:82) , so Λ ( , X t ) = ˆ R + log ρ r − log ρ a M (cid:161) , X − a t (cid:162) d a (cid:46) t ( R − r + ρ ) (cid:46) | t | .Therefore also Λ ( , Y t ) = Λ ( , X t ) (cid:46) | t | , by the rotation-invariance of Λ . Since Λ is left-invariant it suffices to show that Λ ( , h ) (cid:46) d ( , h ) for all h ∈ (cid:72) . Any h ∈ (cid:72) can be written as h = X a Y b [ X c , Y c ] for a , b , c ∈ (cid:82) satisfying | a | , | b | , | c | (cid:46) d ( , h ), so Λ ( , h ) (cid:201) Λ ( , X a ) + Λ ( , Y b ) + Λ ( , X c ) + Λ ( , Y c ) (cid:46) d ( , h ). (cid:3) Proof of Theorem 3.1.
Define a semi-metric ∆ on (cid:72) by setting for every h , h ∈ (cid:72) , ∆ ( h , h ) def = k α Λ (cid:161) s k α ( h ), s k α ( h ) (cid:162) + (cid:113)(cid:161) x ( h ) − x ( h ) (cid:162) + (cid:161) y ( h ) − y ( h ) (cid:162) . (77) Λ is an integral of cut metrics, so ( (cid:72) , ∆ ) embeds isometrically into L (see e.g. [DL97]). Byconstruction, Λ is both left-invariant and invariant under the rotations { R θ : θ ∈ [0, 2 π ]}.Suppose that v = ( a , b , c ) ∈ (cid:72) and let w = ( a , b , 0) so that w ∈ H and v = w Z c . ByLemma 3.5 and the second part of Lemma 3.6, we have ∆ ( , v ) (cid:201) ∆ ( , w ) + ∆ ( , Z c ) (cid:46) | a | + | b | + min (cid:169) (cid:112)| c | , k (cid:170) α .Recalling that α (cid:179) (cid:112) log k is given in (74), this establishes (62).To prove Theorem 3.1, it therefore remains to establish (63), i.e.,1 (cid:201) | c | (cid:201) k =⇒ ∆ ( , v ) (cid:38) | a | + | b | + (cid:112)| c | α . (78)By Lemma 3.7, there is L > ∆ ( h , h ) (cid:201) Ld ( h , h ) for any h , h ∈ (cid:72) . By the firstpart of Lemma 3.6, there is C > ∆ ( , Z c ) (cid:202) C (cid:112) c α for all 1 (cid:201) c (cid:201) k . On one hand,if (cid:107) w (cid:107) (cid:202) C (cid:112)| c | L α , then ∆ ( , v ) (cid:202) (cid:107) w (cid:107) (cid:179) | a | + | b | + (cid:112)| c | α . On the other hand, if (cid:107) w (cid:107) < C (cid:112)| c | L α , then ∆ ( , v ) (cid:202) ∆ ( , Z c ) − ∆ ( , w ) (cid:202) C (cid:112) c α − L (cid:107) w (cid:107) (cid:202) C (cid:112)| c | α (cid:38) | a | + | b | + (cid:112)| c | α .In either case, (78) holds. (cid:3) OLIATED CORONA DECOMPOSITIONS 43
Constructing a bumpy intrinsic graph.
In this section, we prove Proposition 3.3.We start with a brief overview of our strategy. As sketched in Section 1.2.2, we will proveProposition 3.3 by constructing a smooth function ψ : V → (cid:82) whose intrinsic graph isroughly α − –far from a vertical plane at α different scales. Specifically, for a suitablechoice of universal constant ρ > ψ as a sum ψ = (cid:80) α − i = β i . Each ofthe summands β i : V → (cid:82) will itself be a sum of smooth bump functions of amplitude (cid:107) β i (cid:107) L ∞ ( V ) (cid:179) α − ρ − i that are supported on regions whose width ( x –coordinate) is ρ − i andwhose height ( z –coordinate) is roughly α − ρ − i ; their aspect ratio is therefore roughly ρ − i (cid:113) α − ρ − i (cid:179) α .These regions cover V and have disjoint interiors. We will see that the bumpiness of β i atscale α − ρ − i implies the desired lower bounds on v PU , ψ ( t ) when t is near log ( αρ i ).In order to ensure that (cid:107) ∂ ψ ψ (cid:107) L ( U ) is bounded, we construct β i iteratively. For i ∈ (cid:78) ,we denote ψ i = (cid:80) i − j = β j and align the long axis of the bump functions making up β i withthe characteristic curves of Γ ψ i . This ensures that the characteristic curves of Γ ψ crossthe bumps from left to right. Since ∂ ψ f measures the change in f : V → (cid:82) along thecharacteristic curves of Γ ψ and each bump has amplitude roughly α − ρ − i and width ρ − i ,we have | ∂ ψ β i | (cid:46) α − ρ − i / ρ − i (cid:179) α − .This iterative procedure is one of the motivations for the definition of a foliated coronadecomposition. A foliated corona decomposition of an arbitrary intrinsic graph Γ canbe viewed as a sequence of partitions of V into regions as above, where the pieces ofthe partition are aligned with the characteristic curves of Γ . One can use these partitionsto reconstruct Γ as a sum of perturbations, just as we constructed ψ as a sum of bumpfunctions. Theorem 1.17 then states that any intrinsic Lipschitz graph can be constructedby such a process.This construction also demonstrates the importance of the aspect ratio. If the construc-tion is modified so that the bump functions making up β i are supported on regions ofaspect ratio α i , then (cid:107) ∂ ψ β i (cid:107) L ( U ) (cid:179) α − i . If the scales of the bump functions are sufficientlyseparated, then { ∂ ψ β i } i (cid:202) are roughly orthogonal in L ( U ) and (cid:107) ∂ ψ ψ (cid:107) L ( U ) (cid:179) (cid:88) i (cid:202) (cid:107) ∂ ψ β i (cid:107) L ( U ) (cid:179) (cid:88) i (cid:202) α − i .For ψ to be intrinsic λ –Lipschitz, we must have (cid:107) ∂ ψ ψ (cid:107) L ( U ) (cid:46) λ
1, which necessitates that (cid:80) i α − i (cid:202) (cid:46) λ
1. This motivates the α ( Q ) − factor in the weighted Carleson condition (33).We next set some notation in preparation for the proof of Proposition 3.3. If ψ : V → (cid:82) is smooth, then the vector field M ψ def = ∂∂ x − ψ ∂∂ z corresponding to ∂ ψ is smooth (recall the definitions in Section 2.2). The flow lines of M ψ are the characteristic curves of Γ ψ , which foliate V (recall the terminology in Section 2.3).For s ∈ (cid:82) , let Φ ( ψ ) s : V → V be the flow of M ψ , so that Φ ( ψ ) = id V and such that for any v ∈ V , the curve s (cid:55)→ Φ ( ψ ) s ( v ) is a characteristic curve of Γ ψ . Denote ψ ≡ Γ = Γ ψ = V . Given an integer i (cid:202) ψ i ,we will next explain how to construct ψ i + : V → (cid:82) from ψ i : V → (cid:82) . Let G i def = (cid:169) ( m ρ − i , 0, n α − ρ − i ) : m , n ∈ (cid:90) (cid:170) ⊆ V . (79)Label the points in G i arbitrarily as v i ,1 , v i ,2 , . . . and note that the points U ∩ { v i ,1 , v i ,2 , . . .}form a ρ i × α ρ i grid in U . For each j ∈ (cid:78) and s , t ∈ (cid:82) define R i , j ( s , t ) def = Φ ( ψ i ) s ( v i , j Z t ) ∈ V . (80)Each R i , j is a diffeomorphism from (cid:82) to V . For any s , t ∈ (cid:82) , the image R i , j ( s × (cid:82) ) isa vertical line and R i , j ( (cid:82) × t ) is a characteristic curve of Γ ψ i . Using the terminology offoliated patchworks that we will introduce in Section 4, the map R i , j sends rectangles in V to pseudoquads of Γ ψ i (regions in V that are bounded by characteristic curves of Γ ψ i above and below and by vertical line segments on either side). Denote Q i , j def = R i , j (cid:161) [0, ρ − i ] × [0, α − ρ − i ] (cid:162) ⊆ V . (81)Thus, Q i , j is a pseudoquad whose lower-left corner is v i , j . The sets Q i ,1 , Q i ,2 , . . . cover V and have disjoint interiors. They are obtained by cutting V into vertical strips of width ρ − i , then cutting each vertical strip along characteristic curves separated by α − ρ − i .Let β : V → (cid:82) be a smooth bump function supported on the unit square U , such that β and its partial derivatives of order at most 2 are all in the interval [ −
1, 1]. Fix also α , ρ ∈ (cid:78) with ρ >
1. Define β i , j : V → (cid:82) by setting it to be 0 on V (cid:224) Q i , j , and for all R i , j ( s , t ) ∈ Q i , j , β i , j (cid:161) R i , j ( s , t ) (cid:162) def = α − ρ − i β ( ρ i s , 0, α ρ i t ). (82)Thus β i , j is a bump function supported on Q i , j . Write β i def = ∞ (cid:88) j = β i , j , (83)and ψ i + = ψ i + β i . (84)Since Q i ,1 , Q i ,2 , . . . have disjoint interiors, (cid:107) ψ i + (cid:107) L ∞ ( V ) (cid:201) (cid:107) ψ i (cid:107) L ∞ ( V ) + α − ρ − i , so by induc-tion we have (cid:107) ψ i (cid:107) L ∞ ( V ) (cid:201) α − ρ − (cid:201) α − . (85)The inductive construction implies that ψ i ( x + m , 0, z + n ) = ψ i ( x , 0, z ) for all m , n ∈ (cid:90) and x , z ∈ (cid:82) . Thus, for any integer i (cid:202) ψ i satisfies the first and third assertions(periodicity and L ∞ boundedness) of Proposition 3.3. We will show that if ρ is large enough(depending only on β ), then ψ = ψ α satisfies the remaining assertions of Proposition 3.3,namely, the stated upper bounds on ∂ ψ ψ and lower bounds on v PU , ψ ( a ). OLIATED CORONA DECOMPOSITIONS 45
The horizontal perimeter of Γ ψ i . In this section, we prove the second assertion ofProposition 3.3 by bounding (cid:107) ∂ ψ i ψ i (cid:107) L ( U ) . This bound, combined with Proposition 2.1,gives an upper bound on H ( Γ ψ i | U ).Write for simplicity ∂ i def = ∂ ψ i and let D i def = ∂ i + ψ i + − ∂ i ψ i . For f , g ∈ L ( U ) we write 〈 f , g 〉 U def = ˆ U f g d H . Lemma 3.8.
For every ρ (cid:202) and α (cid:202) , ∀ i ∈ (cid:78) , (cid:107) D i (cid:107) L ∞ ( V ) (cid:46) α − , and ∀ m , n ∈ (cid:78) , |〈 D m , D n 〉| (cid:46) α − ρ m − n .Note that Lemma 3.8 implies that for every i ∈ (cid:78) , (cid:107) ∂ ψ i ψ i (cid:107) L ( U ) (cid:46) (cid:112) i α . (86)Thus, (cid:107) ∂ ψ i ψ i (cid:107) L ( U ) (cid:46) i (cid:46) α , i.e., the second assertion of Proposition 3.3 holds true.To deduce (86) from Lemma 3.8 write ∂ ψ i ψ i = i − (cid:88) n = D n , (87)and expand the squares to get (cid:107) ∂ ψ i ψ i (cid:107) L ( U ) = i − (cid:88) n = (cid:107) D n (cid:107) L ( U ) + i − (cid:88) m = i − (cid:88) n = m + 〈 D m , D n 〉 (cid:46) i − (cid:88) n = α − + i − (cid:88) m = ∞ (cid:88) k = α − ρ − k (cid:179) i α − ,where the penultimate step is Lemma 3.8 and the final step holds because ρ (cid:202) i (cid:202) D i = ∂ i + ψ i + − ∂ i ψ i = ( ∂ i + − ∂ i ) ψ i + + ∂ i β i = − β i ∂ψ i + ∂ z + ∂ i β i . (88)We will prove Lemma 3.8 by bounding the terms in the right hand side of (88) separately.To this end, it will be convenient to define as follows a system of flow coordinates on Q i , j .Fix i ∈ (cid:78) ∪ {0} and j ∈ (cid:78) . Write for simplicity ( x , 0, z ) = v i , j , Q = Q i , j and R = R i , j .Denote R − = ( s , t ) : Q → (cid:82) . If ( x , 0, z ) : Q → (cid:82) is the standard coordinate system, thenrecalling the differential equation (50) for characteristic curves, we have x = x + s and z = z + t − ˆ s ψ i (cid:161) R ( σ , t ) (cid:162) d σ .Consequently, (cid:181) ∂ x ∂ s ∂ x ∂ t ∂ z ∂ s ∂ z ∂ t (cid:182) = (cid:181) − ψ i − ´ s ∂ψ i ∂ t ( R ( σ , t )) d σ (cid:182) . (89)In particular, it follows that ∂ s ∂ z = ∂ z ∂ t · ∂ t ∂ z =
1. Also, ∂∂ s = ∂∂ x − ψ i ∂∂ z = ∂ i , (90)so ∂∂ s does not depend on j . Observe that by the definition of β i , for all s , t ∈ [0, ρ − i ] × [0, α − ρ − i ], we have β i (cid:161) R i , j ( s , t ) (cid:162) = α − ρ − i β ( ρ i s , 0, α ρ i t ).It follows that for any m , n ∈ (cid:78) ∪ {0}, we have (cid:176)(cid:176)(cid:176)(cid:176) ∂ m ∂ s m ∂ n ∂ t n β i (cid:176)(cid:176)(cid:176)(cid:176) L ∞ ( Q i , j ) = α − ρ − i ρ mi ( α ρ i ) n (cid:176)(cid:176)(cid:176)(cid:176) ∂ m ∂ x m ∂ n ∂ z n β (cid:176)(cid:176)(cid:176)(cid:176) L ∞ ( U ) . (91)This is especially useful when m + n (cid:201)
2, since in this case (cid:176)(cid:176)(cid:176) ∂ m ∂ x m ∂ n ∂ z n β (cid:176)(cid:176)(cid:176) L ∞ ( U ) (cid:201)
1. Thus, (cid:176)(cid:176)(cid:176)(cid:176) ∂β i ∂ t (cid:176)(cid:176)(cid:176)(cid:176) L ∞ ( Q i , j ) (cid:201) ρ i and (cid:176)(cid:176)(cid:176)(cid:176) ∂ β i ∂ t (cid:176)(cid:176)(cid:176)(cid:176) L ∞ ( Q i , j ) (cid:201) α ρ i . (92)Furthermore, since { Q i , j } ∞ j = cover V , (cid:107) ∂ i β i (cid:107) L ∞ ( V ) = max j ∈ (cid:78) (cid:107) ∂ i β i (cid:107) L ∞ ( Q i , j ) (90) = max j ∈ (cid:78) (cid:176)(cid:176)(cid:176)(cid:176) ∂β i ∂ s (cid:176)(cid:176)(cid:176)(cid:176) L ∞ ( Q i , j ) (91) (cid:201) α − . (93)The following lemma obtains bounds on vertical derivatives that will be used later. Lemma 3.9. If ρ (cid:202) , then for all i ∈ (cid:78) ∪ {0} we have (cid:176)(cid:176)(cid:176)(cid:176) ∂ψ i ∂ z (cid:176)(cid:176)(cid:176)(cid:176) L ∞ ( V ) (cid:201) ρ i − , (94) and (cid:176)(cid:176)(cid:176)(cid:176) ∂ ψ i ∂ z (cid:176)(cid:176)(cid:176)(cid:176) L ∞ ( V ) (cid:201) α ρ i − . (95) Furthermore, if ( s , t ) are the above flow coordinates on Q i , j for some j ∈ (cid:78) , then the follow-ing bound holds point-wise on Q i , j . < e − ρ − (cid:201) ∂ t ∂ z = (cid:181) ∂ z ∂ t (cid:182) − (cid:201) e ρ − <
43 (96)
Proof.
Denote for every integer i (cid:202) m i def = (cid:176)(cid:176)(cid:176)(cid:176) ∂ψ i ∂ z (cid:176)(cid:176)(cid:176)(cid:176) L ∞ ( V ) and µ i def = (cid:176)(cid:176)(cid:176)(cid:176) ∂ ψ i ∂ z (cid:176)(cid:176)(cid:176)(cid:176) L ∞ ( V ) . (97)Thus m = µ =
0. Fix j ∈ (cid:78) and let ( s , t ) be the flow coordinates on Q i , j . We will first usethe above identities to deduce bounds on vertical derivatives of t in terms of m i , µ i , andthen bootstrap these bounds to deduce the desired bounds on m i , µ i themselves.By (89), the following identity holds point-wise on Q i , j . ∂∂ s ∂ z ∂ t = − ∂ψ i ∂ t = − ∂ψ i ∂ z ∂ z ∂ t − ∂ψ i ∂ x ∂ x ∂ t = − ∂ψ i ∂ z ∂ z ∂ t .Consequently, ∂∂ s (cid:181) log ∂ z ∂ t (cid:182) = − ∂ψ i ∂ z .Since ∂ z ∂ t = s =
0, we integrate to get the identity ∂ z ∂ t = exp (cid:181) − ˆ s ∂ψ i ∂ z (cid:161) R i , j ( σ , t ) (cid:162) d σ (cid:182) . (98) OLIATED CORONA DECOMPOSITIONS 47
And, by differentiating (98) we also get ∂ z ∂ t = − ∂ z ∂ t ˆ s ∂ ψ i ∂ z (cid:161) R i , j ( σ , t ) (cid:162) ∂ z ∂ t (cid:161) R i , j ( σ , t ) (cid:162) d σ . (99)For points in Q i , j , we have | s | (cid:201) ρ − i , so it follows from (98) that (cid:175)(cid:175) log ∂ z ∂ t (cid:175)(cid:175) (cid:201) ρ − i m i , i.e., e − ρ − i m i (cid:201) ∂ z ∂ t (cid:201) e ρ − i m i . (100)By substituting (100) into (99) we deduce that (cid:175)(cid:175)(cid:175)(cid:175) ∂ z ∂ t (cid:175)(cid:175)(cid:175)(cid:175) (cid:201) ρ − i e ρ − i m i µ i . (101)Since ∂ z ∂ t · ∂ t ∂ z =
1, it follows from (100) that e − ρ − i m i (cid:201) ∂ t ∂ z (cid:201) e ρ − i m i , (102)and also (cid:175)(cid:175)(cid:175)(cid:175) ∂ t ∂ z (cid:175)(cid:175)(cid:175)(cid:175) = (cid:175)(cid:175)(cid:175)(cid:175) − (cid:181) ∂ z ∂ t (cid:182) − ∂ z ∂ t (cid:175)(cid:175)(cid:175)(cid:175) (100) ∧ (101) (cid:201) ρ − i e ρ − i m i µ i . (103)The bounds (102) and (103) on the vertical derivatives of the flow coordinate t are interms of the bounds m i , µ i on the vertical derivatives of ψ i , but they imply as followsunconditional bounds on m i , µ i (hence also, by (102) and (103) once more, unconditionalbounds on the vertical derivatives of t ). Firstly, observe that (cid:176)(cid:176)(cid:176)(cid:176) ∂β i ∂ z (cid:176)(cid:176)(cid:176)(cid:176) L ∞ ( Q i , j ) (cid:201) (cid:176)(cid:176)(cid:176)(cid:176) ∂β i ∂ t (cid:176)(cid:176)(cid:176)(cid:176) L ∞ ( Q i , j ) (cid:176)(cid:176)(cid:176)(cid:176) ∂ t ∂ z (cid:176)(cid:176)(cid:176)(cid:176) L ∞ ( Q i , j ) (92) ∧ (102) (cid:201) ρ i e ρ − i m i ,and (cid:176)(cid:176)(cid:176)(cid:176) ∂ β i ∂ z (cid:176)(cid:176)(cid:176)(cid:176) L ∞ ( Q i , j ) = (cid:176)(cid:176)(cid:176)(cid:176) ∂∂ z ∂ t ∂ z ∂β i ∂ t (cid:176)(cid:176)(cid:176)(cid:176) L ∞ ( Q i , j ) = (cid:176)(cid:176)(cid:176)(cid:176) ∂ t ∂ z ∂β i ∂ t + (cid:181) ∂ t ∂ z (cid:182) ∂ β i ∂ t (cid:176)(cid:176)(cid:176)(cid:176) L ∞ ( Q i , j )(92) ∧ (102) ∧ (103) (cid:201) ρ − i e ρ − i m i µ i ρ i + e ρ − i m i α ρ i = e ρ − i m i µ i + e ρ − i m i α ρ i .Since { Q i , j } ∞ j = cover V , it follows that (cid:176)(cid:176)(cid:176)(cid:176) ∂β i ∂ z (cid:176)(cid:176)(cid:176)(cid:176) L ∞ ( V ) (cid:201) ρ i e ρ − i m i and (cid:176)(cid:176)(cid:176)(cid:176) ∂ β i ∂ z (cid:176)(cid:176)(cid:176)(cid:176) L ∞ ( V ) (cid:201) e ρ − i m i µ i + e ρ − i m i α ρ i .Since by (84) we have ∂ψ i + ∂ z = ∂ψ i ∂ z + ∂β i ∂ z and ∂ ψ i + ∂ z = ∂ ψ i ∂ z + ∂ β i ∂ z , we deduce that m i + (cid:201) m i + ρ i e ρ − i m i and µ i + (cid:201) µ i + e ρ − i m i µ i + e ρ − i m i α ρ i . (104)By induction, we suppose that (94) and (95) hold for some integer i (cid:202)
0, that is, m i (cid:201) ρ i − and µ i (cid:201) α ρ i − . (105)Since ρ (cid:202)
8, it follows that m i + ∧ (105) (cid:201) (cid:181) ρ + e ρ − (cid:182) ρ i (cid:201) (cid:181) + (cid:112) e (cid:182) ρ i (cid:201) ρ i . Thus (94) holds for all integers i (cid:202)
0. Likewise, µ i + ∧ (105) (cid:201) (cid:195) ρ + e ρ − ρ + e ρ − (cid:33) α ρ i (cid:201) (cid:195) + e + (cid:112) e (cid:33) α ρ i (cid:201) α ρ i ,so (95) also holds for all integers i (cid:202)
0. The remaining assertion (96) follows by substitutingthe above bound on m i into (102). (cid:3) Next, we will use the bounds of Lemma 3.9 to bound { D i } ∞ i = and their derivatives. Lemma 3.10.
Suppose that ρ (cid:202) . For every integer i (cid:202) we have (cid:107) D i (cid:107) L ∞ ( V ) (cid:201) α − , (106) (cid:176)(cid:176)(cid:176)(cid:176) ∂ D i ∂ z (cid:176)(cid:176)(cid:176)(cid:176) L ∞ ( V ) (cid:201) ρ i , (107) (cid:107) ∂ i D i (cid:107) L ∞ ( V ) = α − ρ i . (108) Proof.
Fix j ∈ (cid:78) . Let ( s , t ) be the flow coordinates on Q i , j . By (88) and (90), we have D i = − β i ∂ψ i + ∂ z + ∂β i ∂ s . (109)Therefore, by Lemma 3.9 and (91) we have (cid:107) D i (cid:107) L ∞ ( Q i , j ) (cid:201) α − ρ − i · ρ i + α − = α − .This proves (106) because { Q i , j } ∞ j = cover V .Next, we consider ∂ D i ∂ z . By differentiating (109) we see that ∂ D i ∂ z = − ∂ t ∂ z · ∂β i ∂ t · ∂ψ i + ∂ z − β i ∂ ψ i + ∂ z + ∂ t ∂ z · ∂ β i ∂ s ∂ t .Hence, by Lemma 3.9 and (91) we see that (cid:176)(cid:176)(cid:176)(cid:176) ∂ D i ∂ z (cid:176)(cid:176)(cid:176)(cid:176) L ∞ ( Q i , j ) (cid:201) · ρ i · ρ i + α − ρ − i · α ρ i + · ρ i = ρ i .As before, this proves (107) because { Q i , j } ∞ j = cover V .Finally, we consider ∂ i D i . Note first that for any m ∈ (cid:78) , (cid:176)(cid:176)(cid:176)(cid:176) ∂ ( ∂ m ψ m ) ∂ z (cid:176)(cid:176)(cid:176)(cid:176) ∞ (87) (cid:201) m − (cid:88) n = (cid:176)(cid:176)(cid:176)(cid:176) ∂ D n ∂ z (cid:176)(cid:176)(cid:176)(cid:176) ∞ (107) (cid:201) ρ m − ρ − (cid:201) ρ m − , (110)where we used the assumption ρ (cid:202)
8. Recalling (88) and (90), we have ∂ i D i = ∂∂ s (cid:181) − β i ∂ψ i + ∂ z + ∂β i ∂ s (cid:182) = − ∂β i ∂ s · ∂ψ i + ∂ z − β i ∂∂ s (cid:181) ∂ψ i + ∂ z (cid:182) + ∂ β i ∂ s .Using Lemma 3.9 and (91), it follows that (cid:107) ∂ i D i (cid:107) L ∞ ( Q i , j ) (cid:201) α − ρ i + α − ρ − i (cid:176)(cid:176)(cid:176)(cid:176) ∂∂ s ∂ψ i + ∂ z (cid:176)(cid:176)(cid:176)(cid:176) L ∞ ( Q i , j ) . (111)To bound the last term in (111), we first calculate the Lie bracket (cid:183) ∂∂ z , ∂∂ s (cid:184) = (cid:183) ∂∂ z , ∂∂ x − ψ i ∂∂ z (cid:184) = − ∂ψ i ∂ z · ∂∂ z . OLIATED CORONA DECOMPOSITIONS 49
This implies that ∂∂ s ∂ψ i + ∂ z = ∂∂ z (cid:181) ∂ψ i + ∂ s (cid:182) + ∂ψ i ∂ z · ∂ψ i + ∂ z = ∂∂ z (cid:181) ∂ i ψ i + ∂β i ∂ s (cid:182) + ∂ψ i ∂ z · ∂ψ i + ∂ z = ∂ ( ∂ i ψ i ) ∂ z + ∂ t ∂ z · ∂ β i ∂ s ∂ t + ∂ψ i ∂ z · ∂ψ i + ∂ z .Therefore, by Lemma 3.9, (91), (107), and (110), we conclude that (since ρ (cid:202) (cid:176)(cid:176)(cid:176)(cid:176) ∂∂ s ∂ψ i + ∂ z (cid:176)(cid:176)(cid:176)(cid:176) L ∞ ( Q i , j ) (cid:201) ρ i − + · ρ i + ρ i − · ρ i (cid:201) ρ i .Due to (111), this implies the final desired bound (108) of Lemma 3.10. (cid:3) The first assertion (106) of Lemma 3.10 gives the first assertion of Lemma 3.8. To provethe second assertion of Lemma 3.8, we first bound the variation of D m on each of thepseudoquads { Q n , j } ∞ j = when n (cid:202) m . Lemma 3.11.
Fix two integers n (cid:202) m (cid:202) . For any j ∈ (cid:78) and any w , w (cid:48) ∈ Q n , j , we have | D m ( w ) − D m ( w (cid:48) ) | (cid:46) α − ρ m − n . Proof.
Let R = R n , j and let ( s , t ), ( s (cid:48) , t (cid:48) ) ∈ [0, ρ − n ] × [0, α − ρ − n ] be such that R ( s , t ) = w and R ( s (cid:48) , t (cid:48) ) = w (cid:48) . With respect to flow coordinates on Q n , j , we have ∂ D m ∂ s = ∂ n D m = ∂ m D m + ( ψ m − ψ n ) ∂ D m ∂ z .Since (cid:107) ψ m − ψ n (cid:107) L ∞ ( V ) (cid:201) α − ρ − m + α − ρ − n (cid:201) α − ρ − m , using Lemma 3.10 we get that (cid:176)(cid:176)(cid:176)(cid:176) ∂ D m ∂ s (cid:176)(cid:176)(cid:176)(cid:176) L ∞ ( Q n , j ) (cid:201) α − ρ m + α − ρ − m · ρ m = α − ρ m .Hence, using Lemma 3.9 and Lemma 3.10 we conclude that | D m ( w ) − D m ( w (cid:48) ) | (cid:201) (cid:176)(cid:176)(cid:176)(cid:176) ∂ D m ∂ s (cid:176)(cid:176)(cid:176)(cid:176) L ∞ ( Q n , j ) | s − s (cid:48) | + (cid:176)(cid:176)(cid:176)(cid:176) ∂ D m ∂ z (cid:176)(cid:176)(cid:176)(cid:176) L ∞ ( Q n , j ) (cid:176)(cid:176)(cid:176)(cid:176) ∂ z ∂ t (cid:176)(cid:176)(cid:176)(cid:176) L ∞ ( Q n , j ) | t − t (cid:48) |(cid:201) α − ρ m − n + ρ m · · α − ρ − n (cid:46) α − ρ m − n . (cid:3) Prior to proving Proposition 3.3, we record a quick consequence of Stokes’ theorem.
Lemma 3.12.
Let M ⊆ V be a smooth –dimensional submanifold with corners and letf : V → (cid:82) be a smooth function. Then ˆ M ∂ f f d w = ˆ ∂ M (cid:181) f f (cid:182) · d r . In particular, if g : V → (cid:82) is another smooth function such that f = g on ∂ M , then ˆ M ∂ f f d w = ˆ M ∂ g g d w . Proof.
Since ∇ × (cid:181) f f (cid:182) = ∂ f ∂ x − f ∂ f ∂ z = ∂ f f ,the lemma follows from Stokes’ Theorem. (cid:3) Proof of Lemma 3.8.
The first assertion of Lemma 3.8 was proved in Lemma 3.10, so herewe treat its second assertion, namely that { D n } ∞ n = are almost-orthogonal.Fix m , n ∈ (cid:78) ∪ {0} with n (cid:202) m and j ∈ (cid:78) . Since ψ n + − ψ n = β n = ∂ Q n , j , Lemma 3.12implies that ´ Q n , j D n ( w ) d w =
0. So, fixing an arbitrary basepoint w ∈ Q , we have (cid:175)(cid:175)(cid:175)(cid:175) ˆ Q n , j D m ( w ) D n ( w ) d w (cid:175)(cid:175)(cid:175)(cid:175) = (cid:175)(cid:175)(cid:175)(cid:175) ˆ Q n , j (cid:161) D m ( w ) − D m ( w ) (cid:162) D n ( w ) d w (cid:175)(cid:175)(cid:175)(cid:175) (cid:46) α − ρ m − n H ( Q n , j ),where in the final step we used (106) and Lemma 3.11. Hence, |〈 D m , D n 〉 U | is at most (cid:88) j ∈ (cid:78) Q n , j ⊆ U (cid:175)(cid:175)(cid:175)(cid:175) ˆ Q n , j D m ( w ) D n ( w ) d w (cid:175)(cid:175)(cid:175)(cid:175) (cid:201) (cid:88) j ∈ (cid:78) Q n , j ⊆ U H ( Q n , j ) α − ρ m − n (cid:179) α − ρ m − n . (cid:3) The vertical perimeter of Γ ψ i . Here we will complete the proof of Proposition 3.3.Define φ : V → (cid:82) to be the A –periodic extension of β | U , i.e., φ ( x , 0, z ) def = β ({ x }, 0, { z })for ( x , 0, z ) ∈ V , where { a } = a − (cid:98) a (cid:99) is the fractional part of a ∈ (cid:82) . Because the function v PU , φ : (cid:82) → [0, ∞ )is continuous and not identically zero, there exist η , R , r ∈ (cid:82) with r < R such that ∀ a ∈ I def = [ r , R ], v PU , φ ( a ) (cid:202) η >
0. (112)We will show that if ρ ∈ (cid:78) is large enough (depending only on the initial choice of bumpfunction β ), then the conclusion of Proposition 3.3 holds for the above interval I . To thisend, we will first establish the following point-wise bound on the vertical perimeter ofeach of the perturbations { β i } ∞ i = in terms of the vertical perimeter of φ . Lemma 3.13.
Suppose that ρ > . For every i ∈ (cid:78) ∪ {0} and a ∈ (cid:82) we have v P U , β i ( a ) (cid:202) α v PU , φ (cid:161) a − log ( αρ i ) (cid:162) − ρ i − a . In particular, if ρ (cid:202) r η and a ∈ I + log ( αρ i ) , then v PU , β i ( a ) (cid:202) η α − ρ i − r + log ( αρ i ) (cid:202) η α . Proof of Proposition 3.3 assuming Lemma 3.13.
Fix an integer ρ (cid:202) max{12/(2 r η ), 8} thatwill be specified later and let ψ = ψ α . The first three assertions of Proposition 3.3 wereestablished in the construction of ψ and in the discussion after Lemma 3.8. We willestablish the last three by showing that ∀ a ∈ (cid:82) , v PU , ψ ( a ) (cid:46) min (cid:189) α , 2 a α (cid:190) , (113) OLIATED CORONA DECOMPOSITIONS 51 and ∀ a ∈ α − (cid:91) n = (cid:161) I + log ( αρ n ) (cid:162) , v PU , ψ ( a ) (cid:38) α . (114)For every i ∈ (cid:78) ∪ {0}, by the definition of β i and by (92) and (96) we have (cid:176)(cid:176) β i (cid:176)(cid:176) L ∞ ( V ) (cid:201) α − ρ − i and (cid:176)(cid:176)(cid:176)(cid:176) ∂β i ∂ z (cid:176)(cid:176)(cid:176)(cid:176) L ∞ ( V ) (cid:201) ρ i .Due to Lemma 2.13, for every a ∈ (cid:82) we have v PU , β i ( a ) (cid:201) min (cid:110) a + α − ρ − i , 2 − a + ρ i (cid:111) = α − −| a − log ( αρ i ) | . (115)Consequently, v PU , ψ α ( a ) = v PU , (cid:80) α i = β i ( a ) (cid:201) α (cid:88) i = v P U , β i ( a ) (115) (cid:201) ∞ (cid:88) i = α − −| a − log ( αρ i ) | (cid:46) α − .This proves (113), because by Lemma 2.13 we also have v PU , ψ α ( a ) (cid:46) a (cid:107) ψ α (cid:107) L ∞ ( V ) (85) (cid:201) a α − .It remains to prove (114), as we saw in (73) that this implies the remaining assertions ofProposition 3.3. Fix n ∈ {0, . . . , α −
1} and a ∈ I + log ( αρ n ), so that v PU , β n ( a ) > η /(4 α ) byLemma 3.13. Let s = max{ | r | , | R | }, so that | a − log ( αρ n ) | (cid:201) s . It follows from (115) that v PU , β i ( a ) (cid:201) α − −| log ( αρ n ) − log ( αρ i ) |+| a − log ( αρ n ) | (cid:201) α − ρ −| n − i | s (116)for any i ∈ (cid:78) ∪ {0}. Hence, by combining Lemma 3.13 and (116) we conclude that v PU , ψ α ( a ) = v PU , (cid:80) α i = β i ( a ) (cid:202) v PU , β n ( a ) − n − (cid:88) i = v PU , β i ( a ) − α (cid:88) i = n + v PU , β i ( a ) (cid:202) η α − ∞ (cid:88) k = α − ρ − k s (cid:202) η α − αρ s .Choosing ρ def = (cid:108) max (cid:110) r η , · s η (cid:111)(cid:109) , this completes the proof of Proposition 3.3. (cid:3) Proof of Lemma 3.13.
We will start by introducing some (convenient, though ad hoc) no-tation and making some preliminary observations. For i ∈ (cid:78) ∪ {0} define a (discontinuousin the first variable) map S i : (cid:82) → (cid:82) as follows. If s ∈ (cid:82) , then let m ∈ (cid:90) be the uniqueinteger such that s ∈ [ m ρ − i , ( m + ρ − i ), and set for every t ∈ (cid:82) , S i ( s , t ) def = Φ ( ψ i ) s − m ρ − i ( m ρ − i , 0, t ),where we recall the notation Φ ( · ) · ( · ) for characteristic curves that we set at the start ofSection 3.2. Note that by design x ( S ( s , t )) = s . Observe also that the lines (cid:82) × {0} × {0}and (cid:82) × {0} × {1} are characteristic curves for Γ ψ i , since ψ i vanishes on those lines. Hence S i ( s , 0) = ( s , 0) and S i ( s , 1) = ( s , 1) for all s ∈ [0, 1]. As x ( S ( s , t )) = s for all t ∈ (cid:82) , by thecontinuity of S i in the second variable, this implies that S i ( s , [0, 1]) = { s } × {0} × [0, 1]. So, S i ([0, 1] ) = U . (117) The mapping S i is related as follows to the mappings R i ,1 , R i ,2 , . . . that are given in (80).Suppose as above that s ∈ [ m ρ − i , ( m + ρ − i ) for some m ∈ (cid:90) , and fix t ∈ (cid:82) and n ∈ (cid:90) .Recalling that v i ,1 , v i ,2 , . . . is an enumeration of the points in the grid G i that is givenin (79), let j ∈ (cid:78) be the index for which v i , j = ( m ρ − i , 0, n α − ρ − i ). Then S i ( s , t ) = R i , j ( s − m ρ − i , t − n α − ρ − i ).Recalling the definition (81) of the pseudo-quad Q i , j , this implies that S i (cid:161) [ m ρ − i , ( m + ρ − i ) × [ n α − ρ − i , ( n + α − ρ − i ] (cid:162) = Q i , j .Also, recalling the definitions (82) and (83), it follows that if we define φ i : V → (cid:82) by ∀ ( s , t ) ∈ (cid:82) , φ i ( s , 0, t ) def = α − ρ − i φ ( ρ i s , 0, α ρ i t ), (118)then ∀ ( s , t ) ∈ (cid:82) , β i (cid:161) S i ( s , t ) (cid:162) = φ i ( s , 0, t ). (119)Fix i ∈ (cid:78) ∪ {0}, a ∈ (cid:82) and ( x , 0, z ) ∈ U . Let s = s ( x , z ), t = t ( x , z ), t (cid:48) = t (cid:48) ( x , z , a ) ∈ (cid:82) satisfy S i ( s , t ) = ( x , 0, z ) and S i ( s , t (cid:48) ) = ( x , 0, z − − a ). (120)Due to (96) we have e − ρ − − a (cid:201) t − t (cid:48) (cid:201) e ρ − − a . Hence, | t (cid:48) − ( t − − a ) | (cid:201) ( e ρ − − − a (cid:201) ρ − − a . (121)Now, | β i ( x , 0, z ) − β i ( x , 0, z − − a ) | (119) ∧ (120) = | φ i ( s , 0, t ) − φ i ( s , 0, t − − a ) − φ i ( s , 0, t (cid:48) ) + φ i ( s , 0, t − − a ) |(cid:202) | φ i ( s , 0, t ) − φ i ( s , 0, t − − a ) | − | φ i ( s , 0, t (cid:48) ) − φ i ( s , 0, t − − a ) | (118) (cid:202) | φ i ( s , 0, t ) − φ i ( s , 0, t − − a ) | − ρ i | t (cid:48) − ( t − − a ) | (121) (cid:202) | φ i ( s , 0, t ) − φ i ( s , 0, t − − a ) | − ρ i − − a .In other words, we established the following point-wise estimate for the vertical differencequotients that occur in the definition (61) of (parameterized) vertical perimeter. | β i ( x , 0, z ) − β i ( x , 0, z − − a ) | − a (cid:202) | φ i ( s , 0, t ) − φ i ( s , 0, t − − a ) | − a − ρ i − a .By integrating this inequality over U we get v P U , β i ( a ) (61) (cid:202) ˆ ˆ | φ i ( s ( x , z ), 0, t ( x , z )) − φ i ( s ( x , z ), 0, t ( x , z ) − − a ) | − a d x d z − ρ i − a (89) = ˆ S − i ( U ) | φ i ( s , 0, t ) − φ i ( s , 0, t − − a ) | − a (cid:175)(cid:175)(cid:175)(cid:175) ∂ z ∂ t ( s , t ) (cid:175)(cid:175)(cid:175)(cid:175) d s d t − ρ i − a (96) ∧ (117) (cid:202) ˆ ˆ | φ i ( s , 0, t ) − φ i ( s , 0, t − − a ) | − a d s d t − ρ i − a . (122) OLIATED CORONA DECOMPOSITIONS 53
It therefore remains to note the following identity. ˆ ˆ | φ i ( s , 0, t ) − φ i ( s , 0, t − − a ) | − a d s d t = ˆ ρ i ˆ α ρ i α − ρ − i | φ ( σ , 0, τ ) − φ ( σ , 0, τ − α ρ i − a ) | − a d σ d τ (123) = α ˆ U | φ ( v ) − φ ( v Z − α ρ i − a ) | αρ i − a d v (124) = α v PU , φ (cid:161) a − log ( αρ i ) (cid:162) , (125)where (123) uses the definition (118) and the change of variables ( s , t ) = ( ρ − i σ , α − ρ − i τ ),(124) holds by the periodicity of φ , and (125) is a restatement of the definition (61). (cid:3)
4. P
SEUDOQUADS AND FOLIATED PATCHWORKS
Let Γ be the intrinsic Lipschitz graph of f : V → (cid:82) . A pseudoquad Q is a region of V bounded by two vertical lines and two characteristic curves of Γ , i.e., a region of the form Q = (cid:169) ( x , 0, z ) ∈ V : x ∈ I and g ( x ) (cid:201) z (cid:201) g ( x ) (cid:170) ,where I = [ a , b ] ⊆ (cid:82) is a closed, bounded interval and g , g : (cid:82) → (cid:82) are functions whosegraphs are characteristic. We say that I is the base of Q and we call g and g the lower and upper bounds of Q , respectively. The width of the pseudoquad Q is just the length (cid:96) ( I ) = b − a of its base I = [ a , b ]. But, the height of Q is not always well-behaved, sincecharacteristic curves can join and split. We therefore introduce rectilinear pseudoquads ,which approximate projections of rectangles in vertical planes. If Γ is a vertical plane,its characteristic curves are a family of parallel parabolas; conversely, any pseudoquadbounded by two parallel parabolas is the projection of a rectangle in (cid:72) (a loop composedof two parallel horizontal lines and two vertical lines) to V . Thus, if R = (cid:169) ( x , 0, z ) ∈ V : x ∈ I and h ( x ) (cid:201) z (cid:201) h ( x ) (cid:170) where h , h : (cid:82) → (cid:82) are quadratic functions that differ by a constant, then we call R a parabolic rectangle with width δ x ( R ) def = (cid:96) ( I )and height δ z ( R ) def = h − h .For r > I , let r I be the scaling of I around its center by a factor of r , i.e., r I def = (cid:183) a + b − r (cid:96) ( I )2 , a + b + r (cid:96) ( I )2 (cid:184) .For ρ >
0, let ρ R def = (cid:169) ( x , 0, z ) ∈ V : x ∈ ρ I and z ∈ ρ [ h ( x ), h ( x )] (cid:170) = (cid:189) ( x , 0, z ) ∈ V : x ∈ ρ I and (cid:175)(cid:175)(cid:175)(cid:175) z − h ( x ) + h ( x )2 (cid:175)(cid:175)(cid:175)(cid:175) (cid:201) ρ δ z ( R )2 (cid:190) . (126) For 0 < µ (cid:201) , a µ –rectilinear pseudoquad is a pair ( Q , R ), where Q is a pseudoquad and R is a parabolic rectangle with the same base I as Q such that, if g and g (respectively h and h ) are the lower and upper bounds of Q (respectively R ), thenmax (cid:169) (cid:107) g − h (cid:107) L ∞ (4 I ) , (cid:107) g − h (cid:107) L ∞ (4 I ) (cid:170) (cid:201) µδ z ( R ). (127)We will frequently refer to a µ –rectilinear pseudoquad ( Q , R ) as simply Q , but we defineits width and height to be the width and height of the associated parabolic rectangle, i.e., δ x ( Q ) = δ x ( R ) and δ z ( Q ) = δ z ( R ). Likewise, for ρ (cid:202)
1, we define ρ Q = ρ R . Note that Q need not be contained in 1 Q = R , but the following lemma holds. Lemma 4.1.
Let Q be a µ –rectilinear pseudoquad. Then Q ⊆ Q. In fact, for every t ∈ (cid:82) ,Q Z t δ z ( Q ) ⊆ (cid:112) | t | + · Q . Proof.
Let R , g , g , h , h be as above. Let m g = g + g and m h = h + h . Fix ( x , 0, z ) ∈ Q ,so that g ( x ) (cid:201) z (cid:201) g ( x ). For i ∈ {1, 2}, we have | m h ( x ) − g i ( x ) | (cid:201) | m h ( x ) − h i ( x ) | + | h i ( x ) − g i ( x ) | (cid:201) δ z ( Q )2 + µδ z ( Q ) (cid:201) δ z ( Q ),so | m h ( x ) − ( z + t δ z ( Q )) | (cid:201) (1 + | t | ) δ z ( Q ).Therefore, ( x , 0, z + δ z ( Q )) ∈ (cid:112) | t | + · Q . (cid:3) Continuing with the above notation, define the aspect ratio of Q to be α ( Q ) def = δ x ( Q ) (cid:112) δ z ( Q ) . (128)We use a square root here because the distance in the Heisenberg metric between the topand bottom of Q is proportional to (cid:112) δ z ( Q ); thus this aspect ratio is invariant under theHeisenberg scaling. Let | Q | be the Lebesgue measure of Q as a subset of V ∼= (cid:82) .The following lemma is a direct consequence of Lemma 2.9. Lemma 4.2.
Let a , b ∈ (cid:82) (cid:224) {0} and let g = q ◦ ρ h ◦ s a , b : (cid:72) → (cid:72) be a composition of a shearmap q, a left-translation by h ∈ (cid:72) , and a stretch map s a , b . Let ˆ g : V → V be the mapinduced on V , i.e., ˆ g ( x ) = Π ( g ( x )) for all x ∈ V . Suppose that ( Q , R ) is a µ –rectilinear pseu-doquad for an intrinsic graph Γ . Then ( ˆ Q , ˆ R ) = ( ˆ g ( Q ), ˆ g ( R )) is a µ –rectilinear pseudoquadfor the intrinsic graph ˆ g ( Γ ) , with the following parameters. δ x ( ˆ Q ) = | a | δ x ( Q ), δ z ( ˆ Q ) = | ab | δ z ( Q ), | ˆ Q | = | a b | · | Q | , α ( ˆ Q ) = (cid:115) | a || b | · α ( Q ). Remark . For any µ –rectilinear pseudoquad ( Q , R ), there is a transformation of (cid:72) thatsends R to a square in V and Q to an approximation of the square. That is, since ( Q , R )is rectilinear, there are a vertical plane V and a rectangle S ⊆ V such that R = Π ( S ), andthere is a composition h : (cid:72) → (cid:72) of a stretch map, a translation, and a shear map suchthat h ( V ) = V and h ( S ) = [ −
1, 1] × {0} × [ −
1, 1] ⊆ V . Letting ˆ h ( x ) = Π ( h ( x )) for all x ∈ V ,ˆ h ( R ) = Π (cid:179) h (cid:161) Π ( S ) (cid:162)(cid:180) = Π (cid:161) h ( S ) (cid:162) = [ −
1, 1] × {0} × [ −
1, 1].
OLIATED CORONA DECOMPOSITIONS 55
By Lemma 4.2, ( ˆ h ( Q ), ˆ h ( R )) is rectilinear, so if ˆ g and ˆ g are the lower and upper boundsof ˆ h ( Q ), then | ˆ g ( t ) + | < µ and | ˆ g ( t ) − | < µ for all t ∈ [ −
4, 4].We will prove Theorem 1.17 by constructing a collection of nested partitions of V intopseudoquads. We will describe these partitions by associating a rectilinear pseudoquad toeach vertex of a rooted tree. Let ( T , v ) be a rooted tree with vertex set V ( T ). For v ∈ V ( T ),we let C ( v ) = C ( v ) denote the set of children of v and inductively for n (cid:202) C n ( v ) = (cid:91) w ∈ C n − ( v ) C ( w )be the set of n ’th generation descendants of v . Let D ( v ) = (cid:83) ∞ n = C n ( v ). For v ∈ V ( T ) (cid:224) { v },there is a unique parent vertex w such that v ∈ C ( w ), and we denote this vertex by P ( v ).If w ∈ D ( v ), we say that w is a descendant of v or that v is an ancestor of w and write w (cid:201) v . This is a partial order with maximal element v . Definition 4.4 (rectilinear foliated patchwork) . If Q is a µ –rectilinear pseudoquad, a µ -rectilinear foliated patchwork for Q is a complete rooted binary tree ( ∆ , v ) such that everyvertex v ∈ V ( ∆ ) is associated to a µ –rectilinear pseudoquad ( Q v , R v ) with Q v = Q. Eachvertex v ∈ V ( ∆ ) is either vertically cut or horizontally cut in the following sense.Let w and w (cid:48) be the children of v, let I = [ a , b ] be the base of Q v , and let g and g (respectively h and h ) be the lower and upper bounds of Q v (respectively R v ). (1) If v is vertically cut, then Q w and Q w (cid:48) are the left and right halves of Q v , separatedby the vertical line x = a + b . That is,Q w = (cid:189) ( x , 0, z ) ∈ V : a (cid:201) x (cid:201) a + b g ( x ) (cid:201) z (cid:201) g ( x ) (cid:190) , and Q w (cid:48) = (cid:189) ( x , 0, z ) ∈ V : a + b (cid:201) x (cid:201) b and g ( x ) (cid:201) z (cid:201) g ( x ) (cid:190) . Similarly,R w = (cid:181)(cid:183) a , a + b (cid:184) × {0} × (cid:82) (cid:182) ∩ R v and R w (cid:48) = (cid:181)(cid:183) a + b b (cid:184) × {0} × (cid:82) (cid:182) ∩ R v . We therefore have δ x ( Q w ) = δ x ( Q w (cid:48) ) = δ x ( Q v )2 and δ z ( Q w ) = δ z ( Q w (cid:48) ) = δ z ( Q v ) . (2) If v is horizontally cut, then Q w and Q w (cid:48) are the top and bottom halves of Q v ,separated by a characteristic curve. That is, there is a function c : (cid:82) → (cid:82) whosegraph is characteristic, a quadratic function k : (cid:82) → (cid:82) , and d ∈ (0, ∞ ) such thatQ w = (cid:169) ( x , 0, z ) ∈ V : a (cid:201) x (cid:201) b and g ( x ) (cid:201) z (cid:201) c ( x ) (cid:170) , Q w (cid:48) = (cid:169) ( x , 0, z ) ∈ V : a (cid:201) x (cid:201) b and c ( x ) (cid:201) z (cid:201) g ( x ) (cid:170) , R w = (cid:169) ( x , 0, z ) ∈ V : a (cid:201) x (cid:201) b and k ( x ) − d (cid:201) z (cid:201) k ( x ) (cid:170) , R w (cid:48) = (cid:169) ( x , 0, z ) ∈ V : a (cid:201) x (cid:201) b and k ( x ) (cid:201) z (cid:201) k ( x ) + d (cid:170) . Then δ x ( Q w ) = δ x ( Q w (cid:48) ) = δ x ( Q v ) and δ z ( Q w ) = δ z ( Q w (cid:48) ) = d . Furthermore, Q w and Q w (cid:48) are assumed to be µ –rectilinear; thus max (cid:169) (cid:107) ( k − d ) − g (cid:107) L ∞ (4 I ) , (cid:107) k − c (cid:107) L ∞ (4 I ) , (cid:107) ( k + d ) − g (cid:107) L ∞ (4 I ) (cid:170) (cid:201) µ d . (129) In either case, Q v = Q w ∪ Q w (cid:48) and Q w , Q w (cid:48) have disjoint interiors. Let V V ( ∆ ) ⊆ V ( ∆ ) bethe set of vertically cut vertices and let V H ( ∆ ) ⊆ V ( ∆ ) be the set of horizontally cut vertices. It follows from the above definition that v (cid:201) w if and only if Q v ⊆ Q w . Furthermore, ifthe interior of Q v intersects Q w , then either v (cid:201) w or w (cid:201) v . Lemma 4.5.
For every ε > there exists < µ = µ ( ε ) (cid:201) such that if Q is a µ –rectilinearpseudoquad, then (1 − ε ) δ x ( Q ) δ z ( Q ) (cid:201) | Q | (cid:201) (1 + ε ) δ x ( Q ) δ z ( Q ). (130) If Q is horizontally or vertically cut as in Definition 4.4 and Q (cid:48) is a child of Q, then (cid:181) − ε (cid:182) | Q | (cid:201) | Q (cid:48) | (cid:201) (cid:181) + ε (cid:182) | Q | . (131) If Q is vertically cut, then δ x ( Q (cid:48) ) = δ x ( Q )2 , δ z ( Q (cid:48) ) = δ z ( Q ) , and α ( Q (cid:48) ) = α ( Q )2 . If Q is horizon-tally cut, then δ x ( Q (cid:48) ) = δ x ( Q ) , and (cid:181) − µ (cid:182) δ z ( Q ) (cid:201) δ z ( Q (cid:48) ) (cid:201) (cid:181) + µ (cid:182) δ z ( Q ). (132) Finally, (cid:179) (cid:112) − ε (cid:180) α ( Q ) (cid:201) α ( Q (cid:48) ) (cid:201) (cid:179) (cid:112) + ε (cid:180) α ( Q ). (133) When ε = we can take here µ = .Proof. Suppose that µ (cid:201) ε . Let ( Q , R ) be a µ –rectilinear pseudoquad. Suppose that g and g (respectively h and h ) be the lower and upper bounds of Q (respectively R ) and let I be the base of Q . Then | R | = δ x ( Q ) δ z ( Q ) and (cid:175)(cid:175) | Q | − δ x ( Q ) δ z ( Q ) (cid:175)(cid:175) = (cid:175)(cid:175) | Q | − | R | (cid:175)(cid:175) (cid:201) ˆ I | g − h | d x + ˆ I | g − h | d x (cid:201) µδ x ( Q ) δ z ( Q ), (134)so (130) is satisfied.Let Q (cid:48) be a child of Q . If Q is vertically cut, then the formulas for δ x ( Q (cid:48) ), δ z ( Q (cid:48) ), and α ( Q (cid:48) ) follow from Definition 4.4. As Q (cid:48) is µ –rectilinear, (134) implies that (cid:175)(cid:175)(cid:175)(cid:175) | Q (cid:48) | − | Q | (cid:175)(cid:175)(cid:175)(cid:175) (cid:201) (cid:175)(cid:175) | Q (cid:48) | − δ x ( Q (cid:48) ) δ z ( Q (cid:48) ) (cid:175)(cid:175) + (cid:175)(cid:175) δ x ( Q ) δ z ( Q ) − | Q | (cid:175)(cid:175) (cid:201) µδ x ( Q ) δ z ( Q ) (cid:201) µ | Q | ,so Q satisfies (131) if Q is vertically cut.If Q is horizontally cut, then δ x ( Q (cid:48) ) = δ x ( Q ) by Definition 4.4. Let c , k , d = δ z ( Q (cid:48) ) be asin Definition 4.4 and let t ∈ I . As (cid:107) g i − h i (cid:107) L ∞ ( I ) (cid:201) µδ z ( Q ) for i ∈ {1, 2} and δ z ( Q ) = h − h ,(1 − µ ) δ z ( Q ) (cid:201) g ( t ) − g ( t ) (cid:201) (1 + µ ) δ z ( Q ).By (129), (1 − µ ) · d (cid:201) g ( t ) − g ( t ) (cid:201) (1 + µ ) · d .Then d (cid:201) + µ − µ δ z ( Q ) < δ z ( Q ), so | d − δ z ( Q ) | (cid:201) | d − ( g ( t ) − g ( t )) | + | ( g ( t ) − g ( t )) − δ z ( Q ) | (cid:201) µδ z ( Q ),proving (132). This directly implies Equation (133), and the horizontally cut case of (131)follows from the above calculation and (134). (cid:3) The following two lemmas will be helpful later.
OLIATED CORONA DECOMPOSITIONS 57
Lemma 4.6.
For any quadratic function q : (cid:82) → (cid:82) and any t ∈ (cid:82) , | q ( t ) | (cid:201) (1 + t + t ) (cid:107) q (cid:107) L ∞ ([ − . Proof.
One only needs to note that, since q is quadratic, for any t ∈ (cid:82) we have q ( t ) = q (0) + t · q (1) − q ( − + t · q ( − − q (0) + q (1)2 . (cid:3) Lemma 4.7.
For every r (cid:202) there is µ = µ ( r ) > such that if ∆ is a µ –rectilinear foliatedpatchwork and v , w ∈ V ( ∆ ) satisfy w (cid:201) v, then r Q w ⊆ r Q v .Proof. It suffices to consider the case that w ∈ C ( v ). If v is vertically cut, this holdsvacuously, so suppose that v is horizontally cut. Let g and g (respectively h and h )be the lower and upper bounds of Q v (respectively R v ) and let I be their base. Denote m v = ( h + h )/2. Then r R v is bounded by m v ± r δ z ( Q v )/2.Let c , k , d = δ z ( Q w ) be as in Definition 4.4. By Lemma 4.5, we have d (cid:201) δ z ( Q v ). Then (cid:107) ( k − d ) − h (cid:107) L ∞ (4 I ) (cid:201) (cid:107) k − d − g (cid:107) L ∞ (4 I ) + (cid:107) g − h (cid:107) L ∞ (4 I ) (cid:201) µ d + µδ z ( Q v ) (cid:201) µδ z ( Q v ).Likewise, (cid:107) ( k + d ) − h (cid:107) L ∞ (4 I ) (cid:201) µδ z ( Q v ). By Lemma 4.6, since k , h , h are quadraticfunctions, if µ is at most a sufficiciently small universal constant multiple of r − , thenmax{ (cid:107) ( k − d ) − h (cid:107) L ∞ ( r I ) , (cid:107) ( k + d ) − h (cid:107) L ∞ ( r I ) } (cid:201) δ z ( Q v )16 .By the triangle inequality, (cid:107) k − m v (cid:107) L ∞ ( r I ) (cid:201) δ z ( Q v )16 .Suppose that Q w is the lower half of Q v , so that Q w is bounded by g and c and R w isbounded by k − d and k . Let m w = k − d so that r Q w is bounded by m w ± r d . For x ∈ r I , | m v ( x ) − m w ( x ) | (cid:201) δ z ( Q v )16 + d (cid:201) δ z ( Q v ) (cid:201) r δ z ( Q v )2 − r d (cid:183) m w ( x ) − r d m w ( x ) + r d (cid:184) ⊆ (cid:183) m v ( x ) − r δ z ( Q v )2 , m v ( x ) + r δ z ( Q v )2 (cid:184) .That is, r Q w ⊆ r Q v . The case that Q w is the upper half of Q v is treated analogously. (cid:3) Let ∆ be a µ –rectilinear foliated patchwork for a µ –rectilinear pseudoquad Q . For everysubset of vertices S ⊆ V ( ∆ ), define the weight of S to be W ( S ) def = (cid:88) w ∈ S | Q w | α ( Q w ) = (cid:88) w ∈ S δ z ( Q w ) δ x ( Q w ) | Q w | . (135)We will use this to define a weighted Carleson condition which is a variant of the Carlesonpacking condition that is used in the theory of uniform rectifiability [DS93]. Definition 4.8 (weighted Carleson packing condition) . Suppose that ∆ is a µ –rectilinearfoliated patchwork for a µ –rectilinear pseudoquad Q. We say that ∆ satisfies a weightedCarleson packing condition or that ∆ is weighted Carleson with constant C ∈ (0, ∞ ) if everyv ∈ V ( ∆ ) satisfies W (cid:161) D ( v ) ∩ V V ( ∆ ) (cid:162) (cid:201) C | Q v | , (136) where we recall that D ( v ) are the descendants of v and V V ( ∆ ) are the vertically cut vertices. Remark . Vertical cuts increase W and horizontal cuts decrease it. More precisely,suppose that v , w ∈ V ( ∆ ) and w is a child of v . If v is vertically cut, then by Lemma 4.5(with ε = ), W ({ w }) = α ( Q w ) − | Q w | = α ( Q v ) − | Q w | (cid:202) α ( Q v ) − · (cid:181) − ε (cid:182) | Q v | (cid:202) W ({ v }). (137)When ε → + , W ({ w }) approaches 8 W ({ v }). If v is horizontally cut, then by Lemma 4.5(with ε = ), W ({ w }) = α ( Q w ) − | Q w | (cid:201) ( (cid:112) − ε ) − (cid:181) + ε (cid:182) W ({ v }) (cid:201) W ({ v }), (138)and W ({ w }) approaches W ({ v }) when ε → + .The next lemma implies that even though only V V ( ∆ ) appears in (136), this conditionformally implies bounds on V H ( ∆ ) as well. Lemma 4.10.
Let ∆ be a –rectilinear foliated patchwork for Q with W ( V V ( ∆ )) < ∞ , andlet v be the root of ∆ . ThenW (cid:161) V V ( ∆ ) (cid:162) (cid:46) W (cid:161) V H ( ∆ ) (cid:162) (cid:46) W (cid:161) V V ( ∆ ) (cid:162) + α ( Q ) − | Q | . Proof.
Let T H (respectively T V ) be the set of connected components of the subgraph of ∆ spanned by V H ( ∆ ) (respectively V V ( ∆ )). Let T ∈ T H and let M ( T ) be the maximal vertexof T . Each vertex of T is horizontally cut, so by (138), we have W ( C ( v )) (cid:201) W ({ v }) for all v ∈ V ( T ) and w ∈ C ( v ). Therefore, W ( V ( T )) (cid:179) W ({ M ( T )}), because W (cid:161) V ( T ) (cid:162) = ∞ (cid:88) n = W (cid:161) C n ( M ( T )) ∩ V ( T ) (cid:162) (cid:201) ∞ (cid:88) n = (cid:181) (cid:182) n W ({ M ( T )}) (cid:46) W (cid:161) { M ( T )} (cid:162) .Hence, if we denote S M = { M ( T ) : T ∈ T H }, then W ( V H ( ∆ )) (cid:179) W ( S M ).Now, take T ∈ T V . By (137), we have W ({ w }) (cid:202) W ({ v }) for all v ∈ V ( T ) and w ∈ C ( v ).As W ( V ( T )) < ∞ , it follows that T must be finite. Let m ( T ) = { w ∈ V ( T ) : C ( w ) (cid:54)⊆ V ( T )} bethe lower boundary of T and let S m = (cid:83) T ∈ T V m ( T ).For all v ∈ V ( T ), let A ( v ) = { w ∈ V ( T ) : w (cid:202) v } be the set of ancestors of v in T . By (137), W (cid:161) A ( v ) (cid:162) (cid:201) | A ( v ) |− (cid:88) n = − n W ({ v }) (cid:201) W ({ v }).Every vertex of T is an ancestor of a leaf, so it follows that W (cid:161) V ( T ) (cid:162) (cid:201) W (cid:181) (cid:91) v ∈ m ( T ) A ( v ) (cid:182) (cid:201) (cid:88) v ∈ m ( T ) W (cid:161) A ( v ) (cid:162) (cid:201) W (cid:161) m ( T ) (cid:162) (cid:201) W (cid:161) V ( T ) (cid:162) .Therefore, W ( V V ( ∆ )) (cid:179) W ( S m ).If v ∈ S M and v (cid:54)= v , then P ( v ) is horizontally cut and has a vertically cut child, so P ( v ) ∈ S m . In fact, P ( S M (cid:224) { v }) = S m . Since W ({ v }) (cid:179) W ({ P ( v )}) for all v and since P isa two-to-one map, it follows that W ( S M (cid:224) { v }) (cid:179) W ( S m ). Therefore, W (cid:161) V H ( ∆ ) (cid:162) (cid:179) W ( S M ) (cid:46) W ( S m ) + W ({ v }) (cid:179) W ( V V ( ∆ )) + α ( Q ) − | Q | ,and W (cid:161) V V ( ∆ ) (cid:162) (cid:179) W ( S m ) (cid:179) W ( S M (cid:224) { v }) (cid:201) W ( S M ). (cid:3) OLIATED CORONA DECOMPOSITIONS 59
Suppose that ∆ = ( Q v ) v ∈ V ( ∆ ) is a µ –rectilinear foliated patchwork for Γ = Γ f . For σ > σ –approximating planes for ∆ is a collection of vertical half-spaces ( P v ) v ∈ V H ( ∆ ) such that for every v ∈ V V H ( ∆ ), if f v : V → (cid:82) is the linear function such that Γ f v = P v , then (cid:107) f v − f (cid:107) L (10 Q v ) | Q v | (cid:201) σ δ z ( Q v ) δ x ( Q v ) . (139)The following lemma verifies that the choice of right-hand side in (139) produces acondition that is invariant under stretch automorphisms and shear automorphisms. Lemma 4.11.
Let ∆ = ( Q v ) v ∈ V ( ∆ ) be a µ –rectilinear foliated patchwork for an intrinsicLipschitz graph Γ = Γ f with a set ( P v ) v ∈ V H ( ∆ ) of σ –approximating planes and let r : (cid:72) → (cid:72) be a stretch automorphism or a shear map. Let ˆ r = Π ◦ r : V → V be the map inducedon V . Then ∆ (cid:48) = (( ˆ r ( Q v ), ˆ r ( R v ))) v ∈ V ( ∆ ) is a µ –rectilinear foliated patchwork for r ( Γ ) and ( r ( P v )) v ∈ V H ( ∆ ) is a set of σ –approximating planes for ∆ (cid:48) .Proof. By Lemma 2.10 and Lemma 4.2, r ( Γ ) is an intrinsic Lipschitz graph and the el-ements of ∆ (cid:48) are µ –rectlinear pseudoquads for r ( Γ ). It is straightforward to check thatDefinition 4.4 holds for ∆ (cid:48) . Let v ∈ V H ( ∆ ) and let f v : V → (cid:82) be the affine function such that P v = Γ f v . By Lemma 2.9, there are functions ˆ f and ˆ f v such that r ( Γ ) = Γ ˆ f and r ( P v ) = Γ ˆ f v .If r is a shear map and w ∈ Q v , then ˆ r ( w ) ∈ r ( Q v ), and (cid:175)(cid:175) ˆ f (cid:161) ˆ r ( w ) (cid:162) − ˆ f v (cid:161) ˆ r ( w ) (cid:162)(cid:175)(cid:175) = | f ( w ) − f v ( w ) | .In this case, δ x ( Q v ) = δ x ( r ( Q v )) and δ z ( Q v ) = δ z ( r ( Q v )), so if P v is a σ –approximatingplane for Q v , then r ( P v ) is a σ –approximating plane for r ( Q v ).If r = s a , b for some a , b ∈ (cid:82) , then ˆ r (10 Q v ) = r ( Q v ), and for any w ∈ Q v , (cid:175)(cid:175) ˆ f (cid:161) ˆ r ( w ) (cid:162) − ˆ f v (cid:161) ˆ r ( w ) (cid:162)(cid:175)(cid:175) = b | f ( w ) − f v ( w ) | .In this case, δ x ( r ( Q v )) = a δ x ( Q v ) and δ z ( r ( Q v )) = ab δ z ( Q v ), so by (139), (cid:107) ˆ f v − ˆ f (cid:107) L (10 r ( Q v )) | r ( Q v ) | = a b (cid:107) f v − f (cid:107) L (10 Q v ) a b | Q v | (cid:201) b σ δ z ( Q v ) δ x ( Q v ) = σ δ z ( r ( Q v )) δ x ( r ( Q v )) . (cid:3)
5. F
OLIATED CORONA DECOMPOSITIONS
An intrinsic graph that admits rectilinear foliated patchworks that satisfy a weightedCarleson condition and have approximating planes is said to have a foliated coronadecomposition.
Definition 5.1.
Fix < µ (cid:201) and D : (cid:82) + × (cid:82) + → (cid:82) + . We say that an intrinsic Lipschitzgraph Γ has a ( D , µ )–foliated corona decomposition if for every < µ (cid:201) µ , every σ > and every µ –rectilinear pseudoquad Q ⊆ V , there is a µ –rectilinear foliated patchwork ∆ for Q such that ∆ is D ( µ , σ ) –weighted-Carleson and has a set of σ –approximating planes. The following theorem is a more precise formulation of Theorem 1.17.
Theorem 5.2.
For every < λ < there is a function D λ : (cid:82) + × (cid:82) + → (cid:82) + such that for any < µ (cid:201) , any intrinsic λ –Lipschitz graph has a ( D λ , µ ) –foliated corona decomposition. Definition 5.1 requires the root of the foliated patchwork to be µ –rectilinear; the nextlemma shows that intrinsic Lipschitz graphs contain many µ –rectilinear pseudoquads. Lemma 5.3.
Let µ > , let < λ < , and let Γ = Γ f be an intrinsic λ –Lipschitz graph.There is an α > with the following property. Let Q be a pseudoquad for Γ , let v be a pointin the lower boundary of Q and suppose that v Z s is in the upper boundary. Let r = δ x ( Q ) .If r (cid:112) s (cid:201) α , then there is a parabolic rectangle R such that ( Q , R ) is µ –rectilinear.Proof. Denote L def = λ (cid:112) − λ and α = min (cid:189)(cid:114) µ L , µ (1 − λ )24 (cid:190) .Let g , g : (cid:82) → (cid:82) be the lower and upper bounds of Q and let I be its base. After atranslation, we may suppose that v = and f ( v ) =
0. Then I ⊆ [ − r , r ], g (0) = g (0) = s ,and g (cid:48) (0) = − f ( ) =
0. Let R = I × [0, s ]; we claim that ( Q , R ) is a µ –rectilinear pseudoquad.It suffices to show that for all t ∈ [ − r , 4 r ] and i ∈ {1, 2}, we have | g i ( t ) − g i (0) | (cid:201) µ s . ByLemma 2.7, for all t ∈ [ − r , 4 r ], we have (cid:175)(cid:175) g i ( t ) − (cid:161) g i (0) + t g (cid:48) i (0) (cid:162)(cid:175)(cid:175) (cid:201) r L (cid:201) µ s | g ( t ) | (cid:201) µ s . Lemma 2.3 implies that | g (cid:48) (0) | = | f ( Z s ) − f ( ) | (cid:201) − λ d ( , Z s ) = (cid:112) s − λ (cid:201) µ · sr ,so if | t | (cid:201) r , then | g ( t ) − g (0) | (cid:201) µ · sr · r + µ s (cid:201) µ s . (cid:3) Corollary 5.4.
Continuing with the setting and notation of Lemma 5.3, any –rectilinearpseudoquad Q such that α ( Q ) (cid:201) α is µ –rectilinear.Proof. Let v be in the lower boundary of Q . Then there is an s (cid:202) (1 − µ ) δ z ( Q ) such that v Z s is in the upper boundary. If α ( Q ) (cid:201) α , then δ x ( Q ) (cid:201) α (cid:112) δ z ( Q ) (cid:201) α (cid:112) s ,so Lemma 5.3 implies that Q is µ –rectilinear. (cid:3) The following lemma shows that the choice of µ is not important; we can increase µ at the cost of an increase in D . Lemma 5.5.
For any λ > and < µ < µ (cid:48) (cid:201) , and any D : (cid:82) + × (cid:82) + → (cid:82) + , there existsD (cid:48) : (cid:82) + × (cid:82) + → (cid:82) + such that if Γ is an intrinsic λ –Lipschitz graph that has a ( D , µ ) –foliatedcorona decomposition, then Γ also has a ( D (cid:48) , µ (cid:48) ) –foliated corona decomposition.Proof. Fix 0 < µ < µ (cid:48) and 0 < σ <
1. Let α > µ –rectilinear pseudoquad Q . We wish to construct a rectilinear foliated patchworkfor Q with a set of σ –approximating planes. If α ( Q ) < α , then by Corollary 5.4, Q is µ –rectilinear, and since Γ admits a ( D , µ )–foliated corona decomposition, the desiredpatchwork and set of approximating planes for Q exist.We thus suppose that α ( Q ) (cid:202) α and denote i = (cid:187) log α ( Q ) α (cid:188) + OLIATED CORONA DECOMPOSITIONS 61
We will construct a rectilinear foliated patchwork for Q by first cutting Q vertically i timesinto pseudoquads P , . . . , P i of width 2 − i δ x ( Q ), height δ z ( Q ), and aspect ratio α ( P i ) = δ x ( P i ) (cid:112) δ z ( P i ) = − i α ( Q ) < α P i is µ –rectilinear and thus admits a D ( µ , σ )–weighted Carlesonrectilinear foliated patchwork and a set of σ –approximating planes. Combining thesepatchworks, we obtain an rectilinear foliated patchwork ∆ (cid:48) for Q . Let v be its root vertex.Note that for any 0 (cid:201) m (cid:201) i and any w ∈ C m ( v ), we have α ( Q w ) = − m α ( Q ).It remains to check that ∆ is weighted Carleson. Let p , . . . , p i ∈ V ( ∆ (cid:48) ) be the verticessuch that P j = Q p j . If v ∈ V ( ∆ (cid:48) ) and v (cid:201) p j for some j , then Q v satisfies the weightedCarleson condition (136) with constant at most D ( µ , σ ).Otherwise, v is an ancestor of some p j and Q v = P a ∪ · · · ∪ P b for some a (cid:201) b . For each w ∈ V ( ∆ ), let A ( w ) be the set of ancestors of w . Every ancestor of p j except possibly p j itself is vertically cut, so by (137), the weight of P k ( p j ) decays exponentially. Thus W (cid:161) A ( p j ) (cid:162) = i (cid:88) k = W (cid:161) { P k ( p j )} (cid:162) (137) (cid:201) i (cid:88) k = − k W ({ p j }) (cid:201) W ({ p j }).For each w ∈ V ( ∆ (cid:48) ), let D V ( w ) = D ( w ) ∩ V V ( ∆ ) be the set of vertically cut descendantsof w . As every element of D V ( v ) is a descendant or an ancestor of some p j with a (cid:201) j (cid:201) b , W (cid:161) D V ( v ) (cid:162) (cid:201) b (cid:88) j = a (cid:163) W (cid:161) D V ( p j ) (cid:162) + W (cid:161) A ( p j ) (cid:162)(cid:164) (cid:201) b (cid:88) j = a (cid:163) D ( µ , σ ) | P j | + W ({ p j }) (cid:164) = D ( µ , σ ) | Q v | + · α ( P a ) − | Q v | (cid:201) D ( µ , σ ) | Q v | + α − | Q v | .Therefore, ∆ is (cid:161) D ( µ , σ ) + α − (cid:162) –weighted Carleson. (cid:3)
6. V
ERTICAL PERIMETER AND FOLIATED CORONA DECOMPOSITIONS
In this section we will assume Theorem 5.2 and prove the following theorem, whichbounds the vertical perimeter of half-spaces bounded by intrinsic Lipschitz graphs.
Theorem 6.1.
For any < λ < and r > , if Γ is an intrinsic λ –Lipschitz graph, then (cid:176)(cid:176) v B r ( ) (cid:161) Γ + (cid:162)(cid:176)(cid:176) L ( (cid:82) ) (cid:46) λ r .This coincides with the bound (31) needed in Section 1.2.1. Combined with the re-duction from arbitrary sets to intrinsic Lipschitz graphs described in that section, thiscompletes the proof of Theorem 1.1.6.1. Vertical perimeter for graphs with foliated corona decompositions.
Theorem 6.1is a consequence of the following lemma.
Lemma 6.2.
Suppose that f : V → Γ is intrinsic Lipschitz and denote Γ = Γ f . Fix σ > .Let Q ⊆ V be a –rectilinear pseudoquad. Let ∆ be a foliated patchwork for Q and let ( P v ) v ∈ V H ( ∆ ) be a set of σ –approximating planes. Denoting t = − log δ z ( Q ) , we have (cid:176)(cid:176) v PQ , f (cid:176)(cid:176) L ([ t , ∞ )) (cid:46) σ | Q | W (cid:161) V ( ∆ ) (cid:162) . (140) Note that the intrinsic Lipschitz constant of f does not appear in (140). Indeed, thisbound is invariant under stretch automorphisms; if Γ , Q , and ( ∆ , ( Q v ) v ∈ V ( ∆ ) are as inLemma 6.2, a >
0, and s = s a , a − , then, by Lemma 4.11, s ( Q ) is a pseudoquad in s ( Γ ) = Γ ˆ f ,where ˆ f = a − f ◦ s − . Furthermore, ∆ (cid:48) = ( s ( Q v )) v ∈ V ( ∆ ) is a foliated patchwork for s ( Q ) and( s ( P v )) v ∈ V H ( ∆ ) is a set of σ –approximating planes. Then W ( V ( ∆ (cid:48) )) = a − W ( V ( ∆ )), so | s ( Q ) | W (cid:161) V ( ∆ (cid:48) ) (cid:162) = a | Q | a − W (cid:161) V ( ∆ ) (cid:162) = | Q | W (cid:161) V ( ∆ ) (cid:162) .Moreover, since s preserves Lebesgue measure on (cid:72) , v PQ , f ( t ) = v Ps ( Q ), ˆ f ( t ) for all t . Conse-quently, if (140) holds for f and Q , then it also holds for ˆ f and s ( Q ).Similarly, (140) is invariant under scalings; if a > s = s a , a , and ∆ (cid:48) is as above, then ∆ (cid:48) is a foliated patchwork for s ( Q ) with W ( V ( ∆ (cid:48) )) = a W ( V ( ∆ )). Therefore, if (140) holds for f and Q and ˆ f = a f ◦ s − is the function such that s ( Γ ) = Γ ˆ f , then (cid:176)(cid:176) v Ps ( Q ), ˆ f (cid:176)(cid:176) L ([ t − log a , ∞ )) = a (cid:176)(cid:176) v PQ , f (cid:176)(cid:176) L ([ t , ∞ )) (cid:46) σ ( a | Q | ) (cid:179) a W (cid:161) V ( ∆ ) (cid:162)(cid:180) = σ | s ( Q ) | W (cid:161) V ( ∆ (cid:48) ) (cid:162) ,so (140) holds for ˆ f and s ( Q ).To prove Lemma 6.2, we will need some lemmas on partitions and coherent sets. Acollection { Q , . . . , Q n } of pseudoquads is a partition of Q if Q = (cid:83) ni = Q i and if the Q i overlap only along their boundaries. A coherent subtree of T is a connected subtree suchthat for every v ∈ T , either all children of v are contained in T or none of them are. A coherent subset of V ( ∆ ) is the vertex set of a coherent subtree. Lemma 6.3.
Let ∆ be a rectilinear foliated patchwork for Q and suppose that S ⊆ V ( ∆ ) iscoherent. Let M = max S be the maximal element of S and let min
S be the set of minimalelements of S. DenoteF = F ( S ) def = (cid:169) p ∈ Q M : there are infinitely many v ∈ S such that p ∈ Q v (cid:170) . Then Q M = F (cid:91) (cid:181) (cid:91) w ∈ min S Q w (cid:182) . (141) The interiors of { Q w : w ∈ min S } are pairwise disjoint and disjoint from F . If S is finite,then min S is a partition of Q M .Proof. Let v ∈ min S and let p ∈ int Q v . If u ∈ S and p ∈ Q u , then either u < v or v (cid:201) u . Thefirst is impossible by the minimality of v , so v (cid:201) u . It follows that there are only finitelymany w ∈ S such that p ∈ Q w and no such w is minimal except v . That is, int Q v is disjointfrom F and if u ∈ min S and u (cid:54)= v , then int Q v is disjoint from int Q u .If p ∈ Q M (cid:224) F , then there is a minimal v ∈ S such that p ∈ Q v . Since S is coherent, thechildren of v are not contained in S , so v ∈ min( S ). This implies (141). (cid:3) Lemma 6.4.
Fix < µ (cid:201) and let ( ∆ , ( Q v ) v ∈ ∆ ) be a µ –rectilinear foliated patchwork forQ with W ( V V ( ∆ )) < ∞ . For any < σ (cid:201) δ z ( Q ) , denote S σ = { v ∈ V ( ∆ ) : δ z ( v ) (cid:202) σ } and letF σ = min S σ . Then { Q v } v ∈ F σ is a partition of Q into horizontally cut pseudoquads such that σ (cid:201) δ z ( Q v ) < σ for all v ∈ F σ . OLIATED CORONA DECOMPOSITIONS 63
Proof.
By Definition 4.4 and Lemma 4.5, the height of every pseudoquad of ∆ is equal tothe height of its sibling and at most the height of its parent. Therefore, S σ is coherent. If v ∈ S σ , then W ({ v }) = α ( Q v ) − | Q v | (cid:179) δ z ( Q v ) δ x ( Q v ) − (cid:202) σ δ x ( Q ) − ,which is bounded away from 0, so Lemma 4.10 implies that S σ is finite. By Lemma 6.3, F σ partitions Q .Suppose v ∈ F σ and let w ∈ C ( v ). By the minimality of v , we have v ∈ S σ and w (cid:54)∈ S σ ,so δ z ( Q v ) (cid:202) σ > δ z ( Q w ). Since δ z ( Q w ) < δ z ( Q v ), v is horizontally cut. Furthermore,by Lemma 4.5, σ > δ z ( Q w ) (cid:202) δ z ( Q v ), so v is a horizontally cut pseudoquad such that σ (cid:201) δ z ( Q v ) < σ , as desired. (cid:3) We will use these partitions to decompose the parametric vertical perimeter of f andprove Lemma 6.2. Proof of Lemma 6.2.
By the remarks after Lemma 6.2, condition (140) is invariant underscaling, so we may rescale so that δ z ( Q ) =
1. Let ∆ be a µ –rectilinear foliated patchworkfor Q and let ( P v ) v ∈ V H ( ∆ ) be a set of σ –approximating planes. For each v ∈ V H ( ∆ ), let f v : V → (cid:82) be the linear function such that Γ f v = P v . For i ∈ (cid:78) ∪ {0} let C i = F − i − ⊆ V H ( ∆ )be as in Lemma 6.4, so that { Q v } v ∈ C i is a partition of Q into horizontally-cut pseudoquadswith heights in [2 − i − , 2 − i + ). No vertex of ∆ appears in more than one of the C i ’s.We start by bounding v PQ v , f ( t ) from above for each v ∈ C i for a fixed i ∈ (cid:78) ∪ {0}. Thenwe have 2 − i (cid:201) δ z ( Q v ), so Lemma 4.1 implies that Z − − t Q v ⊆ Q v for any t ∈ [ i , i + f v is constant on vertical lines, v PQ v , f ( t ) = t ˆ Q v (cid:175)(cid:175) f ( w ) − f (cid:161) w Z − − t (cid:162)(cid:175)(cid:175) d w (cid:201) t ˆ Q v (cid:179) | f ( w ) − f v ( w ) | + (cid:175)(cid:175) f v (cid:161) w Z − − t (cid:162) − f (cid:161) w Z − − t (cid:162)(cid:175)(cid:175)(cid:180) d w = t (cid:179) (cid:107) f − f v (cid:107) L ( Q v ) + (cid:107) f − f v (cid:107) L ( Z − − t Q v ) (cid:180) (139) (cid:201) t + | Q v | σ δ z ( Q v ) δ x ( Q v ) (130) (cid:179) σδ z ( Q v ) (cid:179) σα ( Q v ) − | Q v | .Since { Q v } v ∈ C i is a partition of Q , we have v PQ , f ( t ) = (cid:80) v ∈ C i v PQ v , f ( t ) for all t ∈ (cid:82) . Thus (cid:176)(cid:176) v P Q , f (cid:176)(cid:176) L ([ i , i + (cid:201) (cid:88) v ∈ C i (cid:176)(cid:176) v P Q v , f (cid:176)(cid:176) L ([ i , i + (cid:46) (cid:88) v ∈ C i σα ( Q v ) − | Q v | . (142)Consequently, (cid:176)(cid:176) v P Q , f (cid:176)(cid:176) L ([0, ∞ )) = ∞ (cid:88) i = (cid:176)(cid:176) v P Q , f (cid:176)(cid:176) L ([ i , i + (cid:46) σ ∞ (cid:88) i = (cid:181) (cid:88) v ∈ C i α ( Q v ) − | Q v | (cid:182) (cid:201) σ ∞ (cid:88) i = (cid:181) (cid:88) v ∈ C i | Q v | (cid:182) (cid:181) (cid:88) v ∈ C i α ( Q v ) − | Q v | (cid:182) (135) = σ | Q | ∞ (cid:88) i = W ( C i ) (cid:201) σ | Q | W (cid:161) V ( ∆ ) (cid:162) ,where the third step is an application of Hölder’s inequality. (cid:3) Finally, we use Lemma 2.11 and Lemma 2.12 to prove Theorem 6.1.
Proof of Theorem 6.1.
After scaling, it suffices to prove the theorem in the case that r = Γ is the intrinsic graph of an intrinsic λ –Lipschitz function f : V → (cid:82) , then (cid:176)(cid:176) v B (cid:161) Γ + (cid:162)(cid:176)(cid:176) L ( (cid:82) ) (cid:46) λ
1. (143)By the definition (29), ∀ t ∈ (cid:82) , v B (cid:161) Γ + (cid:162) ( t ) = t (cid:175)(cid:175) B ∩ (cid:161) Γ + (cid:52) Γ + Z − t (cid:162)(cid:175)(cid:175) (cid:201) t | B | (cid:46) t .Hence, ∀ a ∈ (cid:82) , (cid:176)(cid:176) v B (cid:161) Γ + (cid:162)(cid:176)(cid:176) L (( −∞ , a ]) = (cid:181) ˆ a −∞ v B (cid:161) Γ + (cid:162) ( t ) d t (cid:182) (cid:46) (cid:181) ˆ a −∞ t d t (cid:182) (cid:46) a ,and therefore we have the following simple a priori bound. ∀ a ∈ (cid:82) , (cid:176)(cid:176) v B (cid:161) Γ + (cid:162)(cid:176)(cid:176) L ( (cid:82) ) (cid:46) a + (cid:176)(cid:176) v B (cid:161) Γ + (cid:162)(cid:176)(cid:176) L (( a , ∞ )) . (144)We will first treat the (trivial) case B ∩ Γ = ∅ , so that either B ⊆ Γ + or B ⊆ Γ − . Withoutloss of generality, suppose that B ⊆ Γ + . This implies that B ⊆ Γ + ∩ Z − t Γ + for any t (cid:202) v B ( Γ + )( t ) =
0, and therefore in this case (143) follows from the case a = B ∩ Γ (cid:54)= ∅ . Fix any point p ∈ B ∩ Γ . Then d ( p , 〈 Y 〉 ) (cid:201) p = vY f ( v ) for some v ∈ V with | f ( v ) | (cid:201)
5, so by Lemma 2.3, we have | f ( ) | (cid:201) | f ( v ) | + | f ( v ) − f ( ) | (cid:201) + − λ d ( p , 〈 Y 〉 ) (cid:46) − λ .Likewise, for any t ∈ (cid:82) , | f ( Z t ) | (cid:201) | f ( ) | + − λ d ( , Z t ) (cid:46) + (cid:112)| t | − λ .For t ∈ (cid:82) , let g t : (cid:82) → (cid:82) be a function such that g t (0) = t and the graph of g t is characteristicfor f . By the estimate above and Lemma 2.7,max x ∈ [ − | g t ( x ) − t | (cid:201) f ( Z t ) + λ (cid:112) − λ (cid:46) + (cid:112)| t | − λ . (145)The right hand side of (145) grows slower than | t | as | t | → ∞ , so there is t = t ( λ ) > t (cid:179) − λ ) works here) such that the pseudoquad Q that is bounded bythe lines x = ± z = g ± t ( x ) is –rectilinear and contains the projection Π ( B ).Theorem 5.2 applied with the choice of parameters µ = and σ = Q hasa foliated patchwork ∆ and a set of 1–approximating planes that satisfy W (cid:161) V V ( ∆ ) (cid:162) (cid:46) λ | Q | (cid:46) λ W (cid:161) V ( ∆ ) (cid:162) (cid:46) W (cid:161) V V ( ∆ ) (cid:162) + α ( Q ) − | Q | (cid:46) λ
1. (146)By Lemma 2.11 and Lemma 6.2, we conclude as follows. (cid:176)(cid:176) v B (cid:161) Γ + (cid:162)(cid:176)(cid:176) L ( (cid:82) ) (cid:46) (cid:112) δ z ( Q ) + (cid:176)(cid:176) v B (cid:161) Γ + (cid:162)(cid:176)(cid:176) L ([ − log δ z ( Q ), ∞ )) (cid:201) (cid:112) δ z ( Q ) + (cid:176)(cid:176) v PQ , f (cid:176)(cid:176) L ([ − log δ z ( Q ), ∞ )) (cid:46) (cid:112) δ z ( Q ) + | Q | W (cid:161) V ( ∆ ) (cid:162) (cid:46) λ OLIATED CORONA DECOMPOSITIONS 65 where the first step is an application of (144) with a = − log δ z ( Q ), the second step isan application of Lemma 2.11 because Q ⊇ Π ( B ), the third step is an application ofLemma 6.2, and the final step holds due to (146) and because | Q | (cid:179) δ z ( Q ) (cid:179) λ (cid:3)
7. T
HE SUBDIVISION ALGORITHM : CONSTRUCTING A FOLIATED CORONA DECOMPOSITION
In this section, we will formulate an iterative subdivision algorithm (Lemma 7.3 below)and prove that, given certain propositions on the geometry of pseudoquads, this algorithmproduces a foliated corona decomposition. In the following sections, we will prove thesegeometric propositions. Together, these arguments establish Theorem 5.2.Fix λ , σ ∈ (0, 1). Let f : V → (cid:82) , and suppose that Γ = Γ f is an intrinsic λ –Lipschitzgraph. Let 0 < µ (cid:201) . To show that Γ admits a foliated corona decomposition, we mustshow that for any µ –rectilinear pseudoquad Q , there is a µ –rectilinear foliated patchwork ∆ for Q which has a set of σ –approximating planes and such that ∆ is weighted-Carleson.In order to describe the subdivision algorithm that produces ∆ , we will introduce the R–extended parametric normalized nonmonotonicity of Γ , denoted by Ω P Γ + , R , which is ameasure on V with density based on how horizontal line segments of length at most R > Γ . If Γ is a plane, for instance, then Ω P Γ + , R =
0, while Ω P Γ + , R has positivedensity when Γ is bumpy at scale R ; we will give a full definition in Section 8 below.This is in the spirit of the quantitative nonmonotonicity used in [CKN11] and [NY18],but it counts different segments, and, like the parametric vertical perimeter, it is definedin terms of the function f . In Section 9, we will show that there is c > Γ such that the following kinematic formula (inequality)holds for every measurable subset U ⊆ V . (cid:88) i ∈ (cid:90) Ω P Γ + ,2 − i ( U ) (cid:201) c | U | . Definition 7.1.
Suppose that η , r , R > and Q is a –rectilinear pseudoquad. We say that Γ is ( η , R )–paramonotone on r Q if its average density is bounded as follows. Ω P Γ + , R δ x ( Q ) ( r Q ) | Q | (cid:201) ηα ( Q ) . (147) This condition is invariant under scalings, stretch maps, and shear maps; see the discussionimmediately after the proof of Lemma 8.6 below.
One of the main results of [CKN11] was that for small η >
0, any η –monotone set isclose to a plane in (cid:72) ; this is a “stability version” of the characterization of monotone setsin [CK10b]. The following proposition, which we will prove in Sections 10–12, states notonly that paramonotone pseudoquads are close to vertical planes in (cid:72) , but also that theircharacteristic curves are close to the characteristic curves of their approximating planes. Proposition 7.2.
There is a universal constant r > such that for any σ > and < ζ (cid:201) ,there are η , R > such that if Γ = Γ f is the intrinsic Lipschitz graph of f : V → (cid:82) , and if Qis a –rectilinear pseudoquad for Γ such that Γ is ( η , R ) –paramonotone on r Q, then (1) There is a vertical plane P ⊆ (cid:72) (a σ –approximating plane) and an affine functionF : V → (cid:82) such that P is the intrinsic graph of F and (cid:107) F − f (cid:107) L (10 Q ) | Q | (cid:201) σ δ z ( Q ) δ x ( Q ) . (148)(2) Let u ∈ Q and let g Γ , g P : (cid:82) → (cid:82) be such that { z = g Γ ( x )} (respectively { z = g P ( x )} )is a characteristic curve for Γ (respectively P ) that passes through u. Then (cid:107) g P − g Γ (cid:107) L ∞ (4 I ) (cid:201) ζδ z ( Q ).It is important to observe that the bounds in Proposition 7.2 do not depend on theintrinsic Lipschitz constant of f . Indeed, this proposition holds when Γ is merely theintrinsic graph of a continuous function, with the caveat that if Γ is not intrinsic Lipschitz,then there may not be a characteristic curve through every point of 4 Q . This is impor-tant because paramonotonicity is invariant under stretch automorphisms; a bound thatdepended on the intrinsic Lipschitz constant of Γ would not be invariant.Proposition 7.2 allows us to construct a µ –rectilinear foliated patchwork and a col-lection of σ –approximating planes by recursively subdividing Q according to a greedyalgorithm. Lemma 7.3.
Let r be as in Proposition 7.2. Fix < µ (cid:201) and σ > . There are η , R > withthe following property. Let Γ be an intrinsic Lipschitz graph and let Q be a µ –rectilinearpseudoquad. There is a µ –rectilinear foliated patchwork ∆ for Q, such that for all v ∈ V ( ∆ ) ,Q v is horizontally cut if and only if Γ is ( η , R ) –paramonotone on r Q, and ∆ admits a set of σ –approximating planes.Proof. Let r , η , and R be positive constants so that Proposition 7.2 is satisfied with ζ = µ .We construct ∆ by a greedy algorithm. Denote the root vertex of ∆ by v and let Q v = Q ;by assumption, it is µ –rectilinear. Suppose by induction that we have already constructeda µ –rectilinear pseudoquad ( Q v , R v ). Let v ∈ V ( ∆ ) be a vertex with children w and w (cid:48) .Let I = [ a , b ] be the base of Q v and let g , g : (cid:82) → (cid:82) be its lower and upper bounds,respectively.Suppose that Γ is not ( η , R )–paramonotone on r Q v . The vertical line x = a + b cuts Q v and R v vertically into two halves. Let Q w and Q w (cid:48) be the halves of Q v and let R w and R w (cid:48) be the halves of R v . Since ( Q v , R v ) is µ –rectilinear, ( Q w , R w ) and ( Q w (cid:48) , R w (cid:48) ) are both µ –rectilinear.Now suppose that Γ is ( η , R )–paramonotone on r Q v . Proposition 7.2 states that thereis a σ –approximating plane P for Q v such that for every u ∈ Q v , any characteristic curveof Γ that passes through u is ζδ z ( Q )–close to the characteristic curve of P that passesthrough u . For i ∈ {1, 2}, let u i = (cid:161) a + b , g i ( a + b ) (cid:162) , and let m be the midpoint of u and u .Let g : (cid:82) → (cid:82) be a function whose graph is a characteristic curve for Γ that passesthrough m . Let Q w and Q w (cid:48) be the pseudoquads with base I that are bounded by thegraphs of g , g , and g .The characteristic curves of P that pass through u , u , and m are parallel evenly-spaced parabolas; let h , h , h : V → (cid:82) be the corresponding quadratic functions andlet d = h − h = h − h be the constant distance between them. Let R w and R w (cid:48) bethe parabolic rectangles with base I that are bounded by these three parabolas. By OLIATED CORONA DECOMPOSITIONS 67
Proposition 7.2, we have (cid:107) g i − h i (cid:107) L ∞ (4 I ) (cid:201) ζδ z ( Q ) for i ∈ {1, 2, 3}. In particular, every x ∈ I satisfies | δ z ( Q ) − d | (cid:201) | δ z ( Q ) − ( g ( x ) − g ( x )) | + | g ( x ) − g ( x ) − d | + | g ( x ) − g ( x ) − d | (cid:201) ζδ z ( Q ),so d (cid:202) δ z ( Q ) and (cid:107) g i − h i (cid:107) L ∞ (4 I ) (cid:201) ζ d = µ d for i ∈ {1, 2, 3}. That is, ( Q w , R w ) and( Q w (cid:48) , R w (cid:48) ) are µ –rectilinear and satisfy Definition 4.4 with k = h . We construct the desiredrectilinear foliated patchwork by repeating this process for every vertex of ∆ . (cid:3) Pseudoquads that are not paramonotone contribute to the nonmonotonicity of Γ , so, asin [NY18], the total number and size of these pseudoquads is bounded by the measure of Γ . In Section 9, we will use an argument based on the Vitali Covering Lemma to prove thatrectilinear foliated patchworks constructed using Lemma 7.3 satisfy a weighted Carlesoncondition, as stated in the following proposition. Proposition 7.4.
Let r > and let < µ (cid:201) r . Let η , R > and let < λ < . Let Γ be an intrinsic λ –Lipschitz graph, let ∆ be a µ –rectilinear foliated patchwork for Γ , andsuppose that for all v ∈ V ( ∆ ) , the pseudoquad Q v is horizontally cut if and only if Γ is ( η , R ) –paramonotone on r Q v . Let W : 2 V ( ∆ ) → [0, ∞ ] be as in (135) . Then for any v ∈ V ( ∆ ) ,W (cid:161) { w ∈ V V ( ∆ ) : w (cid:201) v } (cid:162) (cid:46) η , r , R , λ | Q v | . (149)With these tools at hand, Theorem 5.2 follows directly. Proof of Theorem 5.2 assuming Proposition 7.2 and Proposition 7.4.
Let r be as in Propo-sition 7.2 and write µ = r . Fix 0 < µ (cid:201) µ and σ >
0, and let η , R be as in Lemma 7.3.Since Γ is an intrinsic λ –Lipschitz graph, Lemma 7.3 produces a µ –rectilinear foliatedpatchwork ∆ rooted at Q with a set of σ –approximating planes. By Proposition 7.4, thispatchwork is weighted–Carleson with a constant depending on η , r , R , σ , and λ . Since r > η , R depend only on µ , σ , we obtain Theorem 5.2 by using Lemma 5.5to increase µ = r to µ = . (cid:3) Observe in passing that since in the above proof the patchwork that established The-orem 5.2 was obtained from Proposition 7.2, we actually derived the following morenuanced formulation of Theorem 5.2; it is worthwhile to state it explicitly here becausethis is how it will be used in forthcoming work of the second named author.
Theorem 7.5.
For every < λ < there is a function D λ : (cid:82) + × (cid:82) + → (cid:82) + , and for every < µ (cid:201) and σ > there are η = η ( µ , σ ), R = R ( µ , σ ) > with the following properties.Suppose that Γ ⊆ (cid:72) is an intrinsic λ –Lipschitz graph over V and Q ⊆ V is a µ –rectilinearpseudoquad for Γ . Then there is a µ –rectilinear foliated patchwork ∆ for Q such that ∆ is D λ ( µ , σ ) –weighted-Carleson and has a set of σ –approximating planes. Moreover, forall vertices v ∈ V ( ∆ ) , the associated pseudoquad Q v is horizontally cut if and only if Γ is ( η , R ) –paramonotone on r Q, where r > is the universal constant in Proposition 7.2. We will prove Proposition 7.2 and Proposition 7.4 in the following sections. Specifi-cally, in Section 8, we will define extended nonmonotonicity and extended parametricnormalized nonmonotonicity and prove some of their basic properties. In Section 9, wewill prove that Proposition 7.2 implies Proposition 7.4. Finally, in Sections 10–12, we willprove Proposition 7.2.
8. E
XTENDED NONMONOTONICITY
Extended nonmonotonicity in (cid:82) . In this section, we define the extended nonmono-tonicity and extended parameterized nonmonotonicity of a set E ⊆ (cid:72) . Like the quantitativenonmonotonicity that was defined in [CKN11] and the horizontal width that was definedin [FOR18], these measure how horizontal lines intersect ∂ E .We first define these quantities on subsets of lines, then define them on subsets of (cid:72) byintegrating over the space of horizontal lines. Let L be the space of horizontal lines in (cid:72) .Let N be the Haar measure on L , normalized so that the measure of the set of lines thatintersect the ball of radius r is equal to r .Recall that a measurable subset S ⊆ (cid:82) is monotone [CK10b] if its indicator function isa monotone function (i.e., S is equal to either ∅ , (cid:82) , or some ray). For a measurable set U ⊆ (cid:82) , we define the nonmonotonicity of S on U byNM S ( U ) def = inf (cid:169) H (cid:161) U ∩ ( M (cid:52) S ) (cid:162) : M is monotone (cid:170) ,where, as usual, M (cid:52) S = ( M (cid:224) S ) ∪ ( S (cid:224) M ) is the symmetric difference of M and S .For S ⊆ (cid:82) , we say that S has finite perimeter if ∂ H S is a finite set, where we recall thenotation (35) for measure theoretical boundary, which in the present setting becomes ∂ H S def = (cid:169) x ∈ (cid:82) : 0 < H (cid:161) ( x − ε , x + ε ) ∩ S (cid:162) < ε for all ε > (cid:170) .If S ⊆ (cid:82) is a set of finite perimeter, then there is a unique collection of disjoint closedintervals of positive length I ( S ) = { I ( S ), I ( S ), . . . } such that S (cid:52) (cid:83) I ( S ) has measure zero.For any R >
0, we define as follows a point measure ω S , R supported on the boundaries ofthe intervals in I ( S ) of length at most R . ω S , R def = (cid:88) I ∈ I ( S ) H ( I ) (cid:201) R H ( I ) · ( δ min I + δ max I ).Let (cid:98) ω S , R = ω S , R + ω (cid:82) (cid:224) S , R (cid:98) w i } i ∈ (cid:90) used in [CKN11]. It wasshown in [CKN11] that if δ > S at scale δ i is bounded in terms of a measure (cid:98) w i that counts the set of endpoints of intervals in S or (cid:82) (cid:224) S of length between δ i and δ i + . The main difference between (cid:98) w i and (cid:98) ω S , δ i is that (cid:98) w i ignores intervals of length less than δ i + , but (cid:98) ω S , δ i weights them by their lengths.For U ⊆ (cid:82) , we call (cid:98) ω S , R ( U ) the R–extended nonmonotonicity of S on U . (We will typicallyuse this notation when R > diam U .) We use the term “extended” here because it dependsnot only on S ∩ U , but also on the behavior of S outside U . For example, let U = [ a , b ]and suppose that S ⊆ (cid:82) is a set with locally finite perimeter. If (cid:98) ω S , R ( U ) = R > I ( S ) or I ( (cid:82) (cid:224) S ) with a boundary point in U .That is, U ∩ ∂ H S is empty or, up to a measure-zero set, S = [ c , ∞ ) or S = ( −∞ , c ] for some c ∈ [ a , b ]. Similarly, when S = [ a , b ] and R > b − a , if (cid:98) ω S , R ( U ) is much smaller than b − a ,then either U ∩ ∂ H S is almost empty or U is almost monotone on an R –neighborhood of S . This follows from the following two lemmas. The first lemma is based on the bounds inProposition 4.25 of [CKN11] and Lemma 3.4 of [FOR18]. OLIATED CORONA DECOMPOSITIONS 69
Lemma 8.1.
Let a , b ∈ (cid:82) , let U ⊆ [ a , b ] , and let R (cid:202) b − a. For any finite-perimeter set S ⊆ (cid:82) , NM S ( U ) (cid:201) diam (cid:161) ( a , b ) ∩ ∂ H S (cid:162) (cid:201) (cid:98) ω S , R (cid:161) ( a , b ) (cid:162) . Proof.
Let δ = diam (cid:161) ( a , b ) ∩ ∂ H S (cid:162) . Consider the following set of closed intervals. J def = { I ∈ I ( S ) ∪ I ( (cid:82) (cid:224) S ) : I ∩ ( a , b ) (cid:54)= ∅ }.This set is finite, so we may label its elements J , . . . , J n in increasing order. After changing S on a measure-zero subset, the interiors of the J i ’s are alternately contained in S anddisjoint from S . If n =
1, then NM S ( U ) = δ =
0, so we suppose that n (cid:202)
2. Then δ = min( J n ) − max( J ) = n − (cid:88) i = H ( J i )and (cid:98) ω S , R (cid:161) ( a , b ) (cid:162) (cid:202) n − (cid:88) m = H ( J m ) (cid:202) δ .Regardless of whether J and J n are in or out of S , there is a monotone subset M ⊆ (cid:82) suchthat M agrees with S on J and J n . ThenNM S ( U ) (cid:201) H (cid:161) U ∩ ( S (cid:52) M ) (cid:162) (cid:201) H (cid:161) [ a , b ] (cid:224) ( J ∪ J n ) (cid:162) = δ . (cid:3) A similar reasoning gives the following lower bound. Recall that supp H and int H denote measure-theoretic support and interior, see (34)–(36). Lemma 8.2.
Fix R , a , b ∈ (cid:82) with a < b and R (cid:202) b − a. Let S ⊆ (cid:82) have locally finite perimetersuch that a , b ∈ supp H ( (cid:82) (cid:224) S ) . For any closed interval I ⊆ [ a , b ] , either I ⊆ int H S or (cid:98) ω S , R ( I ) (cid:202) H ( S ∩ I )2 . Proof.
Suppose that I (cid:54)⊆ int H S . Let I , . . . , I n be the intervals in I ( S ) that intersect I .By assumption, each of the intervals I , . . . , I n has at least one endpoint in I . Further-more, since a , b ∈ supp H ( (cid:82) (cid:224) S ), we have I j ⊆ [ a , b ] for all j ∈ {1, . . . , n }. In particular,max j ∈ {1,..., n } (cid:96) ( I j ) (cid:201) R . Up to a null set, we have S ∩ I ⊆ (cid:83) nj = I j , so (cid:98) ω S , R ( I ) (cid:202) n (cid:88) j = H ( I j )2 (cid:202) H ( S ∩ I )2 . (cid:3) These lemmas yield the following description of sets with small extended nonmono-tonicity, which states that points in their measure theoretic boundary must be either veryclose to each other, or very far from each other.
Proposition 8.3.
Let S ⊆ (cid:82) be a set with locally finite perimeter and fix c , d ∈ (cid:82) with c < d .Let R (cid:202) d − c and suppose that < ε < d − c and (cid:98) ω S , R (( c , d )) < ε . Then, ∀ t ∈ [ c + ε , d − ε ] ∩ ∂ H S , diam (cid:161) ( t − R , t + R ) ∩ ∂ H S (cid:162) < ε . (150) Proof.
Fix t ∈ [ c + ε , d − ε ] ∩ ∂ H S . We will prove that this implies that( t − R , t + R ) ∩ ∂ H S ⊆ ( c , d ). (151) Equation (150) is a consequence of the inclusion (151), since by Lemma 8.1,diam (cid:161) ( t − R , t + R ) ∩ ∂ H S (cid:162) (151) (cid:201) diam (cid:161) ( c , d ) ∩ ∂ H S (cid:162) (cid:201) (cid:98) ω S , R (cid:161) ( c , d ) (cid:162) < ε .Suppose by way of contradiction that (151) fails. So, there is u ∈ ( t − R , t + R ) ∩ ∂ H S with u (cid:202) d or u (cid:201) c . We will treat only the case u (cid:202) d since the case u (cid:201) c is analogous.Lemma 8.2 applied with [ a , b ] = [ t , u ] and I = [ t , d ], gives H ( S ∩ [ t , d ])2 (cid:201) (cid:98) ω S , R ([ t , d ]) < ε .If we replace S by (cid:82) (cid:224) S , the Lemma 8.2 gives H ([ t , d ] (cid:224) S )2 (cid:201) (cid:98) ω S , R ([ t , d ]) < ε .So d − t < ε , which contradicts the choice of t . (cid:3) Extended nonmonotonicity in (cid:72) . We have defined NM and (cid:98) ω for subsets of (cid:82) , butthe same definitions are valid for subsets of any line L ∈ L . This lets us define thenonmonotonicity of a subset of (cid:72) by integrating over horizontal lines.When U , E ⊆ (cid:72) are measurable sets, we define the nonmonotonicity of E on U byNM E ( U ) def = ˆ L NM E ∩ L ( U ∩ L ) d N ( L ).(Note that this definition differs from the definition in [CKN11]. Specifically, in [CKN11],this was only defined in the case that U = B r ( x ) for some r ∈ (0, ∞ ) and x ∈ (cid:72) , and wasnormalized by a factor of r − to make it scale-invariant.) Definition 8.4.
Fix R > . Let E ⊆ (cid:72) be a set with finite perimeter. By the kinematic formula(Section 2.5), for almost every L ∈ L , the intersection E ∩ L is a set with finite perimeter, andwe define for U ⊆ (cid:72) , (cid:98) ω E , R ( U , L ) def = (cid:98) ω E ∩ L , R ( U ∩ L ). (152) We then define a measure
ENM E , R on (cid:72) by setting ENM E , R ( U ) def = ˆ L (cid:98) ω E , R ( U , L ) d N ( L ). We call
ENM E , R ( U ) the R –extended nonmonotonicity of E on U , and for ν > we say thatE is ( ν , R )–extended monotone on U if ENM E , R ( U ) (cid:201) ν . Like (cid:98) ω S , R ( · ) , ENM E , R ( U ) dependson the behavior of E in an R–neighborhood of U . If R (cid:201) R (cid:48) , then ENM E , R (cid:201) ENM E , R (cid:48) . When we say that a subset U ⊆ (cid:72) is convex , we will always mean that it is convex as asubset of the vector space (cid:82) . For every g ∈ (cid:72) , the map v (cid:55)→ g v is an affine map from (cid:72) toitself, so convexity is preserved by left multiplication. Lemma 8.5.
Let U ⊆ (cid:72) be a measurable bounded set and let K ⊆ U be convex. Let E ⊆ (cid:72) bea finite-perimeter set. Then, for every R > diam U we have NM E ( K ) (cid:201) ENM E , R ( U ). OLIATED CORONA DECOMPOSITIONS 71
Proof.
Let L ∈ L be a horizontal line. By convexity, the intersection I = L ∩ K is an intervaland (cid:96) ( I ) (cid:201) diam U . By Lemma 8.1,NM E ∩ L ( I ) (cid:201) (cid:98) ω E ∩ L , R ( I ) (cid:201) (cid:98) ω E , R ( U , L ).Integrating both sides of this inequality with respect to N yields the desired bound. (cid:3) We will also define a parametric version of extended nonmonotonicity that is betteradapted to intrinsic Lipschitz graphs. This is based on a different measure on the space ofhorizontal lines, denoted N P , which we next describe.Let W = { x =
0} be the y z –plane and let L P be the set of horizontal lines that are notparallel to W . Each L ∈ L P intersects W in a single point w ( L ), called the intercept of L ,and has a unique slope m ( L ) ∈ (cid:82) such that L = w ( L ) · 〈 X + m ( L ) Y 〉 .The map ( m , w ) : L P → (cid:82) × W is a bijection, and we define N P to be the pullback ofthe Lebesgue measure on (cid:82) × W under this bijection. This measure is preserved by shearmaps and translations, and for any a , b > A ⊆ L P , N P (cid:161) s a , b ( A ) (cid:162) = b N P ( A ). (153)Let E ⊆ (cid:72) . For any R >
0, any U ⊆ V , and any L ∈ L P , we define (cid:98) ω PE , R ( U , L ) def = (cid:98) ω x ( E ∩ L ), R (cid:179) x (cid:161) Π − ( U ) ∩ L (cid:162)(cid:180) . (154)This is similar to (cid:98) ω E , R ( Π − ( U ), L ) in (152), but the projection to the x –coordinate thatappears in (154) changes the measures and lengths involved by a constant factor.When E is a finite-perimeter subset of (cid:72) , we define a measure Ω PE , R on V by setting forany measurable subset U ⊆ V , Ω PE , R ( U ) def = R ˆ L P (cid:98) ω PE , R ( U , L ) d N P ( L ). (155)We call Ω PE , R ( U ) the R – extended parametric normalized nonmonotonicity of E on U . Notethat the definition (155) includes an R − factor that does not appear in Definition 8.4; wewill see that this normalization allows for the kinematic formula (32) to hold.This quantity scales nicely under automorphisms. Lemma 8.6.
Fix a , b ∈ (cid:82) (cid:224) {0} and let g = q ◦ ρ h ◦ s a , b : (cid:72) → (cid:72) be a composition of a shearmap q, a left-translation by h ∈ (cid:72) , and a stretch map s a , b . Let ˆ g : V → V be the mapinduced on V , i.e., ˆ g ( x ) = Π ( g ( x )) for all x ∈ V . Let E ⊆ (cid:72) be a set with finite perimeter.For any measurable U ⊆ V and any r > , if Ω P E , R ( U ) is finite, then Ω Pg ( E ), | a | R ( ˆ g ( U )) = | b | Ω PE , R ( U ), (156) and Ω Pg ( E ), | a | R ( ˆ g ( U )) | ˆ g ( U ) | = b a · Ω P E , R ( U ) | U | . (157) In particular, if g is a composition of a scaling, shear, and translation, i.e., when a = babove, then g preserves the density of Ω PE , R . Proof.
The identity (156) is verified by computing as follows, using (153). Ω Pg ( E ), | a | R (cid:161) ˆ g ( U ) (cid:162) = | a | R ˆ L P (cid:98) ω Pg ( E ), | a | R (cid:161) ˆ g ( U ), L (cid:162) d N P ( L ) = | b | | a | R ˆ L P (cid:98) ω Pg ( E ), | a | R (cid:161) ˆ g ( U ), g ( L ) (cid:162) d N P ( L ) = | b | | a | R ˆ L P | a | (cid:98) ω PE , R ( U , L ) d N P ( L ) = | b | Ω PE , R ( U ),By Lemma 2.8 we have | ˆ g ( U ) | = a | b | · | U | , which implies (157). (cid:3) Suppose that Q is a pseudoquad for an intrinsic Lipschitz graph Γ ⊆ (cid:72) and that g is asin Lemma 8.6. If Γ is ( η , R )–paramonotone on r Q as in Definition 7.1, then the density of Ω P Γ + , R δ x ( Q ) is bounded as follows: Ω P Γ + , R δ x ( Q ) ( r Q ) | Q | (cid:201) ηα ( Q ) .Let ˆ Q = ˆ g ( Q ) and ˆ Γ = g ( Γ ). Then (157) and Lemma 4.2 imply that Ω P ˆ Γ + , R δ x ( ˆ Q ) ( r ˆ Q ) | ˆ Q | (cid:201) η b a α ( Q ) = ηα ( ˆ Q ) ,so ˆ Γ is ( η , R )–paramonotone on r ˆ Q if and only if Γ is ( η , R )–paramonotone on r Q .In general, the measure Ω PE , R is not necessarily locally finite. Indeed, if B ⊆ (cid:72) is a ball,then the set of lines that pass through B has infinite N P –measure. But when Γ is anintrinsic λ –Lipschitz graph, any line with sufficiently large slope intersects Γ exactly once.If E = Γ + and L ∈ L P is a line such that L ∩ E is nonmonotone, then L has bounded slope;it follows that Ω P Γ + , R ( K ) is finite for any compact K ⊆ V . Lemma 8.7.
Let R > and let x ∈ (cid:72) . Suppose that E is a finite-perimeter subset of (cid:72) andlet U ⊆ (cid:72) be measurable. Then ENM E , R ( U ) (cid:46) R Ω PE , R (cid:161) Π ( U ) (cid:162) . Proof.
Let L ∈ L P and let m = m ( L ) be the slope of L , so that the restriction x | L shrinkslengths by a factor of φ ( m ) = (cid:112) + m . Then (cid:98) ω PE , R ( Π ( U ), L ) = (cid:98) ω E , φ ( m ) · R ( Π − ( Π ( U )), L ) φ ( m ) (cid:202) (cid:98) ω E , R ( U , L ) φ ( m ) .For w ∈ W and m ∈ (cid:82) , let L w , m be the line L w , m = w · 〈 X + mY 〉 ∈ L P . Then it follows that R Ω P E , R (cid:161) Π ( U ) (cid:162) = ˆ W ˆ (cid:82) (cid:98) ω P E , R ( Π ( U ), L w , m ) d m d w (cid:202) ˆ W ˆ (cid:82) (cid:98) ω E , R ( U , L w , m ) (cid:112) + m d m d w .For θ ∈ (cid:82) , let R θ : (cid:72) → (cid:72) be the rotation by angle θ around the z –axis. Since N isinvariant under translations and rotations, there is c > OLIATED CORONA DECOMPOSITIONS 73 f : L → (cid:82) , ˆ L f ( M ) d N ( M ) = c ˆ W ˆ π − π f (cid:161) R θ ( L g ,0 ) (cid:162) d θ d g .Any line in L P can be written as R θ ( L g ,0 ) for some θ ∈ (cid:82) and g ∈ W . Specifically, for w ∈ W and m ∈ (cid:82) , let θ ( m ) = arctan m and let g m ( w ) be the W –intercept of R − θ ( m ) ( L w , m )),so that L w , m = R θ ( m ) ( L g m ( w ),0 ). Writing g m in coordinates as g m = (0, b m , c m ), its Jacobianis J g m ( y , z ) = det (cid:195) d b m d y d b m d z d c m d y d c m d z (cid:33) = det (cid:195) cos(arctan m ) 0 dc m d y (cid:33) = (cid:112) + m .Consequently, ˆ W ˆ π − π f (cid:161) R θ ( L g ,0 ) (cid:162) d θ d g = ˆ W ˆ (cid:82) f ( L w , m ) d θ d m J g m ( w ) d m d w = ˆ W ˆ (cid:82) f ( L w , m )(1 + m ) d m d w .ThusENM E , R ( U ) = ˆ L (cid:98) ω E , R ( U , L ) d N ( L ) = c ˆ W ˆ (cid:82) (cid:98) ω E , R ( U , L w , m )(1 + m ) d m d w (cid:201) cR Ω PE , R (cid:161) Π ( U ) (cid:162) ,as desired. (cid:3) In particular, it follows from Lemma 8.7 that if Γ is ( η , R )–paramonotone on r Q , thenENM Γ + , R δ x ( Q ) (cid:161) Π − ( r Q ) (cid:162) (cid:46) R δ x ( Q ) Ω P Γ + , R δ x ( Q ) ( r Q ) (cid:201) δ x ( Q ) | Q | α ( Q ) η R (cid:179) Q η R . (158)9. T HE KINEMATIC FORMULA AND THE PROOF OF P ROPOSITION v ∈ V ( ∆ ) in terms of Ω P . Lemma 9.1.
Let r , η , R, λ , Γ , and ∆ be as in Proposition 7.4. Then for any v ∈ V ( ∆ ) ,W ({ w ∈ V V ( ∆ ) : w (cid:201) v }) (cid:46) η , r , R ∞ (cid:88) i = Ω P Γ + ,2 − i R δ x ( Q v ) ( r Q v ). (159)The second is a kinematic formula bounding Ω P in terms of Lebesgue measure on V . Lemma 9.2.
Let < λ < and let Γ be an intrinsic λ –Lipschitz graph. For any measurableset U ⊆ V , (cid:88) i ∈ (cid:90) Ω P Γ + ,2 − i ( U ) (cid:46) λ | U | . (160)Proposition 7.4 follows from Lemma 9.1 and Lemma 9.2. Proof of Proposition 7.4 assuming Lemma 9.1 and Lemma 9.2.
Fix v ∈ V ( ∆ ) and denote δ = δ x ( Q v ). Due to Lemma 9.1, W ({ w ∈ V V ( ∆ ) : w (cid:201) v }) (cid:46) η , r , R ∞ (cid:88) i = Ω P Γ + ,2 − i R δ ( r Q v ). Let k be the integer such that 2 k − (cid:201) R δ < k . Then ∞ (cid:88) i = Ω P Γ + ,2 − i R δ ( r Q v ) (cid:201) ∞ (cid:88) i = Ω P Γ + ,2 − i + k ( r Q v ) (160) (cid:46) λ | r Q v | .Thus, W ({ w ∈ V V ( ∆ ) : w (cid:201) v }) (cid:46) η , r , λ , R | Q v | . (cid:3) We first establish Lemma 9.1, which we prove using an argument based on the VitaliCovering Lemma. The first step is to construct partitions of Q into pseudoquads withdyadic widths. As in Lemma 6.4, we construct these partitions from coherent subtrees. Lemma 9.3.
Let < µ (cid:201) and let ( ∆ , ( Q v ) v ∈ ∆ ) be a µ –rectilinear foliated patchwork for a µ –rectilinear pseudoquad Q. Fix j ∈ (cid:78) ∪ {0} . For v ∈ V ( ∆ ) , let D V ( v ) ⊆ V ( ∆ ) denote the setof vertically cut descendants of v, and letF j ( v ) def = (cid:110) w ∈ D V ( v ) : δ x ( Q w ) = − j δ x ( Q ) (cid:111) . Then, for any w , w (cid:48) ∈ F j ( v ) , if w (cid:54)= w (cid:48) , then Q w and Q w (cid:48) have disjoint interiors.Proof. Let D ( v ) be the set of descendants of v and let R j = (cid:110) w ∈ D ( v ) : δ x ( Q w ) (cid:202) − j δ x ( Q v ) (cid:111) .By Lemma 4.5, this is a coherent set and F j ( v ) = min R j , so Lemma 6.3 implies that F j ( v )consists of pseudoquads with disjoint interiors. (cid:3) Let v be the root of ∆ (so Q = Q v ). For each j ∈ (cid:78) ∪ {0} we write F j = F j ( v ). Denote I = x ( Q ) and l j = − j (cid:96) ( I ) = − j δ x ( Q ). Let I j ,1 , . . . , I j ,2 j be the partition of I into 2 j intervalsof length l j , so that for any v ∈ V ( ∆ ), there are j , m ∈ (cid:78) ∪ {0} such that x ( Q v ) = I j , m . Wepartition F j into columns as follows. ∀ m ∈ {1, . . . , 2 j }, F j , m def = { w ∈ F j : x ( Q w ) = I j , m }. (161)Each column satisfies the following version of the Vitali Covering Lemma. Lemma 9.4.
For each j ∈ (cid:78) ∪ {0} and m ∈ {1, . . . , 2 j } , there is a (possibly finite) sequence ofvertices D j , m = { v , v , . . .} ⊆ F j , m such that r Q v , r Q v , . . . are pairwise disjoint andW ( D j , m ) (cid:179) r W ( F j , m ).We prove Lemma 9.4 using the following expansion property. Lemma 9.5.
Let r > and < µ (cid:201) r . Let ∆ be a µ –rectilinear foliated patchwork. Letv , w ∈ V ( ∆ ) be vertices such that x ( Q v ) = x ( Q w ) and suppose that r Q v ∩ r Q w is nonempty.If δ z ( Q v ) (cid:202) δ z ( Q w ) , then Q w ⊆ r Q v , and if δ z ( Q w ) (cid:202) δ z ( Q v ) , then Q w ⊆ r Q v .Proof. Write I = [ −
1, 1]. By rescaling and translating, we may suppose without loss ofgenerality that x ( Q v ) = x ( Q w ) = I . Also, we may suppose that Q v is vertically below Q w .We first construct a stack of pseudoquads of width at least 2 that connects Q v and Q w .For u ∈ V ( ∆ ), let A ( u ) = { t ∈ V ( ∆ ) : t (cid:202) u } be the set of ancestors of u . If u (cid:54)= v , let S ( u )be the sibling of u and let P ( u ) be the parent of u . Let J = A ( v ) ∪ A ( w ) ∪ S (cid:161) A ( v ) ∪ A ( w ) (cid:162) . OLIATED CORONA DECOMPOSITIONS 75
Since A ( v ) ∪ A ( w ) spans a connected subtree of ∆ , so does J , and J is a coherent subset of V ( ∆ ). Furthermore, J is finite, so K = min J is a partition of Q .If u ∈ K , then u is either an ancestor of v or w or a sibling of such an ancestor. In eithercase, δ x ( Q u ) (cid:202)
2, and the base of Q u either contains I or its interior is disjoint from I . Let K (cid:48) = { u ∈ K : I ⊆ x ( Q u )}. For each u ∈ K (cid:48) , Q u intersects the z –axis in an interval. We denotethe elements of K (cid:48) by u , . . . , u n , in order of increasing z –coordinate. These pseudoquadsform a stack; each pseudoquad Q u i is vertically adjacent to Q u i + . We suppose that u a = v and u b = w , with a < b .Rectilinearity implies that the boundaries of the Q u i ’s have similar slopes. For each i ∈ {1, . . . , n }, let g i be the lower bound of Q u i and let g i + be its upper bound. These maybe defined on different domains, but all of their domains contain I . For each i ∈ {1, . . . , n },let R u i be the parabolic rectangle associated to Q u i and let d i = δ z ( Q u i ), so that there arequadratic functions h i : (cid:82) → (cid:82) satisfying (cid:176)(cid:176)(cid:176)(cid:176) g i − (cid:181) h i − d i (cid:182)(cid:176)(cid:176)(cid:176)(cid:176) L ∞ ( I ) (cid:201) µ d i and (cid:176)(cid:176)(cid:176)(cid:176) g i + − (cid:181) h i + d i (cid:182)(cid:176)(cid:176)(cid:176)(cid:176) L ∞ ( I ) (cid:201) µ d i .Then (cid:176)(cid:176)(cid:176)(cid:176)(cid:181) h i + − d i + (cid:182) − (cid:181) h i + d i (cid:182)(cid:176)(cid:176)(cid:176)(cid:176) L ∞ ( I ) (cid:201) µ ( d i + d i + ).Hence, for any i , j ∈ {1, . . . , n } with i < j , (cid:176)(cid:176)(cid:176)(cid:176)(cid:176) h j − h i − j − (cid:88) k = i d k + d k + (cid:176)(cid:176)(cid:176)(cid:176)(cid:176) L ∞ ( I ) (cid:201) µ j − (cid:88) k = i ( d k + d k + ).Since h j − h i − (cid:80) j − k = i d k + d k + is quadratic, by Lemma 4.6 it follows that (cid:176)(cid:176)(cid:176)(cid:176)(cid:176) h j − h i − j − (cid:88) k = i d k + d k + (cid:176)(cid:176)(cid:176)(cid:176)(cid:176) L ∞ ([ − r , r ]) (cid:201) r µ j − (cid:88) k = i ( d k + d k + ) (cid:201) j − (cid:88) k = i d i + d i + D = b − (cid:88) k = a d k + d k + x ∈ [ − r , r ] we have34 D (cid:201) h b ( x ) − h a ( x ) (cid:201) D . (162)Suppose that δ z ( Q w ) (cid:201) δ z ( Q v ). For each i ∈ {1, . . . , n }, the definition (126) of r Q states r Q u i = (cid:189) ( x , z ) ∈ V : x ∈ [ − r , r ] and | z − h i ( x ) | (cid:201) r d i (cid:190) .Since δ z ( Q w ) (cid:201) δ z ( Q v ) and r Q v intersects r Q w , there is t ∈ [ − r , r ] such that h b ( t ) − h a ( t ) (cid:201) r ( d b + d a )2 = r ( δ z ( Q v ) + δ z ( Q w ))2 (cid:201) r δ z ( Q v ),and thus by (162) we have D (cid:201) r δ z ( Q v ). Hence, for any ( x , z ) ∈ r Q w we have | h a ( x ) − z | (cid:201) | h a ( x ) − h b ( x ) | + | h b ( x ) − z | (cid:201) D + r δ z ( Q w ) (cid:201) (3 r ) δ z ( Q v )2 , where the penultimate step uses (162) and the assumption ( x , z ) ∈ r Q w , and the final stepuses the upper bound on D that we derived above and the assumption δ z ( Q w ) (cid:201) δ z ( Q v ).It follows that ( x , z ) ∈ r Q v and thus r Q w ⊆ r Q v . If δ z ( Q v ) (cid:201) δ z ( Q w ), then the analogousreasoning shows that r Qv ⊆ r Qw . (cid:3) Proof of Lemma 9.4.
Similarly to the proof of the Vitali covering lemma, we define induc-tively a sequence S , S . . . , of subsets of F j , m as follows. Let S = ∅ . For each i ∈ (cid:78) , let v i be an element of F j , m (cid:224) (cid:83) i − k = S i that maximizes δ z ( Q v i ). Define S i = (cid:40) w ∈ F j , m (cid:224) i − (cid:91) k = S k : r Q v i ∩ r Q w (cid:54)= ∅ (cid:41) .If (cid:83) ik = S k = F j , m , we stop. By construction, r Q v , r Q v , . . . are disjoint. We will show thatthe set D j , m = { v , v , . . .} ⊆ F j , m satisfies the desired properties.We first claim that F j , m = S ∪ S ∪ . . ., where this holds by construction if there are onlyfinitely many v i ’s. So, suppose that there are infinitely many v i ’s and let w ∈ F j , m . Thereare only finitely many elements of F j , m with height greater than δ z ( w ), so there is i ∈ (cid:78) such that δ z ( v i ) < δ z ( w ). By the maximality of δ z ( v i ), this implies that w ∈ S ∪ . . . ∪ S i − .We next show that W ( D j , m ) (cid:179) r W ( F j , m ). As D j , m ⊆ F j , m , we have W ( D j , m ) (cid:201) W ( F j , m ).Conversely, if w ∈ S i , then r Q w intersects r Q v i and δ z ( Q w ) (cid:201) δ z ( Q v i ), so Lemma 9.5implies that Q w ⊆ r Q v i . Since the elements of F j , m are pairwise disjoint (Lemma 9.3)pseudoquads of the same width, we have α ( Q w ) (cid:202) α ( Q v i ) and W ( S i ) = (cid:88) w ∈ S i α ( Q w ) − | Q w | (cid:201) α ( Q v i ) − (cid:88) w ∈ S i | Q w |= α ( Q v i ) − (cid:175)(cid:175)(cid:175) (cid:91) w ∈ S i Q w (cid:175)(cid:175)(cid:175) (cid:46) α ( Q v i ) − (cid:175)(cid:175) r Q v i (cid:175)(cid:175) (cid:179) r W ({ v i }).By summing this bound over j , we conclude W ( F j , m ) = W ( S ) + W ( S ) + . . . (cid:46) r (cid:161) W ({ v }) + W ({ v }) + . . . (cid:162) = r W ( D j , m ). (cid:3) We are now ready to prove Lemma 9.1.
Proof of Lemma 9.1.
It suffices to treat the case where v is the root of ∆ , so Q v = Q . Fix j ∈ (cid:78) ∪ {0} and m ∈ {1, . . . , 2 j }. Let F j , m and D j , m be as in Lemma 9.4.Since, by definition, F j , m consists only of vertices that are vertically cut, by hypothesis, Γ is not ( η , R )–paramonotone on Q w for each w ∈ F j , m , i.e., ∀ w ∈ F j , m , Ω P Γ + , Rl j ( r Q w ) > ηα ( Q w ) − | Q w | = η W ({ w }).Let S m = r I j , m × {0} × (cid:82) ⊆ V . The sets { r Q w } w ∈ D j , m are disjoint subsets of S m ∩ r Q , so W ( F j , m ) (cid:179) r W ( D j , m ) (cid:201) η − (cid:88) w ∈ D j , m Ω P Γ + , Rl j ( r Q w ) (cid:201) η − Ω P Γ + , Rl j ( S m ∩ r Q ).By summing this bound over m ∈ {1, . . . , 2 j } we get W ( F j ) (cid:46) r j (cid:88) m = η − Ω P Γ + , Rl j ( S m ∩ r Q ) (cid:46) r η − Ω P Γ + , Rl j ( r Q ), OLIATED CORONA DECOMPOSITIONS 77 where the last step holds because the scaled intervals r I j ,1 , . . . , r I j ,2 j have bounded overlap(depending on r ). By summing this bound over j , we conclude as follows. W (cid:161) V V ( ∆ ) (cid:162) = ∞ (cid:88) j = W ( F j ) (cid:46) r η − ∞ (cid:88) j = Ω P Γ + , R − j δ x ( Q ) ( r Q ). (cid:3) Next, we prove Lemma 9.2 using the following kinematic formula for intrinsic Lipschitzgraphs. Recall (Section 2.1) that for a measurable subset E ⊆ (cid:72) , we let Per E denote theperimeter measure of E ; this measure is supported on ∂ E , and when E is bounded byan intrinsic Lipschitz graph, it differs from 3–dimensional Hausdorff measure on ∂ E by a multiplicative constant. For any horizontal line L ∈ L , let ∂ H | L E be the measure-theoretic boundary of E in L and let Per E , L be the counting measure on ∂ H | L E . Lemma 9.6.
Fix < λ < . Let ψ : V → (cid:82) be intrinsic λ –Lipschitz and let Γ = Γ ψ be itsintrinsic graph. Let U ⊆ V be a measurable set. For almost every L ∈ L P , the intersectionL ∩ Γ + has locally finite perimeter. If M ⊆ L P is the set of lines that intersect Γ at least twice,then ˆ M Per Γ + , L (cid:161) Π − ( U ) (cid:162) d N P ( L ) (cid:46) λ | U | . (163) Proof.
The measures N P and N are absolutely continuous with respect to each other.Indeed, for each m >
0, if D ⊆ L P is a set of lines with slopes that lie in [ − m , m ], then N P ( D ) (cid:179) m N ( D ). By (56), there is c > A ⊆ (cid:72) ,Per Γ + ( A ) = c ˆ L Per Γ + , L ( A ) d N ( L ).Since Γ + has locally finite perimeter, this implies that for almost every line L ∈ L P , theintersection L ∩ Γ + has locally finite perimeter. For L ∈ L P let m ( L ) be the slope of L as inSection 8. Suppose that p ∈ L ∩ Γ . By (47), if | m ( L ) | > λ / (cid:112) − λ , then L ⊆ p · Cone λ andthus L intersects Γ exactly once. Consequently, | m ( M ) | (cid:201) λ / (cid:112) − λ for every M ∈ M , andhence N P ( D ) (cid:179) λ N P ( D ) for every measurable D ⊆ M . So, by (56) and Lemma 2.5, ˆ M Per Γ + , L (cid:161) Π − ( U ) (cid:162) d N P ( L ) (cid:46) λ Per Γ + (cid:161) Π − ( U ) (cid:162) (cid:179) λ | U | . (cid:3) Proof of Lemma 9.2.
For S ⊆ (cid:82) and R >
0, let I ( S ) and (cid:98) ω S , R = ω S , R + ω (cid:82) (cid:224) S , R be as in Sec-tion 8.1. Divide I ( S ) according to the length of the intervals as follows. ∀ j ∈ (cid:90) , C j ( S ) def = (cid:110) I ∈ I ( S ) : 2 − j − < | I | (cid:201) − j (cid:111) .Let E j ( S ) ⊆ (cid:82) be the set of endpoints of the intervals in C j ( S ). Let λ S , j be the countingmeasure on E j ( S ) and let (cid:98) λ j ( S ) def = λ S , j + λ (cid:82) (cid:224) S , j (cid:80) j ∈ (cid:90) (cid:98) λ S , j (cid:201) Per S . (This isn’t necessarily an equality as the left hand side is influencedonly by bounded intervals while the right hand side could have a contribution from rays.) For each k ∈ (cid:90) , the measure (cid:98) ω S ,2 − k is a point measure supported on the set ∞ (cid:91) j = k (cid:161) E j ( S ) ∪ E j ( (cid:82) (cid:224) S ) (cid:162) ,that weights each point according to the lengths of the intervals it bounds. In particular,supp (cid:161) (cid:98) ω S ,2 − k − (cid:98) ω S ,2 − k − (cid:162) ⊆ E k ( S ) ∪ E k ( (cid:82) (cid:224) S ),and ∀ p ∈ E k ( S ) ∪ E k ( (cid:82) (cid:224) S ), 2 − k − (cid:201) (cid:98) ω S ,2 − k ( p ) − (cid:98) ω S ,2 − k − ( p ) (cid:201) − k .Consequently, if we denote (cid:98) κ S , k def = k (cid:161) (cid:98) ω S ,2 − k − (cid:98) ω S ,2 − k − (cid:162) ,then (cid:98) κ S , j (cid:179) (cid:98) λ S , j and (cid:88) j ∈ (cid:90) (cid:98) κ S , j = (cid:88) j ∈ (cid:90) j + (cid:98) ω S ,2 − j − (cid:88) j ∈ (cid:90) j (cid:98) ω S ,2 − j = (cid:88) j ∈ (cid:90) j (cid:98) ω S ,2 − j .It follows that (cid:88) j ∈ (cid:90) j (cid:98) ω S ,2 − j (cid:179) (cid:88) j ∈ (cid:90) (cid:98) λ S , j (cid:46) Per S . (164)For every measurable E ⊆ (cid:72) and U ⊆ V , and every L ∈ L P , we have (cid:88) j ∈ (cid:90) j (cid:98) ω PE ,2 − j ( U , L ) (154) ∧ (164) (cid:46) Per x ( E ∩ L ) (cid:161) x ( Π − ( U ) ∩ L ) (cid:162) = Per E , L (cid:161) Π − ( U ) (cid:162) . (165)Let M ⊆ L P be the set of lines that intersect Γ at least twice. If L ∈ L P (cid:224) M , then I ( L ∩ Γ + )consists of infinite rays, so (cid:98) ω Γ + , R ( U , L ) = U ⊆ V . Thus, (cid:88) j ∈ (cid:90) Ω P Γ + ,2 − j ( U ) (155) = (cid:88) j ∈ (cid:90) ˆ M j (cid:98) ω P Γ + ,2 − j ( U , L ) d N P ( L ) (165) (cid:46) ˆ M Per Γ + , L ( Π − ( U )) d N P ( L ) (163) (cid:46) λ | U | . (cid:3)
10. O
UTLINE OF PROOF OF P ROPOSITION r > h ∈ (cid:72) , let B r ( h ) ⊆ (cid:72) be the convex hull of B r ( h ) (as a subset of (cid:82) ); when h is omitted, we take it to be . Theconvex hull of B r with respect to the horizontal lines or with respect to all lines in (cid:82) isthe same, and B r ⊆ B r . Proposition 10.1.
Let E ⊆ (cid:72) be a measurable set. For any ε > , there are ν , R > suchthat if E ⊆ (cid:72) is ( ν (cid:48) , R (cid:48) ) –extended monotone on B for some ν (cid:48) , R (cid:48) > that satisfy R (cid:48) (cid:202) R and ν (cid:48) R (cid:48) (cid:201) ν R, then there is a plane P ⊆ (cid:72) such that | B ∩ ( P + (cid:52) E ) | < ε . If Γ is an intrinsic Lipschitz graph and E = Γ + , then we can take P to be a vertical plane. OLIATED CORONA DECOMPOSITIONS 79
Proposition 10.1 is in the spirit of the stability theorem for monotone sets that wasproved in [CKN11], though here we do not need to obtain an explicit dependence of ν , R on ε (in [CKN11] it was important to get power-type dependence). The lack of explicitdependence lets us use a compactness argument that was not available in the contextof [CKN11]. At the same time, Theorem 4.3 of [CKN11] states that if the nonmonotonicityof E is small on the unit ball B , then there is a smaller ball B ε on which E is O ( ε )–close to a plane, while Proposition 10.1 assumes a stronger hypothesis, namely thatENM E , R ( B ) < ν , and obtaines the stronger conclusion that E is close to a plane on thesame ball B . This stronger conclusion is crucial for covering arguments like those inSection 9.In the second step, we prove parts 1 and 2 of Proposition 7.2. By Remark 4.3, after astretch, shear, and translation, we may suppose that Q is a rectilinear pseudoquad for Γ that is close to [ −
1, 1] and Γ is ( η , R )–paramonotone on r Q . For any given c , if R issufficiently large, η is sufficiently small, and Π ( B c ) ⊆ r Q , then, by Lemma 8.7, Γ + has smallextended nonmonotonicity on B c , so Γ + is close to a half-space P + on B c .The difficulty with using this conclusion to prove Proposition 7.2 is that when wenormalized Q , we changed the intrinsic Lipschitz constant of Γ . Consequently, we cannotuse the intrinsic Lipschitz condition to prove the bounds in Proposition 7.2, because thecorresponding function f may take on large values on small subsets of Q . Instead, weintroduce new methods based on analyzing the characteristic curves of Γ .For example, a key step in the proof of part 1 of Proposition 7.2 is to show that (cid:107) f (cid:107) L ( Q ) is bounded. Since f is intrinsic Lipschitz, (cid:107) f (cid:107) L ( Q ) < ∞ , but we need a bound independentof the intrinsic Lipschitz constant. We obtain such a bound by studying how lines intersectthe characteristic curves. Since Q is µ –rectilinear, the top and bottom boundaries of Q are characteristic curves that are close to the top and bottom edges of [ −
1, 1] . If L is ahorizontal line such that Π ( L ) crosses [ −
1, 1] from top to bottom, then Π ( L ) must alsocross the top and bottom boundaries of Q . At these intersection points, the slope of Π ( L ) is less than the slope of the boundary, so the corresponding points of L lie in Γ + . If Γ + ∩ L is close to monotone, then most of the interval between these points lies in Γ + andtherefore, f is bounded on Q ∩ Π ( L ). By integrating over a family of lines that all cross thetop and bottom boundaries, we obtain the desired L bound. Similar arguments basedon characteristic curves lead to part 2 of Proposition 7.2, which completes the proof ofProposition 7.2.11. E XTENDED - MONOTONE SETS ARE CLOSE TO HALF - SPACES
In this section, we will prove Proposition 10.1 by studying limits of ( ε , R )–extendedmonotone sets. Let U ⊆ (cid:72) be measurable and let E , E , . . . ⊆ (cid:72) be a sequence of measur-able sets such that E i is ( i , i )–extended monotone on U . By passing to a subsequence, wemay suppose that E i converges weakly to a function f ∈ L ∞ ( (cid:72) ) taking values in [0, 1]. Wecall f a U –LEM (limit of extended monotones) function .One difficulty of studying f is that it need not take values only in {0, 1}. Indeed, theextended monotonicity ENM E i , i ( B ) only depends on the intersection of E i with linesthat pass through B . These lines do not cover all of (cid:72) , so there are regions of (cid:72) where f can take on arbitrary values. Nevertheless, in Section 11.1, we will show that, after changing f on a measure-zeroset, f ( B ) ⊆ {0, 1}. This will follow from the fact that, by Lemma 8.5,lim i →∞ NM E i ( B ) = B convergesto a subset which is monotone on B . If U is an open set, a subset E ⊆ (cid:72) is said to be monotone on U if NM E ( U ) = F ( B ) =
0. A set that is monotone on B need not be a half-space, butwe will show that if F is such a set, then the measure-theoretic boundary ∂ H F is a unionof horizontal lines that has an approximate tangent plane at every point. That is, for any g ∈ ∂ H F , the blowups g · s n , n ( g − ∂ H F ) converge in the Hausdorff metric to a plane T g as n → ∞ . In fact, at all but countably many points g ∈ ∂ H F , there is a unique horizontalline L g through g that is contained in ∂ H F , and T g is the vertical plane containing L g ;in this case, g has an approximate tangent subgroup in the sense of [MSSC10]. At theremaining points, T g is the horizontal plane centered at g .Finally, in Section 11.3, we prove Proposition 10.1. The proof is somewhat involved,but, as an illustration, we consider the case that f = E , where E is precisely ∞ –extendedmonotone on B . That is, for every line L , either B ∩ ∂ ( L ∩ E ) = ∅ or L ∩ E is a monotonesubset of L .We first claim that for every point b ∈ B ∩ ∂ H E , if the approximate tangent plane T b is vertical and H b is the horizontal plane centered at b , then H b ∩ ∂ H E = H b ∩ T b . Let T ± b be the two half-spaces bounded by T b , labeled so that T + b ∩ B r ( b ) approximates E ∩ B r ( b )at small scales. Let L b = H b ∩ T b be the horizontal line in ∂ H E that passes through b andlet L be a line through b that intersects T b transversally. Then E ∩ L is a monotone set with b ∈ ∂ H ( E ∩ L ), so T + b ∩ L ⊆ E ∩ L and T − b ∩ L ⊆ L (cid:224) E . This holds for every horizontal linethrough b except L b , so L b cuts H b into two half-planes P ± = T ± b ∩ H b such that P + ⊆ E and P − ⊆ H b (cid:224) E .When b (cid:48) ∈ L b is close to b , the plane H b (cid:48) intersects H b along L b and the angle betweenthe two planes is small. As above, there are two half-planes P (cid:48)± = T ± b (cid:48) ∩ H b (cid:48) such that P (cid:48)+ ⊆ E and P (cid:48)− ⊆ H b (cid:48) (cid:224) E . As b (cid:48) varies over points close to b , the half-plane P (cid:48)+ varies overhalf-planes close to P + . Therefore P + is in the interior of E , P − is in the exterior, and H b ∩ ∂ H E = L b .Suppose that L and L are two lines in ∂ H E that intersect B , and suppose by way ofcontradiction that they are not coplanar. By the hyperboloid lemma [CK10b, Lemma 2.4](see Lemma 11.1), for any point q ∈ L except possibly a single point, there is a horizontalline M that connects q to a point r in L . Then r ∈ H q ∩ ∂ H E = L , so L and L intersectand are thus coplanar; this is a contradiction. It follows that B ∩ ∂ H E is contained in aplane. The proof of Proposition 10.1 runs along the same lines, but it takes some furthertechnical work to apply the weaker hypothesis that f is merely an LEM function.One of the key tools in the proof is the following “hyperboloid lemma,” which is statedas Lemma 2.4 in [CK10b]. A pair of horizontal lines L , L ∈ L are said to be skew if L and L are disjoint and the projections π ( L ), π ( L ) ⊆ H ∼= (cid:82) are not parallel. OLIATED CORONA DECOMPOSITIONS 81
Lemma 11.1 (Cheeger–Kleiner hyperboloid lemma [CK10b]) . For any L , L ∈ L we have (1) Suppose that the projections π ( L ), π ( L ) are parallel but π ( L ) (cid:54)= π ( L ) . Then everypoint in L can be joined to L by a unique line. In fact, there is a unique fiber π − ( p ) such that every line joining L to L passes through π − ( p ) . Conversely, forevery a ∈ π − ( p ) , there is a unique line joining L to L that passes through a. (2) If L , L are skew, then there is a hyperbola S ⊆ H with asymptotes π ( L ) and π ( L ) such that every tangent line of S has a unique horizontal lift that intersects L andL . If p ∈ H is the intersection between π ( L ) and π ( L ) and a ∈ L is such that π ( a ) (cid:54)= p, then there is a unique horizontal line that connects a to a point in L . Stability of locally monotone sets.
We begin the proof of Proposition 10.1 by usinga compactness argument to prove the following lemma. Throughout what follows, givena measure space ( S , Σ , µ ) and a measure subset Ω ∈ Σ with µ ( Ω ) >
0, we use the (standard)notation ffl Ω to denote the averaging operator on Ω , i.e., ∀ f ∈ L ( Ω , µ ), Ω f d µ def = µ ( Ω ) ˆ Ω f d µ . Lemma 11.2.
Let U ⊆ (cid:72) be a bounded open set and let E , E , . . . ⊆ (cid:72) be a sequence ofmeasurable sets such that NM E i ( U ) < i for every i ∈ (cid:78) . There is a subsequence ( E i j ) j ∈ (cid:78) anda set F ⊆ U that is monotone on U such that lim j →∞ (cid:175)(cid:175) ( E i j ∩ U ) (cid:52) F (cid:175)(cid:175) = .It follows that for any ε > , there is a δ > such that if E ⊆ (cid:72) is a measurable set and NM E ( U ) < δ , then there is a set F ⊆ U such that | ( E ∩ U ) (cid:52) F | < ε and F is monotone on U .Proof. After passing to a subsequence, we may suppose that the characteristic functions E i converge weakly to a function f ∈ L ∞ ( U ) taking values in [0, 1]. We claim that f is acharacteristic function.By Theorem 4.3 of [CKN11] (see also [NY18, Theorem 63]), for every ε >
0, there are c ( ε ) > δ ( ε ) > p ∈ (cid:72) , α >
0, and NM E ( B α ( p )) < δ ( ε ) α − , then there is ahalf-space P + such that B c ( ε ) α ( p ) | P + ( h ) − E ( h ) | d H ( h ) < ε . (166)(The hypothesis in [CKN11] is that NM E ( B α ( p )) < δ ( ε ), but our definition of NM E ( B α ( p ))differs from the definition in [CKN11] by a normalization factor.)By the Lebesgue density theorem, for almost every point p ∈ U , we havelim s → B s ( p ) | f ( h ) − f ( p ) | d H ( h ) =
0. (167)Let p be such a point and let r > B r ( p ) ⊆ U . By (166), for any 0 < s < r , any ε >
0, and any sufficiently large i ∈ (cid:78) (depending on s , ε ), there is a half-space Q + i with B c ( ε ) s ( p ) | Q + i ( h ) − E i ( h ) | d H ( h ) < ε .Choose a half-space Q + such that for infinitely many i ∈ (cid:78) we have B c ( ε ) s ( p ) | Q + ( h ) − E i ( h ) | d H ( h ) < ε . Then B c ( ε ) s ( p ) | Q + ( h ) − f ( h ) | d H ( h ) < ε . (168)Since the function ( x ∈ [0, 1]) (cid:55)→ x (1 − x ) is nonnegative and 1–Lipschitz, B c ( ε ) s ( p ) f ( h ) (cid:161) − f ( h ) (cid:162) d H ( h ) (cid:201) ε + B c ( ε ) s ( p ) Q + ( h ) (cid:161) − Q + ( h ) (cid:162) d H ( h ) = ε .This holds for all 0 < s < r , solim s → B s ( p ) f ( h )(1 − f ( h )) d H ( h ) = f ( p )(1 − f ( p )) = f ( p ) ∈ {0, 1}.Thus f is equivalent to a characteristic function on U . Let F = f − (1). By weak conver-gence, lim i →∞ | U ∩ ( E i (cid:52) F ) | =
0. For any i ∈ (cid:78) ,NM F ( U ) = ˆ L NM F ∩ L ( U ∩ L ) d N ( L ) (cid:201) ˆ L (cid:179) NM E i ∩ L ( U ∩ L ) + H (cid:161) U ∩ L ∩ ( E i (cid:52) F ) (cid:162)(cid:180) d N ( L ) (cid:46) NM E i ( U ) + | U ∩ ( E i (cid:52) F ) | .Both terms on the right go to zero as i → ∞ , so NM F ( U ) =
0, i.e., F is monotone on U . (cid:3) Corollary 11.3.
Let U ⊆ (cid:72) be a convex bounded open set and let f : (cid:72) → [0, 1] be a U –LEMfunction. There is a monotone set E ⊆ U such that f | U = E up to a measure-zero set.Proof. Suppose that E , E , . . . ⊆ (cid:72) are measurable, E i is ( i , i )–extended monotone on U for all i ∈ (cid:78) , and E i converges weakly to f . By Lemma 8.5, for i > diam U we haveNM E i ( U ) (cid:201) ENM E i , i ( U ) (cid:201) i .So, by Lemma 11.2, f | U = F for some set F ⊆ U that is monotone on U . (cid:3) Locally monotone sets are bounded by rectifiable ruled surfaces.
Here we willdescribe sets that are monotone on an open subset of (cid:72) , which we call locally monotonesets . Note that a locally monotone set need not be a half-space; see Example 9.1 of [CKN11].Regardless, we use the techniques developed in [CK10b] and [CKN11] to describe suchsets.
Proposition 11.4.
Let E ⊆ (cid:72) be a measurable set that is monotone on a convex open setU ⊆ (cid:72) . Then (1) U ∩ ∂ H E has empty interior. (2)
For every p ∈ U ∩ ∂ H E , there is a horizontal line L through p with U ∩ L ⊆ ∂ H E .If this line is not unique, then U ∩ H p ⊆ ∂ H E , and we call p a characteristic point . (3) ∂ H E has an approximate tangent plane T p at every p ∈ U ∩ ∂ H E . The plane T p is horizontal if and only if p is a characteristic point, and there are only countablymany characteristic points in U . OLIATED CORONA DECOMPOSITIONS 83 (4)
If T p is vertical, then it divides (cid:72) into two half-spaces T + p and T − p such that thefollowing holds. For t > , let T ± p ( t ) = { v ∈ T ± p : d ( v , T p ) > t } and for any ε > , letW ± ε , t = B t ( p ) ∩ T ± p ( ε t ). For any < ε < , there is r > such that if < α < r , thenW + ε , α ⊆ int H ( E ) and W − ε , α ⊆ int H ( (cid:72) (cid:224) E ).We rely on the following proposition and lemmas, which adapt results from [CK10b]. Proposition 11.5 (generalization of [CK10b, Proposition 5.8]) . Let E ⊆ (cid:72) be a measurableset that is monotone on a convex open set U ⊆ (cid:72) . Let L be a horizontal line and let p , q ∈ Lbe points such that p (cid:54)= q and the segment [ p , q ] ⊆ L is contained in U . We choose the linearorder on L so that p < q. Suppose that q ∈ int H ( E ) . (1) If p ∈ supp( E ) and r ∈ L ∩ U satisfies p < r < q, then r ∈ int H ( E ) . (2) If p ∈ supp( (cid:72) (cid:224) E ) and r ∈ L ∩ U satisfies p < q < r , then r ∈ int H ( E ) .Proof. In [CK10b], this proposition is proved in the case that U = (cid:72) . We will sketch howthe reasoning in [CK10b] actually proves the full statement of Proposition 11.5.Both cases of Proposition 11.5 are proved by considering paths from a neighborhoodof p to a neighborhood of r . In case (1), one chooses a point v ∈ L such that p < v < r < q and considers once-broken geodesics consisting of line segments from a neighborhood of p to a neighborhood of v and then to a neighborhood of r . We can extend the segments ofthis broken geodesic to line segments connecting a neighborhood of p to a neighborhoodof q . The monotonicity condition is used to show that if I is not in a measure-zero setof lines, then the intersection I ∩ E is monotone on I . If these neighborhoods are smallenough, then I ⊆ U and I ∩ E remains monotone on I , so the desired conclusion holds.In case (2), one chooses a point v ∈ L such that p < q < r < v and considers brokengeodesics consisting of line segments from a neighborhood of p to a neighborhood of v toa neighborhood of r . The monotonicity condition shows that if I is a segment connectinga neighborhood of p to a neighborhood of v , then I ∩ E is monotone on I . As above, if v is sufficiently close to r and if the neighborhoods are sufficiently small, then any suchsegment I is contained in U and thus I ∩ E is monotone on I . This completes the proof ofProposition 11.5. (cid:3) By Proposition 11.5 and the proof of [CK10b, Lemma 4.8], we get the following lemma.
Lemma 11.6 (generalization of [CK10b, Lemma 4.8]) . Let E ⊆ (cid:72) be a measurable set thatis monotone on a convex open set U ⊆ (cid:72) . If L is a horizontal line such that L ∩ U containsat least two points of ∂ H E , then L ∩ U ⊆ ∂ H E .Proof.
Let I = L ∩ U . Let p , q ∈ I ∩ ∂ H E be distinct points. Choose the linear orderon L so that p < q . Let r ∈ I be such that q < r . By part (1) of Proposition 11.5, if r ∈ int H ( E ), then q ∈ int H ( E ), which is a contradiction. Likewise, if r ∈ int H ( (cid:72) (cid:224) E ),then q ∈ int H ( (cid:72) (cid:224) E ), which is a contradiction, so r ∈ ∂ µ E . Thus [ q , ∞ ) ∩ I ⊆ ∂ H E . Bysymmetry, I (cid:224) ( p , q ) = I ∩ (cid:161) ( −∞ , p ] ∪ [ q , ∞ ) (cid:162) ⊆ ∂ H E for any distinct points p , q ∈ I ∩ ∂ H E .Let r , s ∈ I ∩ [ q , ∞ ) be such that r < s . Then r , s ∈ I ∩ ∂ H E , so I (cid:224) ( r , s ) ⊆ ∂ H E . Since ( r , s )and ( p , q ) are disjoint, I ⊆ ∂ H E . (cid:3) Likewise, the following lemma is based on the proof of Lemma 4.9 of [CK10b].
Lemma 11.7 (generalization of [CK10b, Lemma 4.9]) . Let E ⊆ (cid:72) be a measurable set thatis monotone on a convex open set U ⊆ (cid:72) . For every p ∈ U ∩ ∂ H E , there is an open linesegment I such that p ∈ I and I ⊆ ∂ H E .Proof.
Let B ⊆ U be a ball centered at p and let H p be the horizontal plane centered at p . Let B (cid:48) = B (cid:224) { p }. Suppose by way of contradiction that H p ∩ B (cid:48) ∩ ∂ H E = ∅ . Since H p ∩ B (cid:48) is connected, we have H p ∩ B (cid:48) ⊆ int H ( E ) or H p ∩ B (cid:48) ⊆ int H ( (cid:72) (cid:224) E ). Without lossof generality, we assume that H p ∩ B (cid:48) ⊆ int H ( E ).Let L be a line through p and let q , r ∈ L ∩ B be two points on opposite sides of p .Then q , r ∈ int H ( E ), so by part (1) of Proposition 11.5, we have p ∈ int H ( E ). This is acontradiction, so there exists some point q lying in H p ∩ B (cid:48) ∩ ∂ H E . By Lemma 11.6, anopen segment of the line containing p and q is contained in ∂ H E , as desired. (cid:3) The fact that U ∩ ∂ H E has empty interior also follows from the techniques of [CK10b]. Lemma 11.8.
If E and U are as in Lemma 11.7, then U ∩ ∂ H E has empty interior.Proof.
The measure-theoretic version of Lemma 4.12 of [CK10b], whose proof appearsin (part (4) of ) the proof of Theorem 5.1 of [CK10b], asserts that if F ⊆ (cid:72) is monotoneon (cid:72) , then ∂ H F (cid:54)= (cid:72) . That proof relies on the monotonicity of a configuration of linesegments, and it directly shows that there is a large enough universal constant r > B r ( ). Consequently, if B r ( ) ⊆ U , then there is apoint p ∈ B r ( ) such that p (cid:54)∈ ∂ H E . By rescaling and translation, this is true with B r ( )replaced by an arbitrary ball, and thus int H ( E ) ∪ int H ( (cid:82) (cid:224) E ) is dense in U . (cid:3) Lemma 11.8 proves part (1) of Proposition 11.4. Lemma 11.6 and Lemma 11.7 implythe first half of part (2) of Proposition 11.4. Before proving the rest of Proposition 11.4, wemake the following definition.
Definition 11.9.
Let U ⊆ (cid:72) be a convex open set and let A ⊆ (cid:72) . We say that A is U –ruled if for all L ∈ L , if L ∩ U intersects A in two points, then L ∩ U ⊆ A. We call such a line L aU – ruling of A.
Lemmas 11.6–11.8 imply that U ∩ ∂ H E is U –ruled and has empty interior. We willprove the rest of Proposition 11.4 by studying lines in the boundary of such a set. Thefollowing lemma is based on Step B3 in Section 8.2 of [CKN11], which shows that theboundary of a monotone set cannot contain skew lines. Lemma 11.10.
Let M be the line 〈 X 〉 and let M be the line Z 〈 Y 〉 . There exists r > suchthat any B r –ruled set containing ( M ∪ M ) ∩ B r has nonempty interior.Proof. Let r be large enough that [ −
2, 2] ⊆ B r . Let E be a B r –ruled set with B r –rulings M , M ∈ L . By Lemma 11.1, there is a hyperbola S ⊆ H , asymptotic to the x –axis and the y –axis, such that every tangent line of S has a unique horizontal lift that intersects M and M . Indeed, for every t (cid:54)=
0, the points X t ∈ M and Z Y t ∈ M are connected by ahorizontal line ∀ u ∈ (cid:82) , L t ( u ) def = X t (cid:181) − t , 2 t , 0 (cid:182) u . OLIATED CORONA DECOMPOSITIONS 85
For t ∈ [ − − ∪ [1, 2] and u ∈ [0, 1], the point L t ( u ) lies on a horizontal line segmentconnecting two points in E , so L t ( u ) ∈ E . The resulting family of points S def = { L t ( u ) : t ∈ [ − − ∪ [1, 2], u ∈ [0, 1]} ⊆ E consists of two disjoint embedded surfaces. Let w = L (cid:112) (cid:181) (cid:182) = (cid:195) (cid:112)
22 , (cid:112)
22 , 12 (cid:33) .The vertical plane P containing L (cid:112) is tangent to S (considered as a smooth surface in (cid:82) )at w . For every w = ( x , y , z ) ∈ S , the line from w to s − − ( w ) = ( − x , − y , z ) is a horizontalline intersecting the z –axis. Let w (cid:48) = s − − ( w ) and let L be the horizontal line from w to w (cid:48) . Then L intersects S transversally twice, at w and w (cid:48) , so L ∩ B r ⊆ E . Anyhorizontal line close to L also intersects S twice, so E contains a neighborhood of the set L ∩ B r , and thus it has nonempty interior. (cid:3) For any pair of skew lines, there is an automorphism of (cid:72) that sends them to M and M . The next lemma uses this fact to show that nearby skew lines in ∂ H E must havenearly parallel projections. For φ ∈ (cid:82) , let R φ : (cid:72) → (cid:72) be the rotation by angle φ around the z –axis. Lemma 11.11.
Let r be as in Lemma 11.10. Let L , L ∈ L be skew lines and let p ∈ H be the intersection of π ( L ) and π ( L ) . Suppose that the angle between π ( L ) and π ( L ) is θ ∈ (0, π ) . For i ∈ {1, 2} , let q i ∈ (cid:72) be the point where π − ( p ) intersects L i . Suppose thatd ( q , q ) (cid:201) (cid:112) θ r (cid:112) If L , L ∈ L are B ( q ) –rulings of an B ( q ) –ruled set S, then S has nonempty interior.Proof. After applying a translation and rotation and possibly replacing S with s − ( S ), wemay suppose that q = , q = Z h for some h > π ( L ) and π ( L ) form angles of θ with the x –axis. (We cannot control which line forms a positive angle with the x –axisand which line forms a negative angle.) Let t = tan θ ∈ (0, 1) so that the lines π (cid:179) s (cid:112) t , (cid:112) t ( L ) (cid:180) and π (cid:179) s (cid:112) t , (cid:112) t ( L ) (cid:180) are perpendicular. There is an angle φ = ± π such that if f def = R φ ◦ s (cid:112) h , (cid:112) h ◦ s (cid:112) t , (cid:112) t ,then f ( L ) = M and f ( L ) = M , where M , M are the lines in Lemma 11.10. Now, by theball-box inequality and our hypothesis on d ( q , q ),Lip( f − ) = (cid:112) h (cid:112) t (39) (cid:201) d ( q , q ) (cid:112) tan θ /2 (cid:201) d ( q , q ) (cid:112) θ /2 (169) (cid:201) r .Thus, f − ( B r ) ⊆ B r Lip( f − ) ⊆ B , or B r ⊆ f ( B ). Since f ( S ) is a f ( B )–ruled set and M and M are f ( B )–rulings of f ( S ), by Lemma 11.10, f ( S ) has nonempty interior and thus S has nonempty interior. (cid:3) It follows from Lemma 11.8 and Lemma 11.11 that two lines in ∂ H E with differentangles must either intersect or stay at least a definite distance apart. In the terminology of[CKN11], every pair of rulings of ∂ H E must form a degenerate initial condition. Lemma 11.12.
For any ε > , there is δ > such that if S is a B –ruled set with emptyinterior and L , L are B –rulings of S that intersect B δ and such that ∠ ( π ( L ), π ( L )) > ε ,then L and L intersect.Proof. We suppose that 0 < ε < δ = ε r (cid:201) , where r is as in Lemma 11.10.Let p ∈ H be the intersection of the projections π ( L ) and π ( L ). Since π ( B δ ) is the ball B H δ of radius δ in H , the projections intersect B H δ and form an angle of at least ε , so (cid:107) p (cid:107) (cid:201) δ sin ε (cid:201) δε <
14 .For i ∈ {1, 2}, let q i = π − ( p ) ∩ L i . By assumption, L and L intersect B δ ⊆ B δ , so if b i ∈ L i ∩ B δ , then d ( , q i ) (cid:201) d ( , b i ) + d ( b i , q i ) = d ( , b i ) +(cid:107) π ( b i ) − π ( q i ) (cid:107) (cid:201) d ( , b i ) +(cid:107) π ( b i ) (cid:107)+(cid:107) p (cid:107) (cid:201) δ +(cid:107) p (cid:107) .In particular, d ( , b i ) (cid:201) . Hence B ( q ) ⊆ B , so S is a B ( q )–ruled set. Further, d ( q , q ) (cid:201) (cid:107) p (cid:107) + δ < δε (cid:201) (cid:112) ε r .Because S has empty interior, Lemma 11.11 implies that L and L cannot be skew lines,and must therefore intersect. (cid:3) The next lemma completes the proof of part (2) of Proposition 11.4.
Lemma 11.13.
Suppose that U is an open set and that E ⊆ (cid:72) is monotone on U . Let p ∈ U ,and let L and L be two distinct U –rulings of ∂ H E that intersect at p. Then there isa neighborhood A containing p such that A ∩ ∂ H E = A ∩ H p , where we recall that H p denotes the horizontal plane through p. If U is convex, we can take A = U .Proof.
After translating and applying an automorphism, we may suppose that p = , B ⊆ U , and that L and L are the x –axis and y –axis, respectively. Set ε = π and let δ > q ∈ B δ ∩ ∂ H E . By Lemma 11.7, ∂ H E has a ruling M q that passes through q . Wewill show that M q intersects both L and L and that any such line passes through p .For any horizontal line L , let L = π ( L ). Either ∠ ( L , M q ) (cid:202) π or ∠ ( L , M q ) (cid:202) π . Therefore,by Lemma 11.12, M q intersects either L or L . Suppose by way of contradiction that M q intersects L but not L . By Lemma 11.12, this implies that ∠ ( L , M q ) (cid:201) ε . Let r be theintersection of M q with L and let t = d ( p , r ) (see Figure 4). Straightforward trigonometryshows that t < δ and that the intersection of M q and L , if it exists, is at least t cot ε > t from p .Let a be a point on L that is distance t from p . By Lemma 11.1, there is a unique point b ∈ M q such that there is a horizontal line N that passes through a and b . Since a ∈ B δ ( p ),this line intersects B δ ( p ), and since a , b ∈ ∂ H E , N is a ruling of ∂ H E . The points r , p , a , OLIATED CORONA DECOMPOSITIONS 87 r b NM q L pa L F IGURE
4. If line M q intersects the y –axis L but not the x –axis L , theremust be a line N intersecting L and M q as seen above. Lines above areprojected to H by π .and b are the vertices of a quadrilateral Q in (cid:72) whose sides are horizontal lines, so theprojection π ( Q ) has zero signed area. It follows that N crosses L between p and r andthus that ∠ ( L , N ) > π > ε . By Lemma 11.12, N intersects both L and L .If a line K intersects L and L , then K , L , and L must all intersect at p ; otherwise, theprojections of K , L , and L to H would contain a non-degenerate triangle that lifts to ahorizontal closed curve in (cid:72) , but this is impossible since the signed area of the projectionof a horizontal closed curve must vanish. Therefore, N passes through p , which is tosay N = L , but by assumption, L does not intersect M q . This is a contradiction, so M q intersects L and L and thus goes through p .Hence, every point q ∈ B δ ∩ ∂ H E lies on the horizontal plane H p through p . Themeasure-theoretic boundary of E disconnects B δ , so B δ ∩ ∂ H E = B δ ∩ H p .If U is convex, then U ∩ ∂ H E is U –ruled. Any line L through p intersects U ∩ ∂ H E inat least two points, so U ∩ L ⊆ ∂ H E . The union of all such lines is H p . (cid:3) Finally, we prove parts (3) and (4) of Proposition 11.4.
Proof of parts (3) and (4) of Proposition 11.4.
Due to Lemma 11.13, if p is a characteristicpoint, then ∂ H E has a horizontal approximate tangent plane at p . Lemma 11.13 alsoimplies that if p is a characteristic point, then there is a ball B such that B contains nocharacteristic points other than p . That is, the characteristic points form a discrete subsetof (cid:72) ; since (cid:72) is separable, there are only countably many characteristic points.Let p ∈ U ∩ ∂ H E be a non-characteristic point, so that there is a unique line L through p . Let V be the vertical plane that contains L . Fix 0 < ε < . We claim that there is r > < α (cid:201) r , then B α ( p ) ∩ ∂ H E is contained in the εα –neighborhood of V .We translate, rotate, and rescale so that p = , L is the x –axis, and B is a subset of U that contains no characteristic points. Then V = V is the xz –plane. Let Π : (cid:72) → V be theprojection to V along cosets of 〈 Y 〉 , as in Section 2.2, so that Π ( x , y , z ) = ( x , 0, z − x y ).For each point s ∈ B ∩ ∂ H E , there is a unique ruling M s passing through s . ByLemma 11.12, there is δ ∈ (0, 1) such that ∠ ( M s , L ) < ε for every s ∈ B δ ∩ ∂ H E . Let r = min{ δ , ε } and let 0 < α (cid:201) r . Let q ∈ B α ∩ ∂ µ E and suppose by way of contradictionthat d ( q , V ) = | y ( q ) | (cid:202) εα . Without loss of generality, we may suppose that y ( q ) > εα .Let m ∈ (cid:82) be the slope of π ( M q ), so that M q = q · 〈 X + mY 〉 . Let γ ( t ) = q · ( X + mY ) t parametrize M q . Then | m | = | sin ∠ ( M s , L ) | < ε
200 .Since q ∈ B α ⊆ B α , we have Π ( q ) ∈ B α and thus | z ( Π ( q )) | (cid:201) α . By (50), for all t ∈ (cid:82) ,dd t z (cid:179) Π (cid:161) γ ( t ) (cid:162)(cid:180) = − y (cid:161) γ ( t ) (cid:162) = − y ( q ) − mt .Consequently, ∀ t ∈ (cid:82) , z (cid:179) Π (cid:161) γ ( t ) (cid:162)(cid:180) = z ( q ) − y ( q ) t − m t s = αε , it follows that z (cid:179) Π (cid:161) γ ( s ) (cid:162)(cid:180) (cid:201) α − αε s + ε · s (cid:201) − α and z (cid:179) Π (cid:161) γ ( − s ) (cid:162)(cid:180) (cid:202) − α + αε s − ε · s (cid:202) α .So, there is t with | t | < s (cid:201) and z ( Π ( γ ( t ))) =
0, i.e., Π ( γ ( t )) ∈ L . The coset N = γ ( t ) 〈 Y 〉 isthus a horizontal line that intersects M q at γ ( t ) and intersects L at Π ( γ ( t )). Since d (cid:161) , γ ( t ) (cid:162) (cid:201) d ( , q ) + | t | (cid:201) α + (cid:201)
12 ,and d (cid:179) , Π (cid:161) γ ( t ) (cid:162)(cid:180) (cid:201) d (cid:161) , γ ( t ) (cid:162) (cid:201) γ ( t ) and Π ( γ ( t )) belong to B ∩ ∂ H E , so N ∩ B ⊆ ∂ H E . Then M q and N are distinctrulings of ∂ H E passing through γ ( t ), which contradicts the fact that there are no charac-teristic points in B . Therefore, d ( q , V ) < εα for all q ∈ B α ∩ ∂ H E .Let T p = V and let T + p and T − p be the corresponding half-spaces. The argument aboveshows that for any 0 < α (cid:201) r , the sets W ± ε , α are disjoint from ∂ H E , so each set is containedin either int H ( E ) or int H ( (cid:72) (cid:224) E ).Consider W + ε , r and W − ε , r . Every line sufficiently close to the y –axis intersects both ofthese sets, so if both are contained in int H ( E ), then by Proposition 11.5, p ∈ int H ( E ) aswell. Likewise, if both are contained in int H ( (cid:72) (cid:224) E ), then p ∈ int H ( (cid:72) (cid:224) E ). Either of theseconclusions is a contradiction, so one of W + ε , r , W − ε , r is contained in int H ( E ) and the otheris contained in int H ( E ). If necessary, we switch T + p and T − p so that W + ε , r ⊆ int H ( E ).We claim that W + ε , α ⊆ int H ( E ) for every α ∈ (0, r ]. Fix 0 < β (cid:201) r with β < α < β . Then W + ε , α intersects W + ε , β , so if W + ε , β ⊆ int H ( E ), then W + ε , α ⊆ int H ( E ) as well. By induction, W + ε , α ⊆ int H ( E ) for all 0 < α (cid:201) r . Likewise, W − ε , α ⊆ int H ( (cid:72) (cid:224) E ) for all 0 < α (cid:201) r . (cid:3) OLIATED CORONA DECOMPOSITIONS 89
Stability of extended monotone sets.
Here we prove Proposition 10.1. We showthat there are ν > R > E is a set that is ( ν , R )–extended monotone on B , then E is close to a half-space on B . If R (cid:48) (cid:202) R and ν (cid:48) R (cid:48) (cid:201) ν R , then ( ν (cid:48) , R (cid:48) )–extendedmonotonicity implies ( ν , R )–extended monotonicity, so this implies the full proposition.To prove this, it suffices to show that if f is a B –LEM function, then f | B is the charac-teristic function of a half-space. Suppose that f is a weak limit of a sequence ( E i ) i , where E , E , . . . ⊆ (cid:72) are sets such that E i is ( i , i )–extended monotone on B . By Corollary 11.3, f | B is the characteristic function of a locally monotone subset F ⊆ B , but this result onlyuses the fact that each E i is i –monotone on B . In this section, we improve Corollary 11.3by using the stronger hypothesis that the E i are extended monotone sets.The first issue is that ENM E i , R ( B ) only depends on the intersection of E i with linesthrough B . These lines don’t cover all of (cid:72) , so a B –LEM function need not take values in{0, 1} outside B . The following lemma shows that it is takes values in {0, 1} on lines thatintersect the boundary of F transversally. For p ∈ (cid:72) and V ∈ H a horizontal vector, thecoset p 〈 V 〉 is a horizontal line. Let p 〈 V 〉 + = { pV t : t >
0} and let p 〈 V 〉 − = { pV t : t < Lemma 11.14.
Let f be a B –LEM function and let F = f − (1) ∩ B be the correspondinglocally monotone set. Let p ∈ B ∩ ∂ H F be a point with a vertical approximate tangentplane T p and let V ∈ H p be a horizontal vector pointing into T + p . Then,p 〈 V 〉 + ⊆ int H (cid:161) f − (1) (cid:162) and p 〈 V 〉 − ⊆ int H (cid:161) f − (0) (cid:162) . (170) Proof.
Let E i ⊆ (cid:72) be a sequence of sets such that E i is ( i , i )–monotone on B and E i converges weakly to f . Let L = p 〈 V 〉 , L ± = p 〈 V 〉 ± and θ = ∠ ( V , T p ). Let ε = θ and let W ± ε , t be as in Proposition 11.4. For t > L ± intersects W ± ε , t in an interval of length at least t .Fix t > q = pV t . For the first inclusion in (170), the goal is to demonstrate that q ∈ int H ( f − (1)). Let 0 < α < t be a radius such that B α ( p ) ⊆ B , W + ε , α ⊆ F up to a nullset, and W − ε , α ⊆ (cid:72) (cid:224) F up to a null set. For any δ >
0, let K δ ⊆ L be the set of lines of theform q (cid:48) 〈 V (cid:48) 〉 where q (cid:48) ∈ B δ ( q ) and V (cid:48) ∈ H is a horizontal vector such that ∠ ( V , V (cid:48) ) < δ . For K ∈ K δ , let K ± = K ∩ T ± p .Since the lines K δ are all close to L , there is a δ depending on θ and α such that0 < δ < ε and every line K ∈ K δ intersects both W + ε , α and W − ε , α in intervals of length atleast α . We claim that lim i →∞ H (cid:161) ( (cid:72) (cid:224) E i ) ∩ B δ ( q ) (cid:162) = f = B δ ( q ).For each i ∈ (cid:78) define T i def = (cid:110) K ∈ K δ : H (cid:161) K ∩ B ∩ ( E i (cid:52) F ) (cid:162) < α (cid:111) .By Fubini’s theorem, for any measurable subset A ⊆ (cid:72) and any horizontal vector M ∈ H that is not parallel to T p , we have ˆ T p H ( b 〈 M 〉 ∩ A ) sin (cid:161) ∠ ( M , T p ) (cid:162) d H ( b ) (cid:179) H ( A ). (171) Therefore, lim i →∞ N ( T i ) = N ( K δ ), and for almost every K ∈ T i , H (cid:161) K + ∩ F ∩ B α ( p ) (cid:162) (cid:202) H ( K + ∩ W + ε , α ) > α T i , this implies that H (cid:161) K + ∩ E i ∩ B α ( p ) (cid:162) > α H (cid:161) K − ∩ E ci ∩ B α ( p ) (cid:162) > α S i def = (cid:169) K ∈ T i : H (cid:161) K ∩ B δ ( q ) ∩ ( (cid:72) (cid:224) E i ) (cid:162) > (cid:170) .Suppose that i (cid:202) d ( p , q ) + δ + α and K ∈ S i . By (172), (173), and the definition of S i ,there are disjoint intervals I = K − ∩ B α ( p ), I = K + ∩ B α ( p ), and I = K ∩ B δ ( q ) suchthat: I is between I and I ; I ∪ I ∪ I has diameter at most i ; H ( I ∩ ( (cid:72) (cid:224) E i )) > α ; H ( I ∩ E i ) > α ; and H ( I ∩ ( (cid:72) (cid:224) E i )) >
0. Lemma 8.2 implies that (cid:98) ω E i , i ( B , K ) (cid:202) (cid:98) ω E i , i ( B α ( p ), K ) (cid:202) H ( E i ∩ I )2 (cid:202) α
16 .Hence, α N ( S i ) (cid:201) ˆ L (cid:98) ω E i , i ( B , K ) d N ( K ) = ENM E i , i ( B ) (cid:201) i ,so lim i →∞ N ( S i ) = R i def = (cid:169) K ∈ K δ : H (cid:161) K ∩ B δ ( q ) ∩ ( (cid:72) (cid:224) E i ) (cid:162) > (cid:170) .Then N ( R i ) (cid:201) N ( S i ) + N ( K δ (cid:224) T i ), and so lim i →∞ N ( R i ) =
0. By (171), H (cid:161) B δ ( q ) ∩ ( (cid:72) (cid:224) E i ) (cid:162) (cid:179) δ ˆ K δ H (cid:161) K ∩ B δ ( q ) ∩ ( (cid:72) (cid:224) E i ) (cid:162) d N ( K ) (cid:201) ˆ R i δ d N ( K ),where the last inequality follows from the fact that H ( K ∩ B δ ( q )) (cid:201) δ for any horizontalline K . We therefore conclude as follows.lim i →∞ H (cid:161) B δ ( q ) ∩ ( (cid:72) (cid:224) E i ) (cid:162) (cid:201) lim i →∞ δ N ( R i ) = (cid:3) By Lemma 11.7, B ∩ ∂ H F is a union of line segments. Extended monotonicity impliesthat these line segments can be extended to lines. Lemma 11.15.
Let f be a B –LEM function and let F = f − (1) ∩ B be the correspondinglocally monotone set. Let L be a horizontal line. If an open subinterval I ⊆ L is contained inB ∩ ∂ H F , then L ⊆ ∂ H F .Proof.
By Proposition 11.4, ∂ H F has at most countably many characteristic points. Let p ∈ I be non-characteristic. Then the vertical plane T p containing L is the approximatetangent plane to ∂ H F at p . Recalling that H p is the horizontal plane centered at p , everyhorizontal line through p , other than L itself, intersects ∂ H F transversally at p , so byLemma 11.14, we have T + p ∩ H p ⊆ int H ( F ) and T − p ∩ H p ⊆ int H ( (cid:72) (cid:224) F ). Since L lies inthe closures of T + p ∩ H p and T − p ∩ H p , we have L ⊆ supp( F ) ∩ supp( (cid:72) (cid:224) F ) = ∂ H F . (cid:3) OLIATED CORONA DECOMPOSITIONS 91
Finally, we show that if B ∩ ∂ H F is nonplanar, then we can construct an arrangementof lines that leads to a contradiction. Lemma 11.16.
Let f be a B –LEM function. There is a plane Q ⊆ (cid:72) such that f | B = Q + outside a null set. In fact, the same holds true in a larger set. LetS def = ( Q ∩ B ) H (174) be the union of the horizontal lines intersecting Q ∩ B . Then f | S = Q + outside a null set.Proof. Let F = f − (1) ∩ B be the locally monotone set corresponding to f and supposeby way of contradiction that B ∩ ∂ H F is non-planar. By part (2) of Proposition 11.4 andby Lemma 11.15, for every point p ∈ B ∩ ∂ H F , there is a horizontal line M p through p such that M p ⊆ ∂ H F .Reasoning as in Lemma 4.11 of [CK10b] shows that there are two B –rulings of F thatsatisfy one of the cases of Lemma 11.1, i.e., they are a pair of skew lines or a pair of lineswith distinct parallel projections. Indeed, suppose that J and K are B –rulings of F withparallel projections. If π ( J ) (cid:54)= π ( K ), we are done; otherwise, J and K are contained in avertical plane V . Let L be a B –ruling of F not in V , which exists by the assumed non-planarity. Then L is skew to J or K or parallel to V with a distinct projection. It remains totreat the case when any two B –rulings of F have nonparallel projections. Let J and K betwo such rulings. If J and K are disjoint, we are done, so we suppose J and K intersectat a point p and are thus contained in the horizontal plane H p centered at p . If L is a B –ruling of F that is not contained in H p (it exists by assumed non-planarity), then L intersects H p at a single point other than p , so L is skew to either J or K , as desired.This shows that there are two B –rulings L and L of F that are skew or have distinctparallel projections. Let I = L ∩ B and let p ∈ I be a noncharacteristic point suchthat π ( p ) (cid:54)∈ π ( L ). By Lemma 11.1, there is a horizontal line M that goes through p andintersects L at q . This line is not equal to L , so it intersects ∂ H F transversally at p . ByLemma 11.14, this implies that q ∈ int H ( f − (0)) or q ∈ int H ( f − (1)), but q ∈ L ⊆ ∂ H F ,which is a contradiction. Therefore, B ∩ ∂ H F is planar and there is a plane Q such that F ∩ B = Q + ∩ B up to a null set. Since F takes values in {0, 1} inside B , this implies thefirst part of Lemma 11.16.With S as in (174), take w ∈ Q + ∩ S . Then w lies on a horizontal line that intersects Q ∩ B transversally, and Lemma 11.14 implies that w ∈ int H ( f − (1)). It follows that f = Q + ∩ S and likewise that f = Q − ∩ S . (cid:3) The second part of Proposition 10.1 states that extended monotone intrinsic graphsare close to vertical planes. This follows from the fact that neighborhoods of the center ofa horizontal plane cannot be approximated by intrinsic graphs.
Lemma 11.17.
Let V be the xz–plane and let E , E , . . . ⊆ (cid:72) be a sequence of intrinsicgraphs over V such that E + i is ( i , i ) –extended monotone on B and E + i converges weaklyto a function f ∈ L ∞ ( (cid:72) ) as i → ∞ . There is a vertical plane Q ⊆ (cid:72) such that f | B = Q + outside a null set. Furthermore, if S is as in (174) , then f | S = Q + outside a null set. Proof.
For any intrinsic graph Γ and any g ∈ Γ + , we have g Y t ∈ Γ + for every t >
0. Since H is right-invariant, this implies that for any measurable set U ⊆ (cid:78) and any i ∈ (cid:78) , H (cid:161) U ∩ E + i (cid:162) (cid:201) H (cid:161) U ∩ E + i Y t (cid:162) .Therefore, ˆ U f d H (cid:201) ˆ UY t f d H .Consequently, f ( g ) (cid:201) f ( g Y t ) for almost every ( g , t ) ∈ (cid:72) × (0, ∞ ). (175)If f is almost-surely constant on B , we can take Q to be a vertical plane that does notintersect B . We thus suppose that f | B is not almost-surely constant. By Lemma 11.16,there is a plane Q that satisfies f | S = Q + outside a null set, where S is given in (174).Suppose for contradiction that Q is horizontal. Let c ∈ (cid:72) be such that Q = H c = c H andlet p ∈ Q ∩ int( B ) be such that x ( p ) (cid:54)= x ( c ). Let L be the horizontal line from c to p and let V = ( x V , y V , 0) be the horizontal vector such that p = cV . Set q = cV − = c ( − x V , − y V , 0).We claim that there is ε > pY ± ε , qY ± ε } ⊆ S . Choose ε > pY t ∈ B and r t = c ( x V , y V + t , 0) ∈ B ∩ Q for all t ∈ [ − ε , 2 ε ]. Then r t (cid:181) − x V , − y V − t (cid:182) = c ( x V , y V + t , 0) (cid:181) − x V , − y V − t (cid:182) = c (cid:181) − x V , − y V − t x V t (cid:182) = qY − t .It follows that qY − t ∈ r t H ⊆ S . In particular, qY ± ε ∈ S . At the same time, pY ε and pY − ε are on opposite sides of Q ; equation (175) implies that pY ε ∈ Q + and pY − ε ∈ Q − . Likewise, qY ± ε ∈ Q ± . But since c is between p and q , the points pY ε and qY ε are on opposite sidesof Q , which is a contradiction. Therefore, Q is a vertical plane. (cid:3) Proof of Proposition 10.1.
If the first part of the proposition were false, then there wouldexist ε > E i ) ∞ i = such that for any i ∈ (cid:78) , the set E i is( i , i )–extended monotone on B and | B ∩ ( P + (cid:52) E i ) | > ε for every plane P ⊆ (cid:72) . There isa subsequence ( E i ( j ) ) ∞ j = whose characteristic functions converge weakly to a B –LEMfunction f . By Lemma 11.16, there is a plane Q ⊆ (cid:72) such that f = Q + almost everywhereon B . Then lim j →∞ | B ∩ ( Q + (cid:52) E i ( j ) ) | =
0, which is a contradiction.Similarly, if the second part of the proposition were false, then there would exist ε > E i ) ∞ i = over V such that for any i ∈ (cid:78) , the epigraph E + i is ( i , i )–extended monotone on B and | B ∩ ( P + (cid:52) E + i ) | > ε for every vertical plane P ⊆ (cid:72) .Passing to a subsequence, we may suppose that the indicators E + i converge weakly to a B –LEM function f . By Lemma 11.17, there is a vertical plane Q ⊆ (cid:72) such that f = Q + almost everywhere on B . Then lim i →∞ | B ∩ ( Q + (cid:52) E + i ) | =
0, which is a contradiction. (cid:3)
OLIATED CORONA DECOMPOSITIONS 93 L BOUNDS AND CHARACTERISTIC CURVES ON MONOTONE INTRINSIC GRAPHS
Here we complete the proof of Proposition 7.2, which obtains L bounds for paramono-tone pseudoquads and bounds their characteristic curves.Fix 0 < µ (cid:201) and a µ –rectilinear pseudoquad Q in an intrinsic Lipschitz graph Γ = Γ f .Suppose that Γ is ( η , R )–paramonotone on r Q . By Remark 4.3, we can normalize Q and Γ so that the corresponding parabolic rectangle is the square [ −
1, 1] × {0} × [ −
1, 1]; byLemma 8.6 and the discussion immediately after its proof, the normalized pseudoquadremains paramonotone. So, it suffices to prove Proposition 7.2 for such pseudoquads.For t >
0, denote D t = [ − t , t ] × {0} × [ − t , t ] ⊆ V . By our choice of normalization, wehave tQ = D t . Furthermore, D t ⊆ B t and Π ( B t ) ⊆ D t . We will proceed in several steps.(1) First, we will prove in Lemma 12.2 that there is a universal constant κ > (cid:107) f (cid:107) L ( Q ) (cid:201) κ when η is sufficiently small. This relies on Lemma 12.1 that boundsthe tails of f in regions that are bounded above and below by supercharacteristiccurves (projections of horizontal curves in Γ ∪ Γ + ).(2) Next, we will show that Γ is close to a plane on a ball around the origin. Since (cid:107) f (cid:107) L ( Q ) (cid:201) κ , the intersections Γ + ∩ B κ and Γ − ∩ B κ both have positive measure.For any r >
0, we have Π ( B r ) ⊆ r Q , so ENM Γ + , R ( B r ) (cid:46) η R . When η R is sufficientlysmall and r and R are sufficiently large, Proposition 10.1 implies that there is avertical plane P that intersects B κ and approximates Γ on B r , i.e., H (cid:179) ( Γ + (cid:52) P + ) ∩ B r (cid:180) < ε .Furthermore, since (cid:107) f (cid:107) L ( Q ) (cid:201) κ , the slope and y –intercept of π ( P ) are both atmost some universal constant.We then apply an automorphism that sends P to V . Since the slope and y –intercept of P are bounded, there is a universal constant c > q : (cid:72) → (cid:72) (a composition of a left translation in the y –direction and a shear)such that q ( P ) = V and B c − s − c ⊆ q ( B s ) ⊆ B cs + c for all s >
0. We let ˆ Γ = q ( Γ ),ˆ Q = ˆ q ( Q ) = Π ( q ( Q )), and let ˆ f be such that ˆ Γ = Γ ˆ f . Since q preserves H , H (cid:179) ( ˆ Γ + (cid:52) V + ) ∩ B c − r − c (cid:180) < ε . (176)This inequality controls ˆ f on V ∩ B c − r − c , and we choose r large enough that11 ˆ Q ⊆ B c − r − c .(3) By (176), ˆ ˆ Q max (cid:169) | ˆ f ( p ) | (cid:170) d H ( p ) (cid:201) ε ,so a bound on the tails of ˆ f would lead to a bound on (cid:107) ˆ f (cid:107) L (10 ˆ Q ) . We boundthe tails in Lemma 12.4, by finding supercharacteristic curves above and below10 ˆ Q , then applying Lemma 12.1 again. This implies that (cid:107) ˆ f (cid:107) L (10 ˆ Q ) (cid:46) ε when η issufficiently small, which proves the first part of Proposition 7.2.(4) Finally, we bound the characteristic curves of ˆ Γ in Lemma 12.6, by showing thatif ˆ Γ contains characteristic curves that are not nearly parallel to the x –axis, then either (cid:107) ˆ f (cid:107) L is bounded away from zero or Ω P Γ + , R is bounded away from zero. Thiscompletes the proof of Proposition 7.2.We will use the following notation for horizontal lines. Every horizontal line in L P canbe written uniquely as follows for some for some w = (0, y , z ) ∈ (cid:72) and m ∈ (cid:82) . L w , m def = w 〈 X + mY 〉 .Let ρ L w , m : (cid:82) → L w , m be the following parametrization, so that x ( ρ L ( t )) = t for all t ∈ (cid:82) . ∀ t ∈ (cid:82) , ρ L w , m ( t ) def = w ( X + mY ) t .For every x ∈ (cid:82) define g L w , m ( x ) def = z (cid:179) Π (cid:161) ρ L w , m ( x ) (cid:162)(cid:180) = − m x − y x + z . (177)Note that since L w , m is horizontal, we have y ( ρ L w , m ( x )) = − g (cid:48) L w , m ( x ).12.1. Bounding the tails of f . We start by showing that if Q is a rectilinear pseudoquadfor Γ = Γ f such that Γ + is ( η , R )–paramonotone on r Q , as in Proposition 7.2, and Q isnormalized so that the corresponding parabolic rectangle is a 2 × κ such that (cid:107) f (cid:107) L ( Q ) (cid:201) κ when r and R are sufficientlylarge and η is sufficiently small.This step relies on the following lemma, which will also be used in step 3. A super-characteristic curve (respectively subcharacteristic curve ) for Γ is the projection Π ( γ ) of ahorizontal curve γ : I → (cid:72) such that x ( γ ( t )) = t for all t ∈ I and γ ( I ) ⊆ Γ ∪ Γ + (respectively γ ( I ) ⊆ Γ ∪ Γ − ). Such a curve can be written as a graph of the form { z = g ( x )} ⊆ V , where g : I → (cid:82) is differentiable and g (cid:48) ( x ) (cid:201) − f ( x , g ( x )) for all x ∈ I . We then say that g is a function with supercharacteristic graph . Lemma 12.1.
Let g , g : [ −
2, 2] → (cid:82) be functions with supercharacteristic graphs such that sup g ([ −
2, 2]) (cid:201) inf g ([ −
2, 2]) . For (cid:201) r (cid:201) , letU r = {( x , 0, z ) ∈ V : | x | (cid:201) r and g ( x ) (cid:201) z (cid:201) g ( x )}. Denoting H = max{ (cid:107) g (cid:107) L ∞ ([ − , (cid:107) g (cid:107) L ∞ ([ − } , for any t (cid:202) H we have | { v ∈ U : f ( v ) (cid:202) t } | (cid:46) t Ω P Γ + ,4 ( U ). (178)Once we prove Lemma 12.1, we will apply it to the case that Q approximates [ −
1, 1] and g and g are the lower and upper bounds of Q . Proof.
Fix t (cid:202) H and y , m , z ∈ (cid:82) such that | y − t | < t and | m | < t . Let L = L (0, y , z ), m .For any s ∈ [ −
2, 2] we have (cid:175)(cid:175)(cid:175)(cid:175) g (cid:48) L ( s ) + t (cid:175)(cid:175)(cid:175)(cid:175) = (cid:175)(cid:175)(cid:175)(cid:175) y (cid:161) ρ L ( s ) (cid:162) − t (cid:175)(cid:175)(cid:175)(cid:175) < t − t < g (cid:48) L ( s ) < − t on [ −
2, 2].We claim that for any almost every such L we have (cid:98) ω P Γ + ,4 ( U , L ) (cid:202) H (cid:179) x (cid:161) Γ − ∩ L ∩ Π − ( U ) (cid:162)(cid:180) . (180) OLIATED CORONA DECOMPOSITIONS 95
Suppose that Π ( L ) is transverse to the boundary of U and that L ∩ Γ + has finite perimeter;these are true for almost every L . If Π ( L ) does not intersect U , then the right side of (180)is 0 and the inequality holds trivially. We thus suppose in addition that L intersects U . Inthis case, there is some s ∈ [ −
1, 1] such that | g L ( s ) | (cid:201) H . By (179), we have g L ( − = g L ( s ) − ˆ s − g (cid:48) L ( u ) d u > H + ( s + t (cid:202) − H + H = H ,and g L (2) = g L ( s ) + ˆ s g (cid:48) L ( u ) d u < H − (2 − s ) t (cid:201) H − H = − H .Hence, Π ( L ) crosses U negatively (from top to bottom), as depicted in Figure 12.1.Fix i ∈ {1, 2} and suppose that Π ( L ) crosses the graph of g i negatively at ( t , 0, g L ( t )). Let v = ρ L ( t ) be the point on L over the intersection. Then g L ( t ) = g i ( t ) and g (cid:48) L ( t ) < g (cid:48) i ( t ).Since the graph of g i is supercharacteristic, f ( t , g i ( t )) (cid:201) − g (cid:48) i ( t ), and therefore y ( v ) = − g (cid:48) L ( t ) > − g (cid:48) i ( t ) (cid:202) f (cid:161) t , g i ( t ) (cid:162) = f (cid:161) Π ( v ) (cid:162) .That is, v ∈ Γ + . g g L F IGURE
5. Two characteristic curves g and g and a horizontal line L ,projected to V ; the positive y –axis points toward the reader. Since Π ( L )crosses U negatively, the segments of L at the first and last crossings liein Γ + , so the size of the intersection L ∩ Γ − is bounded by (cid:98) ω P Γ + ,4 ( U , LZ t ).Since Π ( L ) is transverse to the boundary of U , the intersection Π ( L ) ∩ U consists of acollection of intervals. Let [ a , b ], . . . , [ a n , b n ] ⊆ (cid:82) be the disjoint intervals such that x ( Π ( L ) ∩ U ) = [ a , b ] ∪ · · · ∪ [ a n , b n ],and these intervals are in ascending order. The projection Π ( L ) does not intersect theleft or right boundary of U , so Π ( L ) crosses the graph of g or g at each a i or b i . Since g L is decreasing and sup g ([ −
2, 2]) (cid:201) inf g ([ −
2, 2]), the crossings of g all have smaller x –coordinate than the crossings of g .Consider S = int H ( x ( L ∩ Γ + )). Since Π ( L ) crosses the graph of g negatively at a and crosses the graph of g negatively at b n , the argument above implies that a , b n ∈ S .Furthermore, for each i ∈ {1, . . . , n }, one of three cases holds. (1) Π ( L ) crosses the graph of g negatively at a i and positively (from bottom to top)at b i .(2) Π ( L ) crosses the graph of g negatively at a i and crosses the graph of g negativelyat b i .(3) Π ( L ) crosses the graph of g positively at a i and negatively at b i .In each case, a i ∈ S or b i ∈ S . By Lemma 8.2 (applied with [ a , b ] = [ a , b n ]), (cid:98) ω S ,4 ([ a i , b i ]) = (cid:98) ω (cid:82) (cid:224) S ,4 ([ a i , b i ]) (cid:202) H ( x ( Γ − ∩ L ) ∩ [ a i , b i ]).Summing over i ∈ {1, . . . , n }, we find that (cid:98) ω S ,4 (cid:179) n (cid:91) i = [ a i , b i ] (cid:180) = (cid:98) ω P Γ + ,4 ( U , L ) (cid:202) H (cid:179) x (cid:161) Γ − ∩ L ∩ Π − ( U ) (cid:162)(cid:180) . (181)This proves (180).Next, let A = U ∩ f − ([ t , ∞ )). By (179), y ( ρ L ( s )) < t for all s ∈ [ −
2, 2], so if Π ( ρ L ( s )) ∈ A ,then ρ L ( s ) ∈ Γ − . Therefore, by (180),12 H (cid:161) x ( Π ( L ) ∩ A ) (cid:162) (cid:201) H (cid:179) x (cid:161) Γ − ∩ L ∩ Π − ( U ) (cid:162)(cid:180) (cid:201) (cid:98) ω P Γ + ,4 ( U , L ).By Fubini’s Theorem, for any y and m as above,12 | A | = ˆ (cid:82) H (cid:161) x ( L (0, y , z ), m ∩ A ) (cid:162) d z (cid:201) ˆ (cid:82) (cid:98) ω P Γ + ,4 ( U , L (0, y , z ), m ) d z .Therefore, recalling the definition (155) of Ω P , we have Ω P Γ + ,4 ( U ) = ˆ L (cid:98) ω P Γ + ,4 ( U , L ) d N P ( L ) (cid:202) ˆ t − t ˆ t t ˆ (cid:82) (cid:98) ω P Γ + ,4 ( U , L (0, y , z ), m ) d z d y d m (cid:202) ˆ t − t ˆ t t | A | d y d m = t | A | .That is, | { v ∈ U : f ( v ) (cid:202) t } | (cid:46) t Ω P Γ + ,4 ( U ).Similarly, | { v ∈ U : f ( v ) (cid:201) − t } | (cid:46) t Ω P Γ + ,4 ( U ).This proves (178). (cid:3) The desired bound on (cid:107) f (cid:107) L ( Q ) follows by integrating (178) with respect to t . Lemma 12.2.
Let f : V → (cid:82) be a continuous function and let Γ be its intrinsic graph. Let ( Q , [ −
1, 1] × {0} × [ −
1, 1]) be a –rectilinear pseudoquad for Γ . Suppose that Ω P Γ + ,4 (2 Q ) (cid:201) .There is a universal constant κ > such that (cid:107) f (cid:107) L ( Q ) (cid:201) κ . OLIATED CORONA DECOMPOSITIONS 97
Proof.
Let g and g be the lower and upper bounds of Q and for 0 (cid:201) r (cid:201)
2, let U r be asin Lemma 12.1. Then Q = U and U ⊆ Q . Let H =
2. Since the graphs of g and g aresupercharacteristic and U ⊆ Q , Lemma 12.1 implies that for any t (cid:202) | { v ∈ Q : f ( v ) (cid:202) t } | (cid:46) t − Ω P Γ + ,4 ( U ) (cid:201) t − Ω P Γ + ,4 (2 Q ) (cid:201) t − .Since the graphs of g and g are also subcharacteristic, for any t (cid:202)
16 we also have | { v ∈ U : f ( v ) (cid:201) − t } | (cid:46) t − .Then (cid:107) f (cid:107) L ( Q ) = ˆ ∞ | { v ∈ Q : | f ( v ) | (cid:202) t } | d t (cid:46) | Q | + ˆ ∞ t − d t (cid:46) (cid:3) Constructing the approximating plane.
Now we will use Lemma 12.2 and theresults of Section 11 to show that if Q is a paramonotone pseudoquad for Γ f , then f isclose on Q to an affine function with bounded coefficients. Lemma 12.3.
Let κ > be the constant in Lemma 12.2, and let C = κ . For any < ε < and r (cid:202) κ + , there are < η < and R > with the following property.Let Γ = Γ f be an intrinsic graph such that ( Q , [ −
1, 1] × {0} × [ −
1, 1]) is a –rectilinearpseudoquad for Γ . Let g and g be the lower and upper bounds of Q, respectively. If Q is ( η , R ) –paramonotone on r Q, then there is a vertical plane P ⊆ (cid:72) such that H (cid:179) B r ∩ (cid:161) P + (cid:52) Γ + (cid:162)(cid:180) < ε . (182) Moreover, P is the graph of an an affine function F : V → (cid:82) of the form F ( w ) = a + bx ( w ) ,whose coefficients satisfy max{ | a | , | b | } (cid:201) C .Proof.
We have δ x ( Q ) = α ( Q ) = (cid:112)
2. Also, 2 (cid:201) | Q | (cid:201)
6. Hence, recalling (147), if Γ is( η , R )–paramonotone on r Q , then assuming R (cid:202) η R < Ω P Γ + ,4 (2 Q ) (cid:201) R Ω P Γ + ,2 R ( r Q ) (cid:201) R ηα ( Q ) − | Q | (cid:201) R η < (cid:107) f (cid:107) L ( Q ) < κ .Since Π ( B r ) ⊆ r Q , (158) implies thatENM Γ + ,2 R ( B r ) (cid:46) η R .By Proposition 10.1, when R is sufficiently large and η R is sufficiently small, there is ahalf-space P + bounded by a vertical plane such that H (cid:179) B r ∩ (cid:161) P + (cid:52) Γ + (cid:162)(cid:180) < ε .If necessary, we may rotate P infinitesimally around the z –axis so that it is not perpendic-ular to V . Then P is the graph of an affine function F : V → (cid:82) . Let a , b ∈ (cid:82) be such that F ( w ) = a + bx ( w ) for all w ∈ V .For all w ∈ V , let ¯ f ( w ) (respectively ¯ F ( w )) be the element of [ − κ , 2 κ ] that is closest to f ( w ) (respectively F ( w )). Since r (cid:202) κ +
6, the intrinsic graphs of ¯ F and ¯ f over Q both liein B r . Therefore, (cid:176)(cid:176) ¯ F − ¯ f (cid:176)(cid:176) L ( Q ) (cid:201) H (cid:179) B r ∩ ( Γ ¯ f (cid:52) Γ ¯ F ) (cid:180) (cid:201) H (cid:179) B r ∩ ( Γ f (cid:52) Γ F ) (cid:180) (cid:201) ε , and thus (cid:176)(cid:176) ¯ F (cid:176)(cid:176) L ( Q ) (cid:201) ε + (cid:176)(cid:176) ¯ f (cid:176)(cid:176) L ( Q ) (cid:201) ε + (cid:107) f (cid:107) L ( Q ) (cid:201) κ . (183) F is affine and [ −
1, 1] × {0} × [ − , ] ⊆ Q , so | { q ∈ Q : | F ( q ) | > κ } | > | a | > κ or | b | > κ ,which implies that (cid:107) ¯ F (cid:107) L ( Q ) > κ in contradiction to (183). So, max{ | a | , | b | } (cid:201) κ . (cid:3) We will next use Lemma 12.3 to construct a new intrinsic Lipschitz graph ˆ Γ that isclose to V on a ball around . Let 0 < ε < r > η , R , C , Γ , f , Q be as in Lemma 12.3, so that there is a vertical plane P approximating Q that is the graph of an affine function F ( w ) = a + bx ( w ) with max{ | a | , | b | } (cid:201) C .Let q = q a , b : (cid:72) → (cid:72) be the map given by ∀ ( x , y , z ) ∈ (cid:72) , q ( x , y , z ) def = Y − a ( x , y − bx , z ) = (cid:179) x , y − a − bx , z + ax (cid:180) .This is a shear map that preserves the x –coordinate and sends P to V . Let ˆ q : V → V bethe map that q induces on V , i.e., ∀ x , z ∈ (cid:82) , ˆ q ( x , 0, z ) = Π (cid:161) q ( x , 0, z ) (cid:162) = (cid:181) x , 0, z + ax + b x (cid:182) . (184)Let ˆ Γ = q ( Γ ) and ˆ Q = ˆ q ( Q ). By Lemma 2.9, ˆ Q is a pseudoquad for ˆ Γ that contains andˆ Γ = Γ ˆ f , where ˆ f ( v ) = f ( ˆ q − ( v )) − a − bx ( v ) = f ( ˆ q − ( v )) − F ( ˆ q − ( v )).Since a , b ∈ [ − C , C ], there is a universal constant c > s > D c − s − c ⊆ ˆ q ( D s ) = s ˆ Q ⊆ D cs + c , (185)and B c − s − c ⊆ q ( B s ) ⊆ B cs + c . (186)Bounds on Γ and Q correspond directly to bounds on ˆ Γ and ˆ Q . For example, shearmaps preserve H , so H (cid:161) B c − r − c ∩ ( V + (cid:52) ˆ Γ + ) (cid:162) (cid:201) H (cid:161) q ( B r ) ∩ ( V + (cid:52) ˆ Γ + ) (cid:162) = H (cid:161) B r ∩ ( P + (cid:52) Γ + ) (cid:162) < ε . (187)Maps induced by shears preserve the Lebesgue measure H on V , so by (185), (cid:107) f − F (cid:107) L (10 Q ) = (cid:107) ˆ f (cid:107) L (10 ˆ Q ) (cid:201) (cid:107) ˆ f (cid:107) L ( D c ) , (188)and by Lemma 8.6, ˆ Γ is ( η , R )–paramonotone on r ˆ Q .12.3. Bounding (cid:107) f − F (cid:107) L (10 Q ) . Next, we bound (cid:107) f − F (cid:107) L (10 Q ) . Lemmas 12.2 and 12.3imply that (cid:107) f − F (cid:107) L ( Q ) can be made arbitrarily small, but the proof of Proposition 7.2requires bounds on (cid:107) f − F (cid:107) L (10 Q ) and (cid:107) ˆ f (cid:107) L ( D ) . By (188), these bounds can be achievedby bounding ˆ f on a sufficiently large set. Lemma 12.4.
For any δ > , there is β = β ( δ ) > with the following property. Let ˆ Γ = Γ ˆ f bean intrinsic Lipschitz graph. Let τ > and suppose that H (cid:161) B τ ∩ ( ˆ Γ + (cid:52) V + ) (cid:162) < βτ , (189) and Ω P ˆ Γ + ,48 τ ( D τ ) < βτ . OLIATED CORONA DECOMPOSITIONS 99
Then (cid:107) ˆ f (cid:107) L ( D τ ) (cid:201) δτ .Proof. By a rescaling, it is enough to treat the case τ =
1. Let U def = (cid:110) L (0, y , z ), m : z ∈ [200, 201] ∧ y ∈ [1, 2] ∧ m ∈ (cid:104) − y
20 , − y (cid:105)(cid:111) .We claim that there is some L ∈ U such that the segment Π ( ρ L ([ −
16, 16])) is a supercharac-teristic curve above D . A similar construction will produce a second supercharacteristiccurve below D , so we can use Lemma 12.1 to bound ˆ f from above.We clip ˆ f between −
24 and 24 and call the result h ; that is, for all w ∈ V , let h ( w ) be theelement of [ −
24, 24] that is closest to ˆ f ( w ). For L ∈ L P and t ∈ (cid:82) , let h L ( t ) = h ( Π ( ρ L ( t ))).Define U = (cid:169) L ∈ U : Π ( ρ L ([ −
16, 16])) is supercharacteristic (cid:170) U = (cid:189) L ∈ U : ˆ − | h L ( t ) | d t > (cid:190) U = (cid:110) L ∈ U : (cid:98) ω P ˆ Γ + ,48 ( D , L ) (cid:202) (cid:111) .We claim that almost every L ∈ U is contained in U ∪ U ∪ U .Let L ∈ U and suppose that x ( L ∩ ˆ Γ + ) is a subset of (cid:82) with locally finite perimeter.This is true for almost every L . Suppose that L (cid:54)∈ U ∪ U . Then Π ( ρ L ([ −
16, 16])) is notsupercharacteristic, so there is some a ∈ [ −
16, 16] such that ρ L ( a ) ∈ ˆ Γ − . Let p be theintersection point of L with V ; by our choice of parameters, x ( p ) ∈ [20, 21]. Also, since m < − , we have y ( ρ L ( t )) > for t (cid:201)
19. Since L (cid:54)∈ U , there are b ∈ [16, 17] and b ∈ [18, 19] such that for i ∈ {1, 2} we have h L ( t ) (cid:201) < y ( ρ L ( b i )) and thus ρ L ( b i ) ∈ ˆ Γ + .Similarly, y ( ρ L ( t )) < − for all t (cid:202)
22, so there is c ∈ [22, 23] such that h L ( c ) > y ( ρ L ( c ))and ρ L ( c ) ∈ ˆ Γ − . There is an element of ∂ H x ( L ∩ ˆ Γ + ) in ( a , b ) and another in ( b , c ). Since a , b , b , c ∈ [ −
24, 24], Lemma 8.1 implies that (cid:98) ω P ˆ Γ + ,48 ( D , L ) (cid:202) b − b (cid:202) L ∈ U .Therefore, U ∪ U ∪ U contains all of U except a null set. We will next show that N P ( U ) and N P ( U ) are bounded by multiples of β .Suppose L = L (0, y , z ), m . As in (177), let g L ( t ) = z ( ρ L ( t )) = − m t − y t + z . For every t ∈ [ −
24, 24], we have | g L ( t ) − | (cid:201) + m t + y | t | (cid:201) + + (cid:201) Π ( ρ L ([ −
24, 24])) ⊆ D . Furthermore, D ⊆ B , so for all v ∈ D and t ∈ [ −
24, 24], wehave vY t ∈ B . Thus (cid:107) h (cid:107) L ( D ) (cid:201) H (cid:161) B ∩ ( ˆ Γ + (cid:52) V + ) (cid:162) < β . (191)Therefore, for any y ∈ [1, 2] and m ∈ (cid:163) − y , − y (cid:164) , ˆ ˆ − | h L ( t ) | d t d z (cid:201) (cid:107) h (cid:107) L ( D ) < β .
00 ASSAF NAOR AND ROBERT YOUNG
It follows that { z ∈ [200, 201] : L (0, y , z ), m ∈ U } has measure at most 24 β and thus N P ( U ) (cid:201) ˆ ˆ − y − y β d m d y (cid:201) β .To bound N P ( U ), observe that N P ( U ) (cid:201) ˆ L (cid:98) ω P ˆ Γ + ,48 ( D , L ) d N P ( L ) = Ω P ˆ Γ + ,48 ( D ) < β .It follows that if β is sufficiently small, then N P ( U ) (cid:202) N P ( U ) − N P ( U ) − N P ( U ) > U is nonempty. That is, there exists a line L ∈ U with parametrization ρ L suchthat S = Π ( L ) ∩ { − (cid:201) x (cid:201)
24} is a supercharacteristic curve. By (190), S is above D and S ⊆ D . By symmetry, there also exists a line L (cid:48) and a supercharacteristic curve S = Π ( L (cid:48) ) ∩ { − (cid:201) x (cid:201)
24} that lies below D and satisfies S ⊆ D .By Lemma 12.1 applied to a rescaling of ˆ Γ , there is C >
24 such that for any t > C , | { v ∈ D : ˆ f ( v ) (cid:202) t } | (cid:46) t − Ω P ˆ Γ + ,48 ( D ) (cid:201) t − β .Applying another symmetry, the analogous reasoning shows that for any t > C , | { v ∈ D : ˆ f ( v ) (cid:201) − t } | (cid:46) t − β .Then, for all sufficiently small β , (cid:107) ˆ f (cid:107) L ( D ) = (cid:107) h (cid:107) L ( D ) + ˆ ∞ (cid:175)(cid:175)(cid:169) v ∈ D : | ˆ f ( v ) | (cid:202) t (cid:170)(cid:175)(cid:175) d t (cid:46) (cid:107) h (cid:107) L ( D ) + C (cid:175)(cid:175)(cid:169) v ∈ D : | ˆ f ( v ) | (cid:202) (cid:170)(cid:175)(cid:175) + ˆ ∞ C t − β d t (cid:201) β + C (cid:175)(cid:175)(cid:169) v ∈ D : | ˆ f ( v ) | (cid:202) (cid:170)(cid:175)(cid:175) + β .But (cid:175)(cid:175)(cid:169) v ∈ D : | ˆ f ( v ) | (cid:202) (cid:170)(cid:175)(cid:175) = (cid:175)(cid:175)(cid:169) v ∈ D : | h ( v ) | = (cid:170)(cid:175)(cid:175) (cid:201) (cid:107) h (cid:107) L ( D ) (cid:201) β
24 ,so (cid:107) ˆ f (cid:107) L ( D ) (cid:46) β . This proves Lemma 12.4, for β at most a constant multiple of δ . (cid:3) We will use the following corollary in the proof of Proposition 7.2.
Corollary 12.5.
Let c be the universal constant in (185) – (188) and let κ be the universalconstant in Lemma 12.2. Denote τ def =
18 max{100, 11 c } and r def = max (cid:169) κ +
6, 144 c τ + c (cid:170) . For any λ > , there are η , R > with the following property. Let Γ = Γ f be an intrinsicLipschitz graph and ( Q , [ −
1, 1] × {0} × [ −
1, 1]) a –rectilinear pseudoquad for Γ . Supposethat Γ is ( η , R ) –paramonotone on r Q, and P , F , and ˆ Γ = Γ ˆ f are as in Lemma 12.3 and theremarks immediately after its proof. Then (cid:107) F − f (cid:107) L (10 Q ) (cid:201) λ | Q | and (cid:107) ˆ f (cid:107) L ( D ) (cid:201) λ . OLIATED CORONA DECOMPOSITIONS 101
Proof.
Set δ = λτ − . Let β = β ( δ ) be as in Lemma 12.4. By Lemma 12.3, there are η , R such that if Γ is ( η , R )–paramonotone on r Q , then H (cid:179) B c τ + c ∩ ( P + (cid:52) Γ + ) (cid:180) < βτ .By (186), this implies that H (cid:179) B τ ∩ ( V + (cid:52) ˆ Γ + ) (cid:180) < βτ .We take R > R and η R < η R , so that ( η , R )–paramonotonicity implies ( η , R )–para-monotonicity. Then, by (185) and the paramonotonicity of Q , Ω P ˆ Γ + ,48 τ ( D τ ) (cid:201) R δ x ( ˆ Q )48 τ Ω P ˆ Γ + , R δ x ( ˆ Q ) ( D τ ) (cid:201) R τ Ω P Γ + , R δ x ( Q ) ( D c τ + c ) (cid:201) R τ | Q | ηα ( Q ) − (cid:46) η .If η is sufficiently small, then Lemma 12.4 implies that (cid:107) ˆ f (cid:107) L ( D max{100,11 c } ) < λ .By (188), this implies that (cid:107) F − f (cid:107) L (10 Q ) (cid:201) λ | Q | . (cid:3) Characteristic curves are close to lines.
Finally, in this section we will show thatthe characteristic curves of ˆ Γ are close to horizontal lines and prove Proposition 7.2. Thekey argument is that when characteristic curves fail to be horizontal, configurations likethose in Figure 6 produce nonmonotonicity. γ ( t ) γ (0) q L γ p F IGURE
6. A characteristic curve γ and a horizontal line L , projected to V . The projection of L crosses γ positively at p , so L passes behind ˆ Γ at p , and L intersects V (shown as parallel horizontal lines) at q . If ˆ f is zeroaway from γ , then L intersects ˆ Γ at least three times (twice near p andonce at q ) and the contribution to (cid:98) ω P is at least x ( q ) − x ( p )2 Lemma 12.6.
For any A > , there are δ = δ ( A ), θ = θ ( A ) > with the following property.Let ˆ Γ = Γ ˆ f be an intrinsic Lipschitz graph. Suppose that Ω P ˆ Γ + ,16 ( D ) < θ and (cid:176)(cid:176) ˆ f (cid:176)(cid:176) L ( D ) < δ .
02 ASSAF NAOR AND ROBERT YOUNG
Let γ : (cid:82) → V be a characteristic curve through and write γ ( t ) = ( t , 0, g ( t )) for t ∈ (cid:82) . Then | g ( t ) | < A for all t ∈ [ −
1, 1] .Proof.
We may suppose that 0 < A <
1. Choose δ = A and θ = A . Our goal is to showthat if (cid:107) ˆ f (cid:107) L ( D ) < δ and if there is t ∈ [ −
1, 1] with | g ( t ) | (cid:202) A , then Ω P ˆ Γ + ,16 ( D ) (cid:202) θ . Afterapplying a symmetry, we may suppose that t > g ( t ) (cid:201) − A , as in Figure 6.Take z ∈ ( − A , 0), y ∈ [ A , A ], m ∈ [ − y , − y ], and w = (0, y , z ). Let L = L w , m . Supposethat Π ( L ) and γ intersect transversally and L ∩ ˆ Γ − has finite perimeter; these hold foralmost every tuple ( y , z , m ). We will show that if ˆ (cid:175)(cid:175) ˆ f (cid:161) t , 0, g L ( t ) (cid:162)(cid:175)(cid:175) d t < A
24 , (192)then (cid:98) ω P ˆ Γ + ,16 ( D , L ) (cid:202)
1, where g L = z ( Π ( ρ L )).Suppose that (192) holds. For t ∈ [ −
8, 8], we have | g L ( t ) | (cid:201) | z | + | m | t + | y t | < + + < Π ( ρ L ([ −
8, 8])) ⊆ D . The graphs of g L and g intersect as depicted in Figure 6. That is, g L (0) = z < g (0), g L is decreasing on [0, 5], and g L (0) − g L (1) = m + y < A , so g L ( t ) (cid:202) g L (1) > g L (0) − A > − A = g ( t ).It follows that the graph of g L crosses γ positively at some point p = ( a , 0, g ( a )), where a ∈ [0, t ]. Since g is characteristic,ˆ f (cid:161) a , 0, g ( a ) (cid:162) = − g (cid:48) ( a ) > − g (cid:48) L ( a ) = y (cid:161) ρ L ( a ) (cid:162) ,so ρ L ( a ) ∈ ˆ Γ − .Let q be the point where L intersects V . Then x ( q ) = − y m ∈ [5, 6]. Since m (cid:201) − A , wehave y ( ρ L ( t )) (cid:202) A for t (cid:201) y ( ρ L ( t )) (cid:201) − A for t (cid:202)
7. By (192), there are b ∈ [1, 2] and b ∈ [3, 4] such that ˆ f (cid:161) b i , 0, g L ( b i ) (cid:162) < A (cid:201) y (cid:161) ρ L ( b i ) (cid:162) .This implies ρ L ( b i ) ∈ ˆ Γ + . Similarly, there is c ∈ [7, 8] such that y ( ρ L ( c )) < ˆ f ( c , 0, g L ( c )) andthus ρ L ( c ) ∈ ˆ Γ − . There is an element of ∂ H (cid:161) x ( L ∩ ˆ Γ + ) (cid:162) in ( a , b ) and another in ( b , c ),and by Lemma 8.1, (cid:98) ω P ˆ Γ + ,16 ( D , L ) (cid:202) b − b (cid:202) m , y , z ) as above, regardless of whether (192) holds, (cid:98) ω P ˆ Γ + ,16 ( D , L ) + A ˆ (cid:175)(cid:175) ˆ f (cid:161) t , 0, g L ( t ) (cid:162)(cid:175)(cid:175) d t (cid:202)
1, (193)since we showed that at least one of the summands on the left hand side of (193) is atleast 1. By integrating (193) with respect to z , we see that for almost every ( m , y ) that OLIATED CORONA DECOMPOSITIONS 103 satisfy y ∈ [ A , A ] and m ∈ [ − y , − y ], we have ˆ − A (cid:98) ω P ˆ Γ + ,16 ( D , L ) d z (cid:202) A − A ˆ − A ˆ | ˆ f ( x , 0, g L ( x )) | d x d z (cid:202) A − A (cid:107) ˆ f (cid:107) L ( D ) (cid:202) A − δ A = A m and y as above, we conclude as follows. Ω P ˆ Γ + ,16 ( D ) (cid:202) ˆ A A ˆ − y − y ˆ − A (cid:98) ω P ˆ Γ + ,16 ( D , L ) d z d m d y (cid:202) A . (cid:3) Part 2 of Proposition 7.2 follows from Lemma 12.6.
Corollary 12.7.
For every < ζ < there are δ = δ ( ζ ) > and θ = θ ( ζ ) > with the followingproperty. Let ˆ Γ = Γ ˆ f be an intrinsic Lipschitz graph such that Ω P ˆ Γ + ,128 ( D ) < θ and (cid:176)(cid:176) ˆ f (cid:176)(cid:176) L ( D ) < δ . Let ˆ Q be a pseudoquad for ˆ Γ with x ( ˆ Q ) = [ −
1, 1] such that ∈ ˆ Q and δ z ( ˆ Q ) = . For u ∈ Q,if g u : (cid:82) → (cid:82) is such that { z = g u ( x )} is a characteristic curve for ˆ Γ that passes through u,then (cid:107) g − z ( u ) (cid:107) L ∞ ([ − (cid:201) ζ . That is, ˆ Q satisfies part 2 of Proposition 7.2 for P = V .Proof. For p ∈ V and t >
0, denote D t ( p ) = pD t . Let A = ζ and let δ , θ > A .Let p ∈ D so that D ( p ) ⊆ D . Then Ω P ˆ Γ + ,8 · ( D ( p )) < θ and (cid:107) ˆ f (cid:107) L ( D ( p )) < δ , soby Lemma 8.6, the rescaling s ( p − ˆ Γ ) satisfies Lemma 12.6. Hence, if γ = { z = g p ( x )}is a characteristic curve for ˆ Γ that passes through p , then (cid:107) g p − z ( p ) (cid:107) L ∞ ([ x ( p ) − x ( p ) + (cid:201) A = ζ .Let g and g be the lower and upper bounds of ˆ Q , respectively. Then g (0) ∈ [ −
3, 0] and g (0) ∈ [0, 3], so (cid:107) g − g (0) (cid:107) L ∞ ([ − (cid:201) ζ and (cid:107) g − g (0) (cid:107) L ∞ ([ − (cid:201) ζ . Therefore, 4 ˆ Q ⊆ D .If u ∈ Q and { z = g u ( x )} is a characteristic curve, then (cid:107) g u − z ( u ) (cid:107) L ∞ ([ − (cid:201) (cid:107) g u − z ( u ) (cid:107) L ∞ ([ x ( u ) − x ( u ) + (cid:201) ζ . (cid:3) Finally, we combine the results of this section to prove Proposition 7.2.
Proof of Proposition 7.2.
By Lemma 2.9 and Lemma 8.6, if Q is a pseudoquad of Γ and h is a composition of a shear map, a translation, and a stretch map, then Q and Γ satisfyProposition 7.2 if and only if ˆ h ( Q ) = Π ( h ( Q )) and h ( Γ ) do. So, by Remark 4.3, it suffices toprove Proposition 7.2 for rectilinear pseudoquads of the form ( Q , [ −
1, 1] × {0} × [ −
1, 1]).Let r be as in Corollary 12.5; we may suppose r > δ = δ ( ζ ), θ = θ ( ζ ) > R = R ( λ , ζ ) > η = η ( λ , ζ ) > Γ is( η , R )–paramonotone on r Q and P , F , and ˆ Γ = Γ ˆ f are as above, then (cid:107) F − f (cid:107) L (10 Q ) (cid:201) λ | Q | and (cid:107) ˆ f (cid:107) L ( D ) (cid:201) δ . (194)
04 ASSAF NAOR AND ROBERT YOUNG
Denote R = max{ R , 128} and η = min{ θ R , η R R }. Since R (cid:202) R , η R (cid:201) η R , and Γ is( η , R )–paramonotone on r Q , it is also ( η , R )–paramonotone, so Q satisfies (194), whichimplies part 1 of Proposition 7.2. Furthermore, Ω P ˆ Γ + ,128 ( D ) (cid:201) R Ω P ˆ Γ + , R ( r Q ) (cid:201) R | Q | α ( Q ) − η < θ .Thus ˆ Γ satisfies the hypotheses of Corollary 12.7, so ˆ Q satisfies part 2 of Proposition 7.2.As ˆ Q is the image of Q under a shear map, part 2 of Proposition 7.2 holds for Q as well. (cid:3) Acknowledgements.
We thank A. Eskenazis for a discussion that led to Remark 1.13. Ourformer colleague Louis Nirenberg passed away as this project was being completed. Overthe years, he made significant efforts (partially in collaboration with A. N.) to answer thequestion that we resolve here, though in hindsight those attempts were doomed to failbecause they aimed to prove (4) with p =
2, which we now know does not hold. His deepmathematical insights, his contagious joie de vivre, and his kindness are dearly missed.R
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International MathematicsResearch Notices , 09 2018. Rny200. A PPENDIX
A. O
N THE IMPLICIT DEPENDENCE ON p IN [LN14b]A version of Theorem 1.3 was stated in [LN14b] with an implicit dependence on theexponent p . In this section, we explain how the arguments in [LN14b] can be used toderive the explicit dependence on p that we needed in Section 1.1.3.Let ( E , (cid:107) · (cid:107) E ) be a Banach space and fix q ∈ [2, ∞ ]. The q –uniform convexity constantof X , denoted K q ( E ), is defined [Bal92, BCL94] as the infimum over K ∈ (0, ∞ ] such that ∀ x , y ∈ E , (cid:179) (cid:107) x (cid:107) qE + K q (cid:107) y (cid:107) q E (cid:180) q (cid:201) (cid:181) (cid:107) x + y (cid:107) q E + (cid:107) x − y (cid:107) q E (cid:182) q . (195)Setting x = K (cid:202)
1. By convexity, (195) always holdswhen K = ∞ or when q = ∞ and K =
1. Thus, (195) quantifies the extent to which thenorm (cid:107) · (cid:107) E is strictly convex. An equivalent (but somewhat less convenient to work with)formulation of this fact (see [Fig76, BCL94]) is that K q ( E ) is bounded above and below byuniversal constant multiples of the infimum over those C > (cid:107) u + v (cid:107) E (cid:201) − C − q (cid:107) u − v (cid:107) q E holds for any two unit vectors u , v ∈ E .Theorem 1.3 is the special case E = (cid:82) , q = < p (cid:201) OLIATED CORONA DECOMPOSITIONS 109
Theorem A.1.
For any p > and q (cid:202) , if ( E , (cid:107)·(cid:107) E ) is a Banach space with K q ( E ) < ∞ , thenevery smooth and compactly supported function f : (cid:72) → E satisfies (cid:181) ˆ ∞ (cid:107) D t v f (cid:107) max{ p , q } L p ( H ; E ) d tt (cid:182) p , q } (cid:46) max (cid:110) ( p − q − , K q ( E ) (cid:111) (cid:107)∇ (cid:72) f (cid:107) L p ( H ; (cid:96) p ( E )) , (196) where we use the (standard) notation ∇ (cid:72) f def = ( X f , Y f ) ∈ E × E for the horizontal gradient.
Theorem A.1 is due to [LN14b], except that it is stated there with a factor that dependsin an unspecified way on p , q , E in place of the quantity max{ K q ( E ), 1/( p − − q }. Thisis because the proof of [LN14b] uses the vector-valued Littlewood–Paley–Stein inequalityof [MTX06], for which explicit bounds on the relevant constants were not available in theliterature at the time when [LN14b] was written. However, such bounds were subsequentlyderived in [HN19] (using in part an argument of [LN14b] itself), so we will next brieflyexplain how to obtain Theorem A.1 by incorporating this input into [LN14b].Let { h t } t > and { p t } t > be the heat and Poisson kernels on (cid:82) , respectively, i.e., ∀ s > h t ( s ) def = (cid:112) π t e − s t and p t ( s ) def = t π ( s + t ) .It will be convenient to denote the time derivatives ∂∂ t h t , ∂∂ t p t by ˙ h t , ˙ p t , respectively, i.e., ∀ s >
0, ˙ h t ( s ) = s − t (cid:112) π t e − s t and ˙ p t ( s ) = s − t π ( s + t ) .By a straightforward evaluation of the integral in (197) below, one checks the followingstandard identity (semigroup subordination; see e.g. [Boc55, Section 4.4]). ∀ s >
0, ˙ p t ( s ) = (cid:112) π ˆ ∞ e − t u (cid:112) u ˙ h u ( s ) d u . (197)Fix φ ∈ L q ( (cid:82) ; E ) and p (cid:202)
1. The following bound holds for any t > (cid:176)(cid:176) t ˙ p t ∗ φ (cid:176)(cid:176) p L q ( (cid:82) , E ) = p (cid:176)(cid:176)(cid:176)(cid:176) ˆ ∞ t e − t u u (cid:112) π u u ˙ h u ∗ φ d u (cid:176)(cid:176)(cid:176)(cid:176) p L q ( (cid:82) , E ) (cid:201) p − t (cid:112) π ˆ ∞ u − e − t u (cid:107) u ˙ h u ∗ φ (cid:107) p L q ( (cid:82) ; E ) d u . (198)The first step of (198) is the representation (197), and the second step of (198) is Jensen’sinequality, because ´ ∞ t exp( − t /(4 u ))/(2 u (cid:112) π u ) d u =
1. Integration of (198) gives ˆ ∞ (cid:107) t ˙ p t ∗ φ (cid:107) p L q ( (cid:82) ; E ) d tt (cid:201) p − (cid:112) π ˆ ∞ (cid:181) ˆ ∞ e − t u d t (cid:182) u − (cid:107) u ˙ h u ∗ φ (cid:107) p L q ( (cid:82) ; E ) d u = p − ˆ ∞ (cid:107) u ˙ h u ∗ φ (cid:107) p L q ( (cid:82) ; E ) d uu . (199)
10 ASSAF NAOR AND ROBERT YOUNG
Now, if q (cid:202) K q ( E ) < ∞ , then it was proved in [HN19] that (cid:181) ˆ ∞ (cid:107) t ˙ h t ∗ φ (cid:107) qL q ( (cid:82) ; E ) d tt (cid:182) q (cid:46) K q ( E ) (cid:107) φ (cid:107) L q ( (cid:82) ; E ) . (200)In combination with (199) we therefore see that also (cid:181) ˆ ∞ (cid:107) t ˙ p t ∗ φ (cid:107) qL q ( (cid:82) ; E ) d tt (cid:182) q (cid:46) K q ( E ) (cid:107) φ (cid:107) L q ( (cid:82) ; E ) . (201) Remark
A.2 . The reason why we passed from the vector-valued Littlewood–Paley–Steininequality (200) for the heat semigroup to its counterpart (201) for the Poisson semigroupis that at the time when [LN14a] was written this was known (with K q ( E ) in (201) replacedby an unspecified constant factor) for the Poisson semigroup due to [MTX06], while thevalidity of (200) was an open question. For this reason, [LN14a] worked with the Poissonsemigroup, so it is simplest to use (201) when we refer below to steps in [LN14a]. However,one could repeat the reasoning of [LN14a] mutatis mutandis while working directly withthe heat semigroup and using (200). The above subordination argument is standard, butwe included the quick derivation to verify that the constants are universal.The case p = q of Theorem A.1 follows by substituting (201) into [LN14b]. Specifically,we are asserting that the implicit constant in [LN14b, Theorem 2.1] is O ( K q ( E )) when p = q . To check this, note that in the proof of [LN14b, Theorem 2.1] the only loss of a factorthat is not a universal constant occurs in [LN14b, equation (18)], which is an instantiationof [LN14b, inequality (15)]; the latter inequality is the same as (201) when p = q , exceptthat the constant factor in the right hand side is now specified to be O ( K q ( E )).The case p > q of Theorem A.1 follows from the case p = q . When p > q , we have K q ( E ) (cid:202) K p ( E ) (for justification of this monotonicity, see [BCL94] or [MN14, Section 6.2])and ( p − − q (cid:201) ( p − − p (since p > q (cid:202) q decreases. We thus suppose from now that 1 < p < q .For M >
1, let β M : (cid:72) → [0, 1] be a smooth bump function that is O (1)–Lipschitz (withrespect to the Carnot–Carathéodory metric d ), satisfies β M ( h ) = h ∈ B M , and hassupp( β M ) ⊆ B M + .For a smooth compactly supported f : (cid:72) → E , consider F M : (cid:72) → L p ( H ; E ) given by ∀ g , h ∈ (cid:72) , F M ( h )( g ) def = β M ( h ) f ( g h ). (202) [HN19] states (200) with the factor K q ( E ) in the right hand side replaced by a parameter m q ( E ) thatis called [Pis86b] the martingale cotype q constant of E . There is no need to state the definition of m q ( E )here because it will not have a role in the ensuing discussion; it suffices to recall that by the martingaleinequality of [Pis75] we have m q ( E ) (cid:46) K q ( E ). So, (199) is a formal consequence of [HN19], but the aboveformulation is essentially (namely, up to O (1)–renorming) equivalent to that of [HN19]. For the reversedirection use the fact that there is a norm ||| · ||| on E that satisfies (cid:107) x (cid:107) E (cid:179) ||| x ||| for all x ∈ E and such that K q ( E , ||| · ||| ) (cid:46) m q ( E ). This renorming statement is essentially due to the deep work [Pis75], except that it isderived in [Pis75] with the weaker property (cid:107) x (cid:107) E (cid:201) ||| x ||| (cid:46) m q ( E ) (cid:107) x (cid:107) E . The existence of such a norm whichis O (1)–equivalent to (cid:107) · (cid:107) E follows by combining [LNP09] and [MN13a], though we checked (details omitted)that one could adapt the reasoning in [Pis75] so as to obtain a proof of this fact which avoids any reference tothe nonlinear considerations of [LNP09, MN13a]. Alternatively, Gilles Pisier has recently showed us (privatecommunication) a derivation of this O (1)–renorming result from the statement of [Pis75, Theorem 3.1]. OLIATED CORONA DECOMPOSITIONS 111
We have ( q − q − (cid:201) (cid:201) K q ( E ), so the case p = q of Theorem A.1 with E replaced by L p ( H ; E ) gives (cid:181) ˆ ∞ (cid:107) D t v F M (cid:107) qL q ( H ; L p ( H ; E )) d tt (cid:182) q (cid:46) K q (cid:161) L p ( H ; E ) (cid:162) (cid:107)∇ (cid:72) F M (cid:107) L q ( H ; (cid:96) q ( L p ( H ; E ))) (cid:46) max (cid:110) ( p − q − , K q ( E ) (cid:111) (cid:107)∇ (cid:72) F M (cid:107) L q ( H ; (cid:96) q ( L p ( H ; E ))) , (203)where the last step uses the fact that, by inequality (4.4) in [Nao14] , we have K q (cid:161) L p ( H ; E ) (cid:162) (cid:46) max (cid:110) ( p − q − , K q ( E ) (cid:111) . (204)To bound the final term in (203) from above, note that by the left invariance of ∇ (cid:72) , ∇ (cid:72) F M ( h )( g ) = (cid:161) X β M ( h ) f ( g h ), Y β M ( h ) f ( g h ) (cid:162) + β M ( h ) ∇ (cid:72) f ( g h ).Hence, for all h ∈ (cid:72) , (cid:107)∇ (cid:72) F M ( h ) (cid:107) (cid:96) q ( L p ( H ; E )) (cid:46) (cid:107) f (cid:107) L ∞ ( H ; E ) B M + ( ) (cid:224) B M ( ) ( h ) + (cid:107)∇ (cid:72) f (cid:107) L p ( H ; E ) B M + ( ) ( h ).So, (cid:107)∇ (cid:72) F M (cid:107) L q ( H ; (cid:96) q ( L p ( H ; E ))) (cid:46) M q (cid:107) f (cid:107) L ∞ ( H ; E ) + M q (cid:107)∇ (cid:72) f (cid:107) L p ( H ; E ) . (205)In order to bound the left hand side of (203) from below, note that by (39), if 0 < t < M and h ∈ B M − (cid:112) t ( ), then hZ t ∈ B M , and therefore β ( h ) = β ( hZ t ) =
1. Hence, ∀ h ∈ B M − (cid:112) t ( ), (cid:107) D t v F M ( h ) (cid:107) L p ( H ; E ) = (cid:107) D t v f (cid:107) L p ( H ; E ) .Consequently, (cid:107) D t v F M (cid:107) L q ( H ; L p ( H ; E )) (cid:202) H (cid:161) B M − (cid:112) t ( ) (cid:162) q (cid:107) D t v f (cid:107) L p ( H ; E ) (cid:179) (cid:179) M − (cid:112) t (cid:180) q (cid:107) D t v f (cid:107) L p ( H ; E ) .Hence, for every 0 < T < M we have (cid:181) ˆ T (cid:107) D t v F M (cid:107) qL q ( H ; L p ( H ; E )) d tt (cid:182) q (cid:38) ( M − T ) q (cid:181) ˆ T (cid:107) D t v f (cid:107) qL p ( H ; E ) d tt (cid:182) q .Combining this with (203) and (205), letting M → ∞ and then T → ∞ , gives Theorem A.1. Remark
A.3 . In the setting of the proof of Theorem A.1, the Hardy–Littlewood–Stein(Poisson semigroup) G –function of a function φ ∈ L q ( (cid:82) ; E ) is the function G q ( φ ) : (cid:82) → (cid:82) that is defined by ∀ x ∈ E , G q ( φ )( x ) def = (cid:181) ˆ ∞ (cid:107) t ˙ p t ∗ φ ( x ) (cid:107) q E d tt (cid:182) q . (206)By [MTX06], if K q ( E ) < ∞ , then for every 1 < p < ∞ , (cid:107) G q ( φ ) (cid:107) L p ( (cid:82) ) (cid:46) p , q , K q ( E ) (cid:107) φ (cid:107) L p ( (cid:82) ; E ) . (207)If the implicit constant in (207) were O (max{ K q ( E ), 1/( p − − q }) for 1 < p < q (this isso when p (cid:202) q by (201) and Jensen’s inequality), then Theorem A.1 would follow by directsubstitution into [LN14b] without the need to consider the above averaging argument Formally, [Nao14, inequality (4.4)] is the dual of (204); see [BCL94, Lemma 5] for the relevant duality.
12 ASSAF NAOR AND ROBERT YOUNG using the auxiliary function F M in (202). However, it seems that the interpolation argu-ment [MTX06] does not yield this dependence. Determining the optimal dependence on p , q , K q ( E ) in the G –function bound (207) remains an interesting open question.The same question for the heat semigroup variant of (207), i.e., with ˙ p t replaced by ˙ h t in (206), is a bigger mystery. That such an inequality for the vector-valued heat semigroupHardy–Littlewood–Stein G –function holds with any dependence on p , q , K q ( E ) was es-tablished recently in [Xu18], but as p → + the dependence of [Xu18] seems suboptimal.Obtaining the analogue of (200) for the n –dimensional heat semigroup (in which case φ is a mapping from (cid:82) n to E ) would be very interesting. In [Xu18], this is achieved with aconstant that is independent of n but has a much worse dependence on K q ( E ).A substitution of Theorem A.1 into the reasoning of [LN14b] yields the following re-statement of the nonembedding result of [LN14b], with explicit dependence on K q ( E ). Theorem A.4.
For q (cid:202) , if E is a Banach space with K q ( E ) < ∞ , then for every n ∈ (cid:78) , theword-ball in (cid:72) of radius n has E –distortion c E ( B n ) (cid:38) (log n ) q K q ( E ) .Since by [BCL94], the Schatten–von Neumann trace class S r has K ( S r ) = (cid:112) r − < r (cid:201)
2, Theorem A.4 implies the lower bound on c S r ( B n ) that we used in Section 1.1.3(recall that the behavior as r → + was important for that application). This also showsthat the following question about a possible strengthening of Theorem A.4 would implythe distortion lower bound (24) that we asked about in Section 1.1.3. In fact, a positiveanswer to this question would be a remarkable geometric result, which, as we explainedin Section 1.1.3, would have strong implications; at present, we do not have sufficientevidence to conjecture that the answer is indeed positive in such great generality. Question A.5.
Can the conclusion of Theorem A.4 be improved to c E ( B n ) (cid:38) (cid:179) log nK qq
Can the conclusion of Theorem A.4 be improved to c E ( B n ) (cid:38) (cid:179) log nK qq ( E ) (cid:180) qq