Square functions and uniform rectifiability
aa r X i v : . [ m a t h . C A ] J a n SQUARE FUNCTIONS AND UNIFORM RECTIFIABILITY
VASILEIOS CHOUSIONIS, JOHN GARNETT, TRIET LE, AND XAVIER TOLSA
Abstract.
In this paper it is shown that an Ahlfors-David n -dimensional measure µ on R d is uniformly n -rectifiable if and only if for any ball B ( x , R ) centered at supp( µ ), Z R Z x ∈ B ( x ,R ) (cid:12)(cid:12)(cid:12)(cid:12) µ ( B ( x, r )) r n − µ ( B ( x, r ))(2 r ) n (cid:12)(cid:12)(cid:12)(cid:12) dµ ( x ) drr ≤ c R n . Other characterizations of uniform n -rectifiability in terms of smoother square functionsare also obtained. Introduction
Given 0 < n < d , a Borel set E ⊂ R d is said to be n -rectifiable if it is contained in acountable union of n -dimensional C manifolds and a set of zero n -dimensional Hausdorffmeasure H n . On the other hand, a Borel measure µ in R d is called n -rectifiable if it is of theform µ = g H n | E , where E is a Borel n -rectifiable set and g is positive and H n integrable on E . Rectifiability is a qualitative notion, but David and Semmes in their landmark works[DS1] and [DS2] introduced the more quantitative notion of uniform rectifiability. Todefine uniform rectifiability we need first to recall the notion of Ahlfors-David regularity.We say a Radon measure µ in R d is n -dimensional Ahlfors-David regular with constant c if(1.1) c − r n ≤ µ ( B ( x, r )) ≤ c r n for all x ∈ supp( µ ), 0 < r ≤ diam(supp( µ )) . For short, we sometimes omit the constant c and call µ n -AD-regular. It follows easilythat such a measure µ must be of the form µ = h H n | supp( µ ) , where h is a positive functionbounded from above and from below.An n -AD-regular measure µ is uniformly n -rectifiable if there exist θ, M > x ∈ supp( µ ) and all r > ρ from the ball B n (0 , r )in R n to R d with Lip( ρ ) ≤ M such that µ ( B ( x, r ) ∩ ρ ( B n (0 , r ))) ≥ θr n . V.C. was funded by the Academy of Finland Grant SA 267047. Also, partially supported by the ERCAdvanced Grant 320501, while visiting Universitat Aut`onoma de Barcelona.J.G. was partially supported by NSF DMS 1217239 and the IPAM long program Interactions BetweenAnalysis and Geometry, Spring 2013.T.L. was partially supported by NSF DMS 1053675 and the IPAM long program Interactions BetweenAnalysis and Geometry, Spring 2013.X.T. was funded by the an Advanced Grant of the European Research Council (programme FP7/2007-2013), by agreement 320501. Also, partially supported by grants 2009SGR-000420 (Generalitat deCatalunya) and MTM-2010-16232 (Spain).
When n = 1, µ is uniformly 1-rectifiable if and only if supp( µ ) is contained in a rectifiablecurve in R d on which the arc length measure satisfies (1.1). A Borel set E ⊂ R d is n -AD-regular if µ = H n | E is n -AD-regular, and it is called uniformly n -rectifiable if, further, H n | E is uniformly n -rectifiable. Thus µ is an uniformly n -rectifiable measure if and only if µ = h H n | E where h > E is an uniformly n -rectifiableclosed set.Uniform rectifiability is closely connected to the geometric study of singular integrals.In [Da1] David proved that if E ⊂ R d is uniformly n -rectifiable, then for any convolutionkernel K : R d \ { } → R satisfying(1.2) K ( − x ) = − K ( x ) and (cid:12)(cid:12) ∇ j K ( x ) (cid:12)(cid:12) ≤ c j | x | − n − j , for x ∈ R d \ { } , j = 0 , , , . . . , the associated singular integral operator T K f ( x ) = R K ( x − y ) f ( y ) d H n | E ( y ) is bounded in L ( H n | E ). David and Semmes in [DS1] proved conversely that the L ( H n | E )-boundednessof all singular integrals T K with kernels satisfying (1.2) implies that E is uniformly n -rectifiable. However if one only assumes the boundedness of some particular singularintegral operators satisfying (1.2), then the situation becomes much more delicate.In [MMV] Mattila, Melnikov and Verdera proved that if E is an 1-AD regular set, theCauchy transform is bounded in L ( H n | E ) if and only if E is uniformly 1-rectifiable. It isremarkable that their proof depends crucially on a special subtle positivity property of theCauchy kernel related to the so-called Menger curvature. See [CMPT] for other examplesof 1-dimensional homogeneous convolution kernels whose L -boundedness is equivalentto uniform rectifiability, again because of Menger curvature. Recently in [NToV] it wasshown that in the codimension 1 case, that is, for n = d −
1, if E is n -AD-regular, thenthe vector valued Riesz kernel x/ | x | n +1 defines a bounded operator on L ( H n | E ) if andonly if E is uniformly n -rectifiable. In this case, the notion of Menger curvature is notapplicable and the proof relies instead on the harmonicity of the kernel x/ | x | n +1 . It is anopen problem if the analogous result holds for 1 < n < d − n -rectifiability in terms ofsquare functions. Our first characterization involves the following difference of densities∆ µ ( x, r ) := µ ( B ( x, r )) r n − µ ( B ( x, r ))(2 r ) n and reads as follows. Theorem 1.1.
Let µ be an n -AD-regular measure. Then µ is uniformly n -rectifiable ifand only if there exists a constant c such that, for any ball B ( x , R ) centered at supp( µ ) , (1.3) Z R Z x ∈ B ( x ,R ) | ∆ µ ( x, r ) | dµ ( x ) drr ≤ c R n . Recall that a celebrated theorem of Preiss [Pr] asserts that a Borel measure µ in R d is n -rectifiable if and only if the density lim r → µ ( B ( x, r )) r n exists and is positive for µ -a.e. x ∈ R d . In a sense, Theorem 1.1 can be considered as a square function version ofPreiss’ theorem for uniform rectifiability. On the other hand, let us mention that the “if”implication in our theorem relies on some of the deep results by Preiss in [Pr].It is also worth comparing Theorem 1.1 to some earlier results from Kenig and Toro[KT], David, Kenig and Toro [DKT] and Preiss, Tolsa and Toro [PTT]. In these works QUARE FUNCTIONS AND UNIFORM RECTIFIABILITY 3 it is shown among other things that, given α >
0, there exists β ( α ) > µ is n -AD-regular and for each compact set K there exists some constant c K such that (cid:12)(cid:12)(cid:12)(cid:12) µ ( B ( x, r )) r n − µ ( B ( x, tr ))( tr ) n (cid:12)(cid:12)(cid:12)(cid:12) ≤ c K r α for 1 < t ≤ x ∈ K ∩ supp( µ ), 0 < r ≤ µ is supported on an C β n -dimensional manifold union a closed set with zero µ -measure. This result can be thought of as the H¨older version of one of the implications inTheorem 1.1.We also want to mention the forthcoming work [ADT] by Azzam, David and Torofor some other conditions on a doubling measure which imply rectifiability. One of theconditions in [ADT] quantifies the difference of the measure at different close scales interms of the Wasserstein distance W . In our case, the square function in Theorem 1.1just involves the difference of the n -dimensional densities of two concentric balls such thatthe largest radius doubles the smallest one.Motivated by the recent work [LM] studying local scales on curves and surfaces, whichwas the starting point of this paper’s research, we also prove smooth versions of Theorem1.1. For any Borel function ϕ : R d → R let ϕ t ( x ) = 1 t n ϕ (cid:16) xt (cid:17) , t > µ,ϕ ( x, t ) := Z (cid:0) ϕ t ( y − x ) − ϕ t ( y − x ) (cid:1) dµ ( y ) , whenever the integral makes sense. If ϕ is smooth, let ∂ ϕ ( x, t ) = t∂ t ϕ t ( x )and define e ∆ µ,ϕ ( x, t ) := Z ∂ ϕ ( y − x, t ) dµ ( y ) , again whenever the integral makes sense. Our second theorem characterizes uniform n -rectifiable n -AD-regular measures using the square functions associated with ∆ µ,ϕ and e ∆ µ,ϕ . Theorem 1.2.
Let ϕ : R d → R be of the form e −| x | N , with N ∈ N , or (1 + | x | ) − a , with a > n/ . Let µ be an n -AD-regular measure in R d . The following are equivalent: (a) µ is uniformly n -rectifiable. (b) There exists a constant c such that for any ball B ( x , R ) centered at supp( µ ) , (1.4) Z R Z x ∈ B ( x ,R ) | ∆ µ,ϕ ( x, r ) | dµ ( x ) drr ≤ c R n . (c) There exists a constant c such that for any ball B ( x , R ) centered at supp( µ ) , (1.5) Z R Z x ∈ B ( x ,R ) | e ∆ µ,ϕ ( x, r ) | dµ ( x ) drr ≤ c R n . The functions ϕ t above are radially symmetric and (constant multiples of) approximateidentities on any n -plane containing the origin. The definitions of ∆ µ,ϕ ( x, t ) and e ∆ µ,ϕ ( x, t ) VASILEIOS CHOUSIONIS, JOHN GARNETT, TRIET LE, AND XAVIER TOLSA arise from convolving the measure µ with the kernels ϕ t ( x ) − ϕ t ( x ) and ∂ ϕ ( x, t ), respec-tively. Note that ϕ t ( x ) − ϕ t ( x ) is a discrete approximation to ∂ ϕ ( x, t ). Note also thatthe quantities ∆ µ ( x, t ) , ∆ µ,ϕ ( x, t ) and e ∆ µ,ϕ ( x, r ) are identically zero whenever µ = H n | L , L is an n -plane, and x ∈ L .For each integer k >
0, let e ∆ kµ,ϕ ( x, t ) = Z ∂ kϕ ( y − x, t ) dµ ( y ) , where ∂ kϕ ( x, t ) = t k ∂ kt ϕ t ( x ) . Similarly, let ∆ kµ,ϕ ( x, t ) = Z D k [ ϕ t ] ( y − x ) dµ ( y ) , where D k [ ϕ t ]( x ) = D k − [ Dϕ t ]( x ) , and Dϕ t ( x ) = ϕ t ( x ) − ϕ t ( x ) . By arguments analogous to the ones of Theorem 1.2, we obtain the following equivalentsquare function conditions for uniform rectifiability.
Proposition 1.3.
Let ϕ : R d → R be of the form e −| x | N , with N ∈ N , or (1 + | x | ) − a ,with a > n/ . Let µ be an n -AD-regular measure in R d and k > . The following areequivalent: (a) µ is uniformly n -rectifiable. (b) There exists a constant c k such that for any ball B ( x , R ) centered at supp µ , (1.6) Z R Z x ∈ B ( x ,R ) | ∆ kµ,ϕ ( x, r ) | dµ ( x ) drr ≤ c k R n . (c) There exists a constant c k such that for any ball B ( x , R ) centered at supp µ , (1.7) Z R Z x ∈ B ( x ,R ) | e ∆ kµ,ϕ ( x, r ) | dµ ( x ) drr ≤ c k R n . Proposition 1.3 is in the same spirit as the characterization of Lipschitz function spacesin Chapter V, Section 4 of [St].There are other characterizations of uniform n -rectifiability via square functions in theliterature. Among the most relevant of these is a condition in terms of the β -numbers ofPeter Jones. For x ∈ supp( µ ) and r > , consider the coefficient β µ ( x, r ) = inf L Z B ( x,r ) dist( y, L ) r n +1 dµ ( y ) , where the infimum is taken over all n -planes L . Like ∆ µ ( x, r ), β µ ( x, r ) is a dimensionalcoefficient, but while β µ ( x, r ) measures how close supp( µ ) is to some n -plane, ∆ µ ( x, r )measures the oscillations of µ . In [DS1], David and Semmes proved that µ is uniformly n -rectifiable if and only if β µ ( x, r ) dx drr is a Carleson measure on supp( µ ) × (0 , ∞ ), thatis, (1.3) is satisfied with ∆ µ ( x, r ) replaced by β µ ( x, r ).The paper is organized as follows. In Section 2 we provide the preliminaries for theproofs of Theorems 1.1 and 1.2. In Section 3 we show first that the boundedness of thesmooth square functions in (1.4) and (1.5) implies uniform rectifiability. Using suitableconvex combinations, we then show that (1.3) implies (1.4), and thus establish one of theimplications in Theorem 1.1. In Section 4 we prove that uniform n -rectifiability implies QUARE FUNCTIONS AND UNIFORM RECTIFIABILITY 5 (1.4) and (1.5), and thereby complete the proof of Theorem 1.2. In Section 5 we provethat (1.3) holds if µ is uniform n -rectifiable; this is the most delicate part of the paperbecause of complications which arise from the non-smoothness of the function r − n χ B (0 ,r ) − (2 r ) − n χ B (0 , r ) . Finally, in Section 6 we outline the proof for Proposition 1.3.Throughout the paper the letter C stands for some constant which may change its valueat different occurrences. The notation A . B means that there is some fixed constant C such that A ≤ CB , with C as above. Also, A ≈ B is equivalent to A . B . A .2. Preliminaries
The David cubes.
Below we will need to use the David lattice D of “cubes”associated with µ (see [Da2, Appendix 1], for example). Suppose for simplicity that µ ( R d ) = ∞ . In this case, D = S j ∈ Z D j and each set Q ∈ D j , which is called a cube,satisfies µ ( Q ) ≈ − jn and diam( Q ) ≈ − j . In fact, we will assume that c − − j ≤ diam( Q ) ≤ − j . We set ℓ ( Q ) := 2 − j . For R ∈ D , we denote by D ( R ) the family of all cubes Q ∈ D which are contained in R . In the case when µ ( R d ) < ∞ and diam(supp( µ )) ≈ − j , then D = S j ≥ j D j . The other properties of the lattice D are the same as in the previous case.2.2. The α coefficients. The so called α coefficients from [To1] play a crucial role in ourproofs. They are defined as follows. Given a closed ball B ⊂ R d which intersects supp( µ ),and two finite Borel measures σ and ν in R d , we setdist B ( σ, ν ) := sup n(cid:12)(cid:12)(cid:12)R f dσ − R f dν (cid:12)(cid:12)(cid:12) : Lip( f ) ≤ , supp f ⊂ B o , where Lip( f ) stands for the Lipschitz constant of f . It is easy to check that this is indeed adistance in the space of finite Borel measures supported in the interior of B . See [Chapter14, Ma] for other properties of this distance. Given a subset A of Borel measures, we setdist B ( µ, A ) := inf σ ∈A dist B ( µ, σ ) . We define α nµ ( B ) := 1 r ( B ) n +1 inf c ≥ ,L dist B ( µ, c H n | L ) , where r ( B ) stands for the radius of B and the infimum is taken over all the constants c ≥ n -planes L . To simplify notation, we will write α ( B ) instead of α nµ ( B ).Given a cube Q ∈ D , let B Q be a ball with radius 10 ℓ ( Q ) with the same center as Q .We denote α ( Q ) := α ( B Q ) . We also denote by c Q and L Q a constant and an n -plane minimizing α ( Q ). We assumethat L Q ∩ B Q = ∅ .The following is shown in [To1]. Theorem 2.1.
Let µ be an n -AD-regular measure in R d . If µ is uniformly n -rectifiable,then there exists a constant c such that (2.1) X Q ⊂ R α ( Q ) µ ( Q ) ≤ c µ ( R ) for all R ∈ D . VASILEIOS CHOUSIONIS, JOHN GARNETT, TRIET LE, AND XAVIER TOLSA
The weak constant density condition.
Given µ satisfying (1.1), we denote by G ( C, ε ) the subset of those ( x, r ) ∈ supp( µ ) × (0 , ∞ ) for which there exists a Borel measure σ = σ x,r satisfying(1) supp( σ ) = supp( µ ),(2) the AD -regularity condition (1.1) with constant C ,(3) | σ ( B ( y, t )) − t n | ≤ εr n for all y ∈ supp( µ ) ∩ B ( x, r ) and all 0 < t < r. Definition 2.2.
A Borel measure µ satisfies the weak constant density condition (WCD)if there exists a positive constant C such that the set G ( C, ε ) c := [supp( µ ) × (0 , ∞ )] \ G ( C, ε )is a Carleson set for every ε >
0, that is, for every ε > C ( ε ) suchthat(2.2) Z R Z B ( x,R ) χ G ( C,ε ) c ( x, r ) dµ ( x ) drr ≤ C ( ε ) R n for all x ∈ supp( µ ) and R > Theorem 2.3.
Let n ∈ (0 , d ) be an integer. An n -AD-regular measure µ in R d is uniformly n -rectifiable if and only if it satisfies the weak constant density condition. David and Semmes in [DS1, Chapter 6] showed that if µ is uniformly n -rectifiable,then it satisfies the WCD. In [DS2, Chapter III.5], they also proved the converse in thecases when n = 1 , , d −
1. The proof of the converse for all codimensions was obtainedvery recently in [To2]. The arguments rely on two essential and deep ingredients: theso called bilateral weak geometric lemma of David and Semmes [DS2], and the (partial)characterization of uniform measures by Preiss [Pr].3.
Boundedness of square functions implies uniform rectifiability
In this section we assume that either ϕ ( x ) = e −| x | N , with N ∈ N , or ϕ ( x ) = (1+ | x | ) − a ,with a > n/
2, as in Theorem 1.2. We will show that if (1.4) or (1.5) holds, then µ isuniformly n -rectifiable.We denote by e U ( ϕ, c ) the class of n -AD-regular measures with constant c such that f ( r, x ) = ϕ r ∗ µ ( x ) is constant on (0 , ∞ ) × supp( µ ). Lemma 3.1.
Let µ be an n -AD-regular measure such that ∈ supp( µ ) . For all ε > there exists δ > such that if Z δ − δ Z x ∈ ¯ B (0 ,δ − ) | e ∆ µ,ϕ ( x, r ) | dµ ( x ) dr ≤ δ, then dist B (0 , ( µ, e U ( ϕ, c )) < ε. Proof.
Suppose that there exists an ε >
0, and for each m ≥ n -AD-regularmeasure µ m such that 0 ∈ supp( µ m ),(3.1) Z m /m Z x ∈ ¯ B (0 ,m ) | e ∆ µ m ,ϕ ( x, r ) | dµ m ( x ) dr ≤ m , QUARE FUNCTIONS AND UNIFORM RECTIFIABILITY 7 and(3.2) dist B (0 , ( µ m , e U ( ϕ, c )) ≥ ε. By (1.1), we can replace { µ m } by a subsequence converging weak * (i.e. when testedagainst compactly supported continuous functions) to a measure µ and it is easy to checkthat 0 ∈ supp( µ ) and that µ is also n -dimensional AD-regular with constant c . We claimthat(3.3) Z ∞ Z x ∈ R d | e ∆ µ,ϕ ( x, r ) | dµ ( x ) dr = 0 . The proof of (3.3) is elementary. Fix m and let η >
0. Because of (1.1) and the decayconditions assumed for ϕ there exists A > m so that(3.4) sup /m ≤ t ≤ m Z ¯ B (0 , m ) Z | x − y | >A | ∂ ϕ ( x − y, t ) | dν ( y ) dν ( x ) < ηm whenever ν satisfies (1.1) with constant c . Set K = [1 /m , m ] × ¯ B (0 , m ) and let e χ be a continuous function with compact support such that χ B (0 ,A ) ≤ e χ ≤ . Then, writing ψ t ( x ) = ∂ ϕ ( x, t ) we have by (3.4) Z Z K | ((1 − e χ ) ψ t ) ∗ µ ( x ) | dµ ( x ) dt < η, and by (3.1) Z Z K | ( e χψ t ) ∗ µ m ( x ) | dµ m ( x ) dt < η + 1 m . Now { y → e χ ( x − y ) ψ t ( x − y ) , ( t, x ) ∈ K } is an equicontinuous family of continuousfunctions supported inside a fixed compact set, which implies that ( e χψ t ) ∗ µ m ( x ) convergesto ( e χψ t ) ∗ µ ( x ) uniformly on K . It therefore follows that(3.5) Z Z K | ψ t ∗ µ ( x ) | dµ ( x ) dt ≤ η +lim sup m Z m /m Z x ∈ ¯ B (0 ,m ) | ( e χψ t ) ∗ µ m ( x ) | dµ m ( x ) dt ≤ η. Since η is arbitrary the left side of (3.5) vanishes, and since this holds for any m ≥
1, ourclaim (3.3) proved.Our next objective consists in showing that µ ∈ e U ( ϕ, c ). To this end, denote by G thesubset of those points x ∈ supp( µ ) such that Z ∞ | e ∆ µ,ϕ ( x, r ) | dr = 0 . It is clear now that G has full µ -measure. For x ∈ G , given 0 < R < R , we have | ϕ R ∗ µ ( x ) − ϕ R ∗ µ ( x ) | = (cid:12)(cid:12)(cid:12)(cid:12)Z R R r ∂ r ( ϕ r ∗ µ ) ( x ) drr (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z R R e ∆ µ,ϕ ( x, r ) drr (cid:12)(cid:12)(cid:12)(cid:12) ≤ R Z R R | e ∆ µ,ϕ ( x, r ) | dr = 0 . (3.6) VASILEIOS CHOUSIONIS, JOHN GARNETT, TRIET LE, AND XAVIER TOLSA
Therefore given x, y ∈ G and R > R > | ϕ R ∗ µ ( x ) − ϕ R ∗ µ ( y ) | = | ϕ R ∗ µ ( x ) − ϕ R ∗ µ ( y ) | ≤ k∇ ( ϕ R ∗ µ ) k ∞ | x − y | . Notice that ∇ ( ϕ R ∗ µ )( x ) = Z ∇ ϕ R ( x − y ) dµ ( y ) , and by decomposing this integral into annuli centered at x , using the fast decay of ∇ ϕ R at ∞ and the fact that µ ( B ( x, r )) ≤ c r n for all r >
0, we easily see that(3.8) k∇ ( ϕ R ∗ µ ) k ∞ ≤ cR , with c depending on c . Thus as R → ∞ the right side of (3.7) tends to 0 and we concludethat ϕ R ∗ µ ( x ) = ϕ R ∗ µ ( y ).By continuity, since G has full µ measure, it follows that f ( r, x ) = ϕ r ∗ µ ( x ) is constanton (0 , ∞ ) × supp( µ ). In other words, µ ∈ e U ( ϕ, c ). However, by condition (3.2), letting m → ∞ , we have dist B (0 , ( µ, e U ( ϕ, c )) ≥ ε, because dist B (0 , ( · , e U ( ϕ, c )) is continuous under the weak * topology, see [Ma, Lemma14.13]. So µ e U ( ϕ, c ), which is a contradiction. (cid:3) By renormalizing the preceding lemma we get:
Lemma 3.2.
Let µ be an n -AD-regular measure such that x ∈ supp( µ ) . For all ε > and r > there exists a constant δ > such that if Z δ − rδ r Z x ∈ ¯ B ( x ,δ − r ) | e ∆ µ,ϕ ( x, t ) | dµ ( x ) dt ≤ δ r n +1 , then dist B ( x ,r ) ( µ, e U ( ϕ, c )) < ε r n +1 . Proof.
Let T : R d → R d be an affine map which maps B ( x , r ) to B (0 , σ = r n T µ , where as usual T µ ( E ) := µ ( T − ( E )), and apply thepreceding lemma to σ . (cid:3) Definition 3.3.
Given n >
0, a Borel measure µ in R d is called n -uniform if there existsa constant c > µ ( B ( x, r )) = c r n for all x ∈ supp( µ ) and r > U ( c ) the collection of all n -uniform measures with constant c . Bythe following lemma, it turns out that e U ( ϕ, · ) and U ( · ) coincide. Lemma 3.4.
Let f : [0 , ∞ ) → [0 , ∞ ) be defined either by f ( x ) = e − x N , for some N ∈ N ,or by f ( x ) = (1 + x ) − a , for some a > n/ . Let µ be a n -dimensional AD-regular Borelmeasure in R d . Then µ is n -uniform if and only if there exists some constant c > suchthat (3.9) Z f (cid:18) | x − y | t (cid:19) dµ ( y ) = c t n for all x ∈ supp( µ ) and t > . QUARE FUNCTIONS AND UNIFORM RECTIFIABILITY 9
For f ( x ) = e − x this lemma is due to De Lellis (see pp. 60-61 of [DeL]) and our proofclosely follows his argument. Proof.
It is clear that (3.9) holds if µ is n -uniform. Now assume (3.9). Write Df ( x ) = x f ′ ( x ). Then(3.10) span (cid:8) D m f : m ≥ (cid:9) is dense in L ((0 , ∞ )) , By the Weierstrass approximation theorem and our particular choice of f .Let B be the set of g ∈ L ((0 , ∞ )) for which there is a constant c g such that Z g (cid:18) | x − y | t (cid:19) dµ ( y ) = c g t n . Then f ∈ B , by the hypothesis (3.9). Differentiating (3.9) with respect to t shows that Df ( x ) = x f ′ ( x ) ∈ B with constant − cn independent of x . Then by induction and theassumption (3.10) B contains a dense subset of L ((0 , ∞ )). Since B is closed in L ((0 , ∞ )),it follows that χ (0 , ∈ B and the lemma is proved. (cid:3) Lemma 3.5.
Let µ be an n -AD-regular measure in R d such that x ∈ supp( µ ) . For all ε > , there exists a constant δ := δ ( ε ) > such that if, for some r > , Z δ − rδ r Z x ∈ ¯ B ( x ,δ − r ) | e ∆ µ,ϕ ( x, t ) | dµ ( x ) dtt ≤ δ n +4 r n , then there exists some constant c > such that (3.11) | µ ( B ( y, t )) − c t n | < εr n for all y ∈ B ( x , r ) ∩ supp( µ ) and < t ≤ r .Proof. Let ε >
0. By Cauchy-Schwarz, we have Z δ − rδ r Z x ∈ ¯ B ( x ,δ − r ) | e ∆ µ,ϕ ( x, t ) | dµ ( x ) dt ≤ "Z δ − rδ r Z x ∈ ¯ B ( x ,δ − r ) | e ∆ µ,ϕ ( x, t ) | dµ ( x ) dtt / "Z δ − rδ r Z x ∈ ¯ B ( x ,δ − r ) t dµ ( x ) dt / ≤ c (cid:2) δ n +4 r n (cid:3) / (cid:2) δ − r µ ( B ( x , δ − r )) (cid:3) / ≤ c (cid:2) δ ( n +4) / r n/ (cid:3) (cid:2) δ − ( n +2) / r ( n +2) / (cid:3) = c δ r n +1 . Hence for any ε > δ is small enough then by Lemma 3.2,dist B ( x , r ) ( µ, e U ( ϕ, c )) < ε r n +1 and there exists σ ∈ U ( c ) such that dist B ( x , r ) ( µ, σ ) < ε r n +1 for a suitable constant c .Let y ∈ B ( x , r ) and for 0 < s ≤ r consider a smooth bump function e χ y,s such that χ B ( y,s ) ≤ e χ y,s ≤ χ B ( y,s (1+ η )) and k∇ e χ y,s k ∞ ≤ csη , where η is some small constant to be determined later. For y ∈ B ( x , r ) and for 0 < s ≤ r , we have (cid:12)(cid:12)(cid:12)(cid:12)Z e χ y,s ( x ) dµ ( x ) − Z e χ y,s ( x ) dσ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ k∇ e χ y,s k ∞ dist B ( x , r ) ( µ, σ ) ≤ c ε r n +1 η s . (3.12)Therefore by (3.12) and Lemma 3.4, for 0 < t ≤ r , µ ( B ( y, t )) ≤ Z e χ y,t ( x ) dµ ( x ) ≤ Z e χ y,t ( x ) dσ ( x ) + c ε r n +1 η t ≤ c t n (1 + η ) n + c ε r n +1 η t , (3.13)and µ ( B ( y, t )) ≥ Z e χ y, t η ( x ) dµ ( x ) ≥ Z e χ y, t η ( x ) dσ ( x ) − c ε r n +1 η t ≥ c t n (1 + η ) n − c ε r n +1 η t . (3.14)Choosing η and ε appropriately, we get that for some small ε := ε ( ε , η ),(3.15) | µ ( B ( y, t )) − c t n | ≤ ε (cid:18) r n +1 t + t n (cid:19) . Hence if t > ε / r , then because t n ≤ r n +1 /t , | µ ( B ( y, t )) − c t n | ≤ c ε r n +1 ε / r ≤ c ε / r n . On the other hand, if t ≤ ε / r , then by the AD-regularity of µ , | µ ( B ( y, t )) − c t n | ≤ µ ( B ( y, t )) + c t n ≤ c ( ε / ) n r n . Therefore, since lim ε → ,η → ε = 0, (3.11) holds if ε and η are sufficiently small. (cid:3) Lemma 3.6.
Let µ be an n -AD-regular measure. Assume that | e ∆ µ,ϕ ( x, r ) | dµ ( x ) drr is aCarleson measure on supp( µ ) × (0 , ∞ ) . Then the weak constant density condition holdsfor µ .Proof. Let ε > A := A ε ⊂ R d × R consist of those pairs ( x, r ) such that (3.11)does not hold. We have to show that Z R Z x ∈ B ( z,R ) χ A ( x, r ) dµ ( x ) drr ≤ c ( ε ) R n for all z ∈ supp( µ ), r > x, r ) ∈ A , then Z δ − rδ r Z y ∈ ¯ B ( x,δ − r ) | e ∆ µ,ϕ ( y, t ) | dµ ( y ) dtt ≥ δ n +4 r n , QUARE FUNCTIONS AND UNIFORM RECTIFIABILITY 11 where δ = δ ( ε ) is as in Lemma 3.5. Then by Chebychev’s inequality, Z R Z x ∈ B ( z,R ) χ A ( x, r ) dµ ( x ) drr ≤ Z R Z x ∈ B ( z,R ) δ n +4 r n Z δ − rδ r Z y ∈ ¯ B ( x,δ − r ) | e ∆ µ,ϕ ( y, t ) | dµ ( y ) dtt ! dµ ( x ) drr ≤ Z δ − R Z | y − z |≤ (1+ δ − ) R | e ∆ µ,ϕ ( y, t ) | Z δ − tδ t µ ( B ( y, δ − r )) δ n +4 r n +1 dr dµ ( y ) dtt . But since Z δ − tδ t µ ( B ( y, δ − r )) δ n +4 r n +1 dr ≤ c δ − n +2) Z δ − tδ t drr ≤ c δ − n +3) , we then get Z R Z x ∈ B ( z,R ) χ A ( x, r ) dµ ( x ) drr ≤ c δ − n +3) Z δ − R Z | y − z |≤ (1+ δ − ) R | e ∆ µ,ϕ ( y, t ) | dµ ( y ) dtt ≤ c δ − n − R n , which is what we needed to show. (cid:3) As an immediate corollary of Theorem 2.3 and Lemma 3.6 we obtain the following.
Theorem 3.7. If µ is an n -AD-regular measure in R d and if c is a constant such that forany ball B ( x , R ) with center x ∈ supp( µ ) , Z R Z x ∈ B ( x ,R ) | e ∆ µ,ϕ ( x, r ) | dµ ( x ) drr ≤ c R n , then µ is uniformly n -rectifiable. We denote by U ( ϕ, c ) the family of n -AD-regular measures with constant c in R d suchthat ∆ µ,ϕ ( x, r ) = 0 for all r > x ∈ supp( µ ) . By an argument similar to the proof of Lemma 3.1 we obtain the following.
Lemma 3.8.
Let µ be an n -AD-regular measure with constant c in R d such that x ∈ supp( µ ) . For all ε > and r > there exists a constant δ > such that if Z δ − rδ r Z x ∈ ¯ B ( x ,δ − r ) | ∆ µ,ϕ ( x, t ) | dµ ( x ) dt ≤ δ r n +1 , then dist B ( x ,r ) ( µ, U ( ϕ, c )) < ε r n +1 . The details of the proof are left for the reader.
Lemma 3.9. If µ ∈ U ( ϕ, c ) then µ is supported on an n -rectifiable set. Proof.
Since µ ∈ U ( ϕ, c ) we have(3.16) ϕ − k ∗ µ ( x ) − ϕ k ∗ µ ( x ) = 0 for all k > x ∈ supp( µ ).Now consider the function F : R d → R defined by F ( x ) = X k> − k (cid:16) ϕ − k ∗ µ ( x ) − ϕ k ∗ µ ( x ) (cid:17) . Taking into account that | ϕ − k ∗ µ ( x ) − ϕ k ∗ µ ( x ) | ≤ c for all x ∈ R d and k ∈ N , it isclear that F ( x ) < ∞ for all x ∈ R d , and so F is well defined. Moreover, by (3.16) we have F = 0 on supp( µ ).Now we claim that F ( x ) > x ∈ R d \ supp( µ ). Indeed, it follows easily thatlim k →∞ ϕ − k ∗ µ ( x ) = 0 for all x ∈ R d \ supp( µ ),while, by the n -AD-regularity of µ ,lim inf k →∞ ϕ k ∗ µ ( x ) ≥ c c − for all x ∈ R d .Thus if x ∈ R d \ supp( µ ) we have ϕ − k ∗ µ ( x ) − ϕ k ∗ µ ( x ) = 0 for all large k >
0, whichimplies that F ( x ) > µ ∈ U ( ϕ, c ), supp( µ ) = F − (0). Next we will show F − (0)is a real analytic variety. Notice that the lemma will follow from this assertion becausesupp( µ ) has locally finite H n measure, so that the analytic variety F − (0) is n -dimensionaland any n -dimensional real analytic variety is n -rectifiable.To prove that the zero set of F is a real analytic variety, it is enough to check that ϕ − k ∗ µ − ϕ k ∗ µ is a real analytic function for each k >
0, because the zero set of areal analytic function is a real analytic variety and the intersection of any family of realanalytic varieties is again a real analytic variety; see [Na]. So it is enough to show that ϕ r ∗ µ is a real analytic function for every r > ϕ ( x ) = e −| x | N , consider the function f : C d → C defined by f ( z , . . . , z d ) = 1 r n Z exp − r − N (cid:18) d X i =1 ( y i − z i ) (cid:19) N ! dµ ( y ) . It is easy to check that f is well defined and holomorphic in the whole C d , and thus ϕ r ∗ µ = f | R d is real analytic.In the case ϕ ( x ) = (1 + | x | ) − a , a > n/
2, for ( z , . . . , z d ) ∈ C d we take f ( z , . . . , z d ) = 1 r n Z (cid:18) r − d X i =1 ( y i − z i ) (cid:19) − a dµ ( y ) . This is a holomorphic function in the open set V = n z ∈ C d : | Im z i | < r d / for 1 ≤ i ≤ d o . Indeed, for z ∈ V , we haveRe (cid:18) r − d X i =1 ( y i − z i ) (cid:19) = 1+ r − d X i =1 (cid:0) ( y i − Re z i ) − (Im z i ) (cid:1) ≥ − r − d X i =1 (Im z i ) > . QUARE FUNCTIONS AND UNIFORM RECTIFIABILITY 13
Thus f is well defined and holomorphic in V , and so ϕ r ∗ µ = f | R d is real analytic. (cid:3) Theorem 3.10. If µ ∈ U ( ϕ, c ) then µ is n -uniform.Proof. If µ ∈ U ( ϕ, c ), then ϕ r ∗ µ = ϕ r ∗ µ ( x ) for all x ∈ supp( µ ) and all r >
0, andconsequently(3.17) ϕ k r ∗ µ ( x ) = ϕ r ∗ µ ( x ) for all 1 ≤ r <
2, all k ∈ Z , and all x ∈ supp( µ ) . By the preceding lemma µ is of the form µ = ρ H n ⌊ E, where ρ is some positive function on E bounded from above and below and E ⊂ R d is an n -rectifiable set. This implies that the densityΘ n ( x, µ ) = lim ε → µ ( B ( x, ε ))(2 ε ) n exists at µ -a.e. x ∈ R d ; see [Ma, Theorem 16.2]. It then follows easily thatlim ε → ϕ ε ∗ µ ( x ) exists at µ -a.e. x ∈ R d and with (3.17) this implies that ϕ R ∗ µ ( x ) = ϕ R ∗ µ ( x ) for all R , R > µ -a.e. x ∈ R d .Using an argument analogous to the proof of Lemma 3.1 we then conclude that ϕ R ∗ µ ( x ) = ϕ R ∗ µ ( y ) for all R , R > x, y ∈ supp( µ ) . Therefore, by Lemma 3.4, µ is n -uniform. (cid:3) Using Lemma 3.8 and Theorem 3.10, we can, with minor changes in their proofs, obtainanalogues of Lemmas 3.5 and 3.6 with e ∆ µ,ϕ replaced by ∆ µ,ϕ . Hence we concluded thefollowing. Theorem 3.11. If µ is an n -AD-regular measure in R d and there exists a constant c suchthat for any ball B ( x , R ) centered at supp( µ ) Z R Z x ∈ B ( x ,R ) | ∆ µ,ϕ ( x, r ) | dµ ( x ) drr ≤ c R n , then µ is uniformly n -rectifiable. Corollary 3.12.
Suppose that for any ball B ( x , R ) centered at supp( µ ) Z R Z x ∈ B ( x ,R ) | ∆ µ ( x, r ) | dµ ( x ) drr ≤ c R n . Then µ is uniformly n -rectifiable.Proof. We will show that (1.3) implies (1.4), by taking a suitable convex combination, andthen apply Theorem 3.11.For
R > e ϕ R : (0 , ∞ ) → (0 , ∞ ) such that(3.18) 1 R n e − s R = Z ∞ r n χ [0 ,r ] ( s ) e ϕ R ( r ) dr = Z ∞ s e ϕ R ( r ) r n dr, for s > . Differentiating with respect to s we get − sR n +2 e − s R = − e ϕ R ( s ) s n . Hence (3.18) is solved for
R > s > e ϕ R ( s ) = 2 s n +1 R n +2 e − s R . Using (3.18) we can now write, for x ∈ supp( µ ), and any R > Z ∞ | ∆ µ,ϕ ( x, R ) | dRR = Z ∞ | ( ϕ R − ϕ R ) ∗ µ ( x ) | dRR = Z ∞ (cid:12)(cid:12)(cid:12) (cid:18)Z ∞ r n χ [0 ,r ] ( | · | ) e ϕ R ( r ) dr ) (cid:19) ∗ µ ( x ) − (cid:18)Z ∞ r n χ [0 ,r ] ( | · | ) e ϕ R ( r ) dr (cid:19) ∗ µ ( x ) (cid:12)(cid:12)(cid:12) dRR . By a change of variables we get Z ∞ r n χ [0 ,r ] ( | y − x | ) e ϕ R ( r ) dr = Z ∞ r ) n χ [0 , r ] ( | y − x | ) e ϕ R ( r ) dr. Therefore, using Cauchy-Schwarz and the fact that R ∞ e ϕ R ( r ) dr .
1, we obtain Z ∞ | ∆ µ,ϕ ( x, R ) | dRR = Z ∞ (cid:12)(cid:12)(cid:12)(cid:12)Z ∞ (cid:18) r n χ B (0 ,r ) ( · ) − r ) n χ B (0 , r ) ( · ) (cid:19) ∗ µ ( x ) e ϕ R ( r ) dr (cid:12)(cid:12)(cid:12)(cid:12) dRR . Z ∞ Z ∞ | ∆ µ ( x, r ) | e ϕ R ( r ) dr dRR . Z ∞ (cid:18)Z ∞ e ϕ R ( r ) dRR (cid:19) | ∆ µ ( x, r ) | dr. Moreover, Z ∞ e ϕ R ( r ) dRR = 2 Z ∞ (cid:16) rR (cid:17) n +1 e − r R dRR = 2 r Z ∞ t n +1 e − t dt . r . Hence we infer that Z ∞ | ∆ µ,ϕ ( x, r ) | drr . Z ∞ | ∆ µ ( x, r ) | drr , which shows that (1.3) implies (1.4). (cid:3) Uniform rectifiabilty implies boundedness of smooth square functions
Let h : R d → R be a smooth function for which there exist positive constants c and ε such that(4.1) | h ( x ) | ≤ c (1 + | x | ) n + ε and |∇ h ( x ) | ≤ c (1 + | x | ) n +1+ ε , for all x ∈ R d . Furthermore assume that Z h ( y − x ) d H n | L ( y ) = 0for every n -plane L and every x ∈ L . For r >
0, denote h r ( x ) = 1 r n h (cid:16) xr (cid:17) . QUARE FUNCTIONS AND UNIFORM RECTIFIABILITY 15
Theorem 4.1.
Let µ be an n -AD-regular measure in R d . If µ is uniformly n -rectifiable,then there exists a constant c such that (4.2) Z R Z x ∈ B ( x ,R ) | h r ∗ µ | dµ ( x ) drr ≤ c R n , for all x ∈ supp( µ ) , R > .Proof. It is immediate to check that the estimate (4.2) holds if and only if for all R ∈ D (4.3) X Q ∈D : Q ⊂ R Z Q Z ℓ ( Q ) ℓ ( Q ) | h r ∗ µ ( x ) | drℓ ( Q ) dµ ( x ) ≤ c µ ( R ) . Let x ∈ B Q and ℓ ( Q ) ≤ r ≤ ℓ ( Q ). If x ∈ B Q ∩ L Q (recall that L Q is the n -planeminimizing α ( Q )), we have Z h r ( y − x ) d H n | L Q ( y ) = 0 . Hence (cid:12)(cid:12)(cid:12)(cid:12)Z h r ( y − x ) dµ ( y ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z h r ( y − x ) d ( µ − c Q H n | L Q )( y ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z X k ≥ e χ k ( y ) h r ( y − x ) d ( µ − c Q H n | L Q )( y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ X k ≥ (cid:12)(cid:12)(cid:12)(cid:12)Z e χ k ( y ) h r ( y − x ) d ( µ − c Q H n | L Q )( y ) (cid:12)(cid:12)(cid:12)(cid:12) , where e χ k , k ≥
0, are bump smooth functions such that • P k ≥ e χ k = 1 • k∇ e χ k k ∞ ≤ ℓ ( Q k ) − , • χ A ( x, k r, k +1 r ) ≤ e χ k ≤ χ A ( x, k − r, k +2 r ) for k ≥
1, and • χ B ( x,r ) ≤ e χ ≤ χ B ( x, r ) . As usual A ( x, r , r ) = { y : r ≤ | y − x | < r } . Moreover for m ∈ N , Q m denotes theancestor of Q such that ℓ ( Q m ) = 2 m ℓ ( Q ).Set F k ( y ) = h r ( x − y ) e χ k ( y ), and notice that supp F k ⊂ B Q k +2 . Then (cid:12)(cid:12)(cid:12)(cid:12)Z h r ( y − x ) dµ ( y ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ X k ≥ (cid:12)(cid:12)(cid:12)(cid:12)Z F k ( y ) d ( µ − c Q k +2 H n | L Qk +2 )( y ) (cid:12)(cid:12)(cid:12)(cid:12) + X k ≥ (cid:12)(cid:12)(cid:12)(cid:12)Z F k ( y ) d ( c Q H n | L Q − c Q k +2 H n | L Qk +2 )( y ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ X k ≥ k∇ F k k ∞ α ( Q k +2 ) ℓ ( Q k +2 ) n +1 + X k ≥ k∇ F k k ∞ dist B Qk +2 ( c Q H n | L Q , c Q k +2 H n | L Qk +2 ):= I + I (4.4) For y ∈ supp F k using (4.1) it follows easily that | h r ( y − x ) | . ℓ ( Q ) n (cid:18) ℓ ( Q ) ℓ ( Q k ) (cid:19) n + ε and |∇ h r ( y − x ) | . ℓ ( Q ) n +1 (cid:18) ℓ ( Q ) ℓ ( Q k ) (cid:19) n +1+ ε . Hence k∇ F k k ∞ . ℓ ( Q k ) 1 ℓ ( Q ) n (cid:18) ℓ ( Q ) ℓ ( Q k ) (cid:19) n + ε + 1 ℓ ( Q ) n +1 (cid:18) ℓ ( Q ) ℓ ( Q k ) (cid:19) n +1+ ε . ℓ ( Q ) ε ℓ ( Q k ) n +1+ ε . (4.5)We can now estimate I : I . X k ≥ α ( Q k +2 ) ℓ ( Q k ) n +1 ℓ ( Q ) ε ℓ ( Q k ) n +1+ ε = X k ≥ α ( Q k +2 ) (cid:18) ℓ ( Q ) ℓ ( Q k ) (cid:19) ε . X P ∈D : R ⊃ Q α ( P ) (cid:18) ℓ ( Q ) ℓ ( P ) (cid:19) ε . (4.6)For I , using also [To1, Lemma 3.4], we get I . X k ≥ ℓ ( Q ) ε ℓ ( Q k ) n +1+ ε X ≤ j ≤ k +2 α ( Q j ) ! ℓ ( Q k +2 ) n +1 . X k ≥ ℓ ( Q ) ℓ ( Q k ) ! ε X ≤ j ≤ k +2 α ( Q j ) ! . X R ∈D : R ⊃ Q X P ∈D : Q ⊂ P ⊂ R α ( P ) (cid:18) ℓ ( Q ) ℓ ( R ) (cid:19) ε = X P ∈D : P ⊃ Q α ( P ) X R ∈D : R ⊃ P (cid:18) ℓ ( Q ) ℓ ( R ) (cid:19) ε ≈ X P ∈D : P ⊃ Q α ( P ) (cid:18) ℓ ( Q ) ℓ ( P ) (cid:19) ε (4.7)Therefore by (4.4), (4.6) and (4.7), for x ∈ B Q ∩ L Q and ℓ ( Q ) ≤ r ≤ ℓ ( Q ),(4.8) (cid:12)(cid:12)(cid:12)(cid:12)Z h r ( y − x ) dµ ( y ) (cid:12)(cid:12)(cid:12)(cid:12) . X P ∈D : P ⊃ Q α ( P ) (cid:18) ℓ ( Q ) ℓ ( P ) (cid:19) ε . On the other hand, given an arbitrary x ∈ Q , let x ′ be its orthogonal projection on L Q (notice that x ′ ∈ B Q ). We have (cid:12)(cid:12)(cid:12)(cid:12)Z h r ( y − x ) dµ ( y ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)Z h r ( y − x ′ ) dµ ( y ) (cid:12)(cid:12)(cid:12)(cid:12) + Z B Q | h r ( y − x ) − h r ( y − x ′ ) | dµ ( y )+ Z R d \ B Q | h r ( y − x ) − h r ( y − x ′ ) | dµ ( y ):= I + I + I . (4.9) QUARE FUNCTIONS AND UNIFORM RECTIFIABILITY 17
For ℓ ( Q ) ≤ r ≤ ℓ ( Q ), by (4.8),(4.10) I . X P ∈D : P ⊃ Q α ( P ) (cid:18) ℓ ( Q ) ℓ ( P ) (cid:19) ε . We can now estimate I and I using (4.1). First(4.11) I . Z B Q | x − x ′ | ℓ ( Q ) n +1 dµ ( y ) . dist( x, L Q ) ℓ ( Q ) n +1 ℓ ( Q ) n = dist( x, L Q ) ℓ ( Q ) . Moreover, noticing that if y / ∈ B Q and ξ ∈ [ y − x, y − x ′ ] we have that | y − x | ≈ | ξ | , I . Z | x − x ′ | ℓ ( Q ) n +1 sup ξ ∈ [ y − x,y − x ′ ] |∇ ( h r )( ξ ) | dµ ( y ) . | x − x ′ | ℓ ( Q ) n +1 Z ℓ ( Q ) n +1+ ε ( ℓ ( Q ) + | y − x | ) n +1+ ε dµ ( y ) . dist( x, L Q ) ℓ ( Q ) ε ℓ ( Q ) − − ε = dist( x, L Q ) ℓ ( Q ) . (4.12)Hence by (4.9), (4.10), (4.11) and (4.12), we get the following pointwise estimate for x ∈ Q and ℓ ( Q ) ≤ r ≤ ℓ ( Q ):(4.13) | h r ∗ µ ( x ) | . dist( x, L Q ) ℓ ( Q ) + X P ∈D : P ⊃ Q α ( P ) (cid:18) ℓ ( Q ) ℓ ( P ) (cid:19) ε . Therefore, X Q ∈D : Q ⊂ R Z Q Z ℓ ( Q ) ℓ ( Q ) | h r ∗ µ ( x ) | drℓ ( Q ) dµ ( x ) . X Q ∈D : Q ⊂ R Z Q Z ℓ ( Q ) ℓ ( Q ) (cid:18) dist( x, L Q ) ℓ ( Q ) (cid:19) drℓ ( Q ) dµ ( x )+ X Q ∈D : Q ⊂ R Z Q Z ℓ ( Q ) ℓ ( Q ) X P ∈D : P ⊃ Q α ( P ) (cid:18) ℓ ( Q ) ℓ ( P ) (cid:19) ε drℓ ( Q ) dµ ( x ) . X Q ∈D : Q ⊂ R Z (cid:18) dist( x, L Q ) ℓ ( Q ) (cid:19) dµ ( x )++ X Q ∈D : Q ⊂ R X P ∈D : P ⊃ Q α ( P ) (cid:18) ℓ ( Q ) ℓ ( P ) (cid:19) ε X P ∈D : P ⊃ Q (cid:18) ℓ ( Q ) ℓ ( P ) (cid:19) ε µ ( Q ) , where we used Cauchy-Schwarz for the last inequality. By [To1, Lemmas 5.2 and 5.4], X Q ∈D : Q ⊂ R Z Q dist( x, L Q ) ℓ ( Q ) dµ ( x ) . µ ( R ) . Finally, X Q ∈D : Q ⊂ R X P ∈D : P ⊃ Q α ( P ) (cid:18) ℓ ( Q ) ℓ ( P ) (cid:19) ε X P ∈D : P ⊃ Q (cid:18) ℓ ( Q ) ℓ ( P ) (cid:19) ε µ ( Q ) . X Q ∈D : Q ⊂ R X P ∈D : Q ⊂ P ⊂ R α ( P ) (cid:18) ℓ ( Q ) ℓ ( P ) (cid:19) ε µ ( Q )+ X Q ∈D : Q ⊂ R X P ∈D : P ⊃ R α ( P ) (cid:18) ℓ ( Q ) ℓ ( P ) (cid:19) ε µ ( Q ) . X P ∈D : P ⊂ R α ( P ) X Q ∈D : Q ⊂ P (cid:18) ℓ ( Q ) ℓ ( P ) (cid:19) ε µ ( Q ) + X Q ∈D : Q ⊂ R (cid:18) ℓ ( Q ) ℓ ( R ) (cid:19) ε µ ( Q ) . X P ∈D : P ⊂ R α ( P ) µ ( P ) + µ ( R ) . µ ( R ) , where the last inequality follows from Theorem 2.1. (cid:3) The proof of Theorem 1.2 follows from Theorems 3.11, 3.7 and Theorem 4.1.5.
Uniform rectifiabilty implies boundedness of square functions: thenon-smooth case
By Corollary 3.12 we already know that condition (1.3) implies the uniform n -rectifiabilityof µ , assuming µ to be n -AD-regular. So to complete the proof of Theorem 1.1 it remainsto show that (1.3) holds for any ball B ( x , R ) centered at supp( µ ) if µ is uniformly n -rectifiable. To this end, we would like to argue as in the preceding section, setting φ r = 1 r n χ B (0 ,r ) ( x ) , x ∈ R d , and h r = φ r − φ r . The main obstacle is the lack of smoothness of h r . To solve this problem we will decompose h r using wavelets as follows.Consider a family of C compactly supported orthonormal wavelets in R n . Tensorproducts of Daubechies compactly supported wavelets with 3 vanishing moments willsuffice for our purposes, see e.g. [Mal, Section 7.2.3]. We denote this family of functionsby { ψ ǫI } I ∈D ( R n ) , ≤ ǫ ≤ n − , where D ( R n ) is the standard grid of dyadic cubes in R n . Each ψ ǫI is a C function supported on 5 I , which satisfies k ψ ǫI k = 1, and moreover k ψ ǫI k ∞ . ℓ ( I ) n/ , k∇ ψ ǫI k ∞ . ℓ ( I ) n/ for all I ∈ D ( R n ) and 1 ≤ ǫ ≤ n − ℓ ( I ) is the sidelength of the cube I . Recall that any function f ∈ L ( R n ) can bewritten as f = X I ∈D ( R n ) h f, ψ ǫI i ψ ǫI . QUARE FUNCTIONS AND UNIFORM RECTIFIABILITY 19
To simplify notation and avoid using the ǫ index, we consider 2 n − D ( R n ) andwe denote by e D ( R n ) their union. Then we can write f = X I ∈ e D ( R n ) h f, ψ I i ψ I , with the sum converging in L ( R n ).In particular, we have(5.1) e h := χ B n (0 , − n χ B n (0 , = X I ∈ e D ( R n ) a I ψ I , where B n (0 , r ) stands for the ball centered at 0 with radius r in R n and a I = D χ B n (0 , − n χ B n (0 , , ψ I E . So we have 1 r n χ B n (0 ,r ) ( x ) − r ) n χ B n (0 , r ) ( x ) = X I ∈ e D ( R n ) a I r n ψ I (cid:16) xr (cid:17) . Notice that we have been talking about wavelets in R n although the ambient space of themeasure µ and the function h r is R d , with d ≥ n . We identify R n with the “horizontal”subspace of R d given by R n × { } × . . . × { } and we consider the following circularprojection Π : R d → R n . For x = ( x , . . . , x d ) ∈ R d we denote x H := ( x , . . . , x n ) and x V = ( x n +1 , . . . , x d ). If x H = 0 we setΠ( x ) = | x || x H | x H . If x H = 0, we set Π( x ) = ( | x | , , . . . , | x | = | Π( x ) | .Notice also that h r ( x ) = 1 r n χ B n (0 ,r ) (Π( x )) − r ) n χ B n (0 , r ) (Π( x )) = X I ∈ e D ( R n ) a I r n ψ I (cid:18) Π( x ) r (cid:19) . Thus,(5.2) h r ∗ µ ( x ) = X I ∈ e D ( R n ) a I r n ψ I (cid:18) Π( · ) r (cid:19) ∗ µ ( x ) . Observe that the functions ψ I are smooth, and so one can guess that the α coefficientsof [To1] will be useful to estimate ψ I (cid:16) Π( · ) r (cid:17) ∗ µ ( x ). Concerning the coefficients a I we have: Lemma 5.1.
For I ∈ e D ( R n ) , we have: (a) If I ∩ (cid:0) ∂B n (0 , ∪ ∂B n (0 , (cid:1) = ∅ , then a I = 0 . (b) If ℓ ( I ) & , then | a I | . ℓ ( I ) − − n/ . (c) If ℓ ( I ) . , then | a I | . ℓ ( I ) n/ . Proof.
The first statement follows from the fact that the wavelets ψ I have zero mean in R n and that e h = χ B n (0 , − n χ B n (0 , is constant on supp ψ I if 5 I ∩ (cid:0) ∂B n (0 , ∪ ∂B n (0 , (cid:1) = ∅ .The statement (c) is immediate: | a I | = (cid:12)(cid:12)(cid:12)(cid:12)Z R n e h ψ I dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ k ψ I k . ℓ ( I ) n/ k ψ I k = ℓ ( I ) n/ . Finally (b) follows from the smoothness of ψ I and the fact that e h has zero mean. Indeed, | a I | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z B n (0 , e h ( x ) ( ψ I ( x ) − ψ I (0)) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ k∇ ψ I k ∞ Z | e h | dx . ℓ ( I ) n/ . (cid:3) By estimating ψ I (cid:16) Π( · ) r (cid:17) ∗ µ ( x ) in terms of the α ( Q )’s, using some arguments in thespirit of the ones in [MT], below we will prove the following. Theorem 5.2.
Let µ be an n -AD-regular measure in R d . If µ is uniformly n -rectifiable,then there exists a constant c such that (5.3) Z R Z x ∈ B ( x ,R ) | h r ∗ µ ( x ) | dµ ( x ) drr ≤ c R n , for all x ∈ supp( µ ) , R > . Preliminaries for the proof of Theorem 5.2.
It is immediate to check that theestimate (5.3) holds if and only if for all R ∈ D (5.4) X Q ∈D : Q ⊂ R Z Q Z ℓ ( Q ) ℓ ( Q ) | h r ∗ µ ( x ) | drℓ ( Q ) dµ ( x ) ≤ c µ ( R ) . Let δ > α (1000 Q ) ≤ δ . Otherwise we have | h r ∗ µ ( x ) | . ≤ α (1000 Q ) δ and, by Theorem 2.1, X Q ∈D ( R ) α (1000 Q ) ≥ δ ℓ ( Q ) Z Q Z ℓ ( Q ) ℓ ( Q ) | h r ∗ µ ( x ) | drdµ ( x ) . δ X Q ∈D ( R ) α (1000 Q ) µ ( Q ) . δ µ ( R ) . (5.5)Since the functions h r are even, we have h r ∗ µ ( x ) = Z h r ( y − x ) dµ ( y ) . Recalling (5.2), we get h r ∗ µ ( x ) = 1 r n X I ∈ e D ( R n ) a I Z ψ I (cid:18) Π( y − x ) r (cid:19) dµ ( y ) . QUARE FUNCTIONS AND UNIFORM RECTIFIABILITY 21
By Lemma 5.1, a I = 0 whenever 5 I ∩ (cid:0) ∂B n (0 , ∪ ∂B n (0 , (cid:1) = ∅ . Therefore the precedingsum ranges over those I such that 5 I ∩ (cid:0) ∂B n (0 , ∪ ∂B n (0 , (cid:1) = ∅ and the domain ofintegration of each ψ I (cid:16) Π( · ) r (cid:17) is Π − ( r · I ).Notice that 5 I stands for the cube from R n concentric with I with side length equal to5 ℓ ( I ). On the other hand, given a set A ⊂ R d , we write r · A = { r · x ∈ R d : x ∈ A } . So r · I = r · (5 I ) is a cube in R d with side length 5 rℓ ( I ) which is not concentric with I unless I is centered at the origin.We set h r ∗ µ ( x ) = 1 r n X I ∈ e D ( R n ): ℓ ( I ) ≥ / a I Z ψ I (cid:18) Π( y − x ) r (cid:19) dµ ( y )+ 1 r n X I ∈ e D ( R n ): ℓ ( I ) < / a I Z ψ I (cid:18) Π( y − x ) r (cid:19) dµ ( y )=: F r ( x ) + G r ( x ) , (5.6)so that X Q ∈D : Q ⊂ R Z Q Z ℓ ( Q ) ℓ ( Q ) | h r ∗ µ ( x ) | drℓ ( Q ) dµ ( x ) . X Q ∈D : Q ⊂ R,α (1000 Q ) ≥ δ Z Q Z ℓ ( Q ) ℓ ( Q ) | h r ∗ µ ( x ) | drℓ ( Q ) dµ ( x )+ X Q ∈D : Q ⊂ R,α (1000 Q ) ≤ δ Z Q Z ℓ ( Q ) ℓ ( Q ) | F r ( x ) | drℓ ( Q ) dµ ( x )+ X Q ∈D : Q ⊂ R,α (1000 Q ) ≤ δ Z Q Z ℓ ( Q ) ℓ ( Q ) | G r ( x ) | drℓ ( Q ) dµ ( x )=: I + I + I . (5.7)As shown in (5.5), we have I . δ µ ( R ) . Thus to prove Theorem 5.2 it is enough to show that I + I ≤ c ( δ ) µ ( R ).5.2. Estimate of the term I in (5.7) . We first need to estimate F r ( x ). To this end,we take Q ∈ D and r > x ∈ Q and ℓ ( Q ) ≤ r < ℓ ( Q ). We also assume that L Q (the best approximating plane for α ( Q )) is parallel to R n .Let I ∈ e D ( R n ) be such that ℓ ( I ) ≥ /
100 and 5 I ∩ (cid:0) ∂B n (0 , ∪ ∂B n (0 , (cid:1) = ∅ . Let P := P ( I ) ∈ D be some cube containing Q such that ℓ ( P ) ≈ rℓ ( I ) ≈ ℓ ( Q ) ℓ ( I ). Let also φ P be a smooth bump function such that χ P ≤ φ P ≤ χ B P , k∇ φ P k ∞ ≤
1, and φ P = 1 on x + Π − ( r · I ). Then Z ψ I (cid:18) Π( y − x ) r (cid:19) dµ ( y ) = Z P ψ I (cid:18) Π( y − x ) r (cid:19) dµ ( y ) = Z φ P ( y ) ψ I (cid:18) Π( y − x ) r (cid:19) dµ ( y ) . Lemma 5.3.
Let I ∈ e D ( R n ) be such that ℓ ( I ) ≥ / and I ∩ (cid:0) ∂B n (0 , ∪ ∂B n (0 , (cid:1) = ∅ and let P = P ( I ) as above. We have (5.8) (cid:12)(cid:12)(cid:12)(cid:12)Z ψ I (cid:18) Π( y − x ) r (cid:19) dµ ( y ) (cid:12)(cid:12)(cid:12)(cid:12) . (cid:18) ℓ ( Q ) ℓ ( P ) (cid:19) n/ dist( x, L Q ) ℓ ( P ) + X S ∈D : Q ⊂ S ⊂ P α (2 S ) ℓ ( P ) n . Proof.
Without loss of generality we assume that x = 0. Let L be the plane parallel to L Q passing through 0 (that is, L = R n ) and denote by Π ⊥ the orthogonal projection onto L . Then Z ψ I (cid:18) Π( y ) r (cid:19) dµ ( y ) = Z φ P ( y ) ψ I (cid:18) Π( y ) r (cid:19) dµ ( y )= Z φ P ( y ) (cid:18) ψ I (cid:18) Π( y ) r (cid:19) − ψ I (cid:18) Π ⊥ ( y ) r (cid:19)(cid:19) dµ ( y )+ Z φ P ( y ) ψ I (cid:18) Π ⊥ ( y ) r (cid:19) d ( µ − c P H n | L )( y )+ c P Z L φ P ( y ) ψ I (cid:18) Π ⊥ ( y ) r (cid:19) d H n ( y )=: A + A + A . (5.9)Since ψ I (cid:16) Π ⊥ ( y ) r (cid:17) = ψ I (cid:0) yr (cid:1) for y ∈ L , and φ P = 1 on r · I , we get(5.10) A = c P Z I ψ I (cid:16) yr (cid:17) dy = 0 . We now proceed to estimate A : | A | ≤ (cid:12)(cid:12)(cid:12)(cid:12)Z φ P ( y ) ψ I (cid:18) Π ⊥ ( y ) r (cid:19) d ( µ − c P H n | L P )( y ) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) c P Z φ P ( y ) ψ I (cid:18) Π ⊥ ( y ) r (cid:19) d ( H n | L P − H n | L )( y ) (cid:12)(cid:12)(cid:12)(cid:12) . (cid:13)(cid:13)(cid:13)(cid:13) ∇ (cid:18) φ P ψ I (cid:18) Π ⊥ ( · ) r (cid:19)(cid:19)(cid:13)(cid:13)(cid:13)(cid:13) ∞ (cid:0) α ( P ) ℓ ( P ) n +1 + dist H ( L P ∩ B P , L ∩ B P ) ℓ ( P ) n (cid:1) , (5.11) QUARE FUNCTIONS AND UNIFORM RECTIFIABILITY 23 from the definition of the α numbers and the fact that c P ≈
1. Using the gradient boundsfor the functions φ P and ψ I , and the fact that ℓ ( P ) ≈ rℓ ( I ) ≈ ℓ ( Q ) ℓ ( I ) , we get (cid:13)(cid:13)(cid:13)(cid:13) ∇ (cid:18) φ P ψ I (cid:18) Π ⊥ ( · ) r (cid:19)(cid:19)(cid:13)(cid:13)(cid:13)(cid:13) ∞ . k∇ φ P k ∞ k ψ I k ∞ + (cid:13)(cid:13)(cid:13)(cid:13) ∇ (cid:18) ψ I (cid:18) Π ⊥ ( · ) r (cid:19)(cid:19)(cid:13)(cid:13)(cid:13)(cid:13) ∞ . ℓ ( P ) 1 ℓ ( I ) n/ + 1 ℓ ( I ) n/ r ≈ ℓ ( Q ) 1 ℓ ( I ) n/ ≈ ℓ ( Q ) (cid:18) ℓ ( Q ) ℓ ( P ) (cid:19) n/ . (5.12)We also remark that in the previous estimate we used the fact that k Π ⊥ k ∞ ≤
1, whichdoes not hold for the spherical projection Π.Furthermore, by [To1, Lemma 5.2 and Remark 5.3],dist H ( L P ∩ B P , L ∩ B P ) ≤ dist H ( L P ∩ B P , L Q ∩ B P ) + dist(0 , L Q ) ≤ X S ∈D : Q ⊂ S ⊂ P α ( S ) ℓ ( P ) + dist(0 , L Q ) . (5.13)Therefore, by (5.11), (5.12), and (5.13),(5.14) | A | . (cid:18) ℓ ( Q ) ℓ ( P ) (cid:19) n/ ℓ ( P ) n X S ∈D : Q ⊂ S ⊂ P α ( S ) + dist(0 , L Q ) ℓ ( P ) . We now estimate the term A : | A | = (cid:12)(cid:12)(cid:12)(cid:12)Z φ P ( y ) (cid:18) ψ I (cid:18) Π( y ) r (cid:19) − ψ I (cid:18) Π ⊥ ( y ) r (cid:19)(cid:19) dµ ( y ) (cid:12)(cid:12)(cid:12)(cid:12) . k∇ ψ I k ∞ r Z B P | Π( y ) − Π ⊥ ( y ) | dµ ( y ) . ℓ ( Q ) (cid:18) ℓ ( Q ) ℓ ( P ) (cid:19) n/ Z B P | Π( y ) − Π ⊥ ( y ) | dµ ( y ) . (5.15)It is easy to check that(5.16) | Π( y ) − Π ⊥ ( y ) | . dist( y, L ) . Furthermore, as in (5.13), for y ∈ B P ,dist( y, L ) ≤ dist(0 , L Q ) + dist( y, L Q ) ≤ dist(0 , L Q ) + dist( y, L P ) + dist H ( L P ∩ B P , L Q ∩ B P ) . dist(0 , L Q ) + dist( y, L P ) + X S ∈D : Q ⊂ S ⊂ P α ( S ) ℓ ( P ) . (5.17) Therefore, by (5.15), (5.16), and (5.17), | A | . ℓ ( Q ) (cid:18) ℓ ( Q ) ℓ ( P ) (cid:19) n/ (cid:16) dist(0 , L Q ) ℓ ( P ) n + Z B P dist( y, L P ) dµ ( y ) + ℓ ( P ) n +1 X S ∈D : Q ⊂ S ⊂ P α ( S ) (cid:17) . (cid:18) ℓ ( Q ) ℓ ( P ) (cid:19) n/ dist(0 , L Q ) ℓ ( P ) + X S ∈D : Q ⊂ S ⊂ P α (2 S ) ℓ ( P ) n , (5.18)where we used that, by [To1, Remark 3.3], Z B P dist( y, L P ) dµ ( y ) . α (2 P ) ℓ ( P ) n +1 . The lemma follows from the estimates (5.9), (5.10), (5.14), and (5.18). (cid:3)
Lemma 5.4.
We have | F r ( x ) | . dist( x, L Q ) ℓ ( Q ) + X S ∈D : S ⊃ Q α (2 S ) ℓ ( Q ) ℓ ( S ) . (5.19) Proof.
Recalling that P = P ( I ) ⊃ Q , by (5.8), | F r ( x ) | . ℓ ( Q ) n X I ∈ e D ( R n ): ℓ ( I ) ≥ / | a I | (cid:18) ℓ ( Q ) ℓ ( P ( I )) (cid:19) n/ dist( x, L Q ) ℓ ( P ( I )) + X S ∈D : Q ⊂ S ⊂ P ( I ) α (2 S ) ℓ ( P ( I )) n . Using (b) from Lemma 5.1, | F r ( x ) | . X I ∈ e D ( R n ): ℓ ( I ) ≥ / ,P ( I ) ⊃ Q ℓ ( Q ) ℓ ( P ( I )) dist( x, L Q ) ℓ ( P ( I )) + X S ∈D : Q ⊂ S ⊂ P ( I ) α (2 S ) . X P ∈D : P ⊃ Q ℓ ( Q ) ℓ ( P ) dist( x, L Q ) ℓ ( P ) + X S ∈D : Q ⊂ S ⊂ P α (2 S ) = X P ∈D : P ⊃ Q dist( x, L Q ) ℓ ( Q ) ℓ ( P ) + X S ∈D : S ⊃ Q α (2 S ) X P ∈D : P ⊃ S ℓ ( Q ) ℓ ( P ) . dist( x, L Q ) ℓ ( Q ) + X S ∈D : S ⊃ Q α (2 S ) ℓ ( Q ) ℓ ( S ) . (cid:3) Lemma 5.5.
The term I in (5.7) satisfies I . µ ( R ) . QUARE FUNCTIONS AND UNIFORM RECTIFIABILITY 25
Proof.
By (5.19),(5.20) I . X Q ∈D ( R ) Z Q dist( x, L Q ) ℓ ( Q ) + X S ∈D : S ⊃ Q α (2 S ) ℓ ( Q ) ℓ ( S ) dµ ( x ) . By Cauchy-Schwartz,(5.21) X S ∈D : S ⊃ Q α (2 S ) ℓ ( Q ) ℓ ( S ) ≤ X S ∈D : S ⊃ Q α (2 S ) ℓ ( Q ) ℓ ( S ) · X S ∈D : S ⊃ Q ℓ ( Q ) ℓ ( S ) . Since P S ∈D : S ⊃ Q ℓ ( Q ) ℓ ( S ) . I . X Q ∈D ( R ) Z Q (cid:18) dist( x, L Q ) ℓ ( Q ) (cid:19) dµ ( x ) + X Q ∈D ( R ) X S ∈D : S ⊃ Q α (2 S ) ℓ ( Q ) ℓ ( S ) µ ( Q )=: S + S . (5.22)By [To1, Lemmas 5.2 and Lemma 5.4] and Theorem 2.1, we obtain S . µ ( R ). We nowdeal with the term S : S = X Q ∈D ( R ) X S ∈D : Q ⊂ S ⊂ R α (2 S ) ℓ ( Q ) ℓ ( S ) µ ( Q ) + X Q ∈D ( R ) X S ∈D : S ) R α (2 S ) ℓ ( Q ) ℓ ( S ) µ ( Q )=: S + S . (5.23)Using just that α (2 S ) . S . X Q ∈D ( R ) X S ∈D : S ⊃ R ℓ ( Q ) ℓ ( S ) µ ( Q ) . X Q ∈D ( R ) µ ( Q ) ℓ ( Q ) ℓ ( R ) . µ ( R ) . Finally, using Fubini and Theorem 2.1,(5.25) S ≤ X S ∈D ( R ) α (2 S ) X Q ∈D : Q ⊂ S ℓ ( Q ) ℓ ( S ) µ ( Q ) . X S ∈D ( R ) α (2 S ) µ ( S ) . µ ( R ) . By (5.22), (5.23), (5.24), and (5.25) we obtain I . µ ( R ). (cid:3) Estimate of the term I in (5.7) . It remains to show that I . µ ( R ). Recall thatthe cubes in the sum corresponding to I in (5.7) satisfy α (1000 Q ) ≤ δ .We need now to estimate G r ( x ) (see (5.6)) for x ∈ Q and ℓ ( Q ) ≤ r < ℓ ( Q ). Recallthat(5.26) G r ( x ) = 1 r n X I ∈ e D ( R n ): ℓ ( I ) < / a I Z ψ I (cid:18) Π( y − x ) r (cid:19) dµ ( y ) . The arguments will be more involved than the ones we used for F r ( x ).To estimate G r ( x ) we now introduce a stopping time condition for P ∈ D : P belongsto G if(1) P ⊂ Q , and(2) P S ∈D : P ⊂ S ⊂ Q α (100 S ) ≤ δ . The maximal cubes in
D \ G may vary significantly in size, even if they are neighbors, andthis would cause problems. For this reason we use a quite standard smoothing procedure.We define(5.27) ℓ ( y ) := inf P ∈G ( ℓ ( P ) + dist( y, P )) , y ∈ R d , and(5.28) d ( z ) := inf y ∈ Π − ( z ) ℓ ( y ) , z ∈ R n . Lemma 5.6.
The function ℓ ( · ) is -Lipschitz, and the function d ( · ) is -Lipschitz.Proof. For simplicity we assume that x = 0. The function ℓ ( · ) is 1-Lipschitz, as theinfimum of the family of 1-Lipschitz functions { ℓ ( P ) + dist( · , P ) } P ∈G .Let us turn our attention to d ( · ). Let z, z ′ ∈ R n and ε >
0. Let y ∈ Π − ( z ) such that ℓ ( y ) ≤ d ( z ) + ε . Consider the points y = | z ′ || y | y and z = | z ′ || z | z. Notice that Π( y ) = z . Let L y be the n -plane parallel to R n which contains y andconsider the point { y ′ } = Π − ( z ′ ) ∩ L y . That is, y ′ is the point which fulfils the followingproperties: | y ′ | = | z ′ | , y ′ H = | y ′ H || z ′ | z ′ , y ′ V = y V . Observe that | y | = | z | = | z ′ | = | y ′ | . Since y V = y ′ V , this implies that | y H | = | y ′ H | . Furthermore, | y − y ′ | = | y H − y ′ H | = (cid:12)(cid:12)(cid:12)(cid:12) | y H || y | z − | y ′ H || y ′ | z ′ (cid:12)(cid:12)(cid:12)(cid:12) = | y ′ H || y ′ | | z − z ′ | ≤ | z − z ′ | . Moreover, | y − y | = | z − z | = (cid:12)(cid:12)(cid:12)(cid:12) z − | z ′ || z | z (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12) | z | − | z ′ | (cid:12)(cid:12) ≤ | z − z ′ | . Hence, | y − y ′ | ≤ | y − y | + | y − y ′ | ≤ | z − z ′ | + | z − z ′ |≤ | z − z ′ | + | z − z | + | z − z ′ | ≤ | z − z ′ | . Then, using that ℓ is 1-Lipschitz and (5.3), d ( z ′ ) ≤ ℓ ( y ′ ) ≤ | y − y ′ | + ℓ ( y ) ≤ | z − z ′ | + d ( z ) + ε. Since ε > d ( z ′ ) ≤ | z − z ′ | + d ( z ). In the same way onegets that d ( z ) ≤ | z − z ′ | + d ( z ′ ). (cid:3) For δ small enough, the condition α (1000 Q ) ≤ δ guarantees that any cube P ⊂ Q such that ℓ ( P ) = ℓ ( Q ) belongs to G , in particular ℓ ( y ) ≤ ℓ ( Q ) for all y ∈ Q .Furthermore since G = ∅ , we deduce that ℓ ( y ) , d ( z ) < ∞ for all y ∈ R d , z ∈ R n .Now we consider the family F of cubes I ∈ D ( R n ) such that(5.29) r diam( I ) ≤ z ∈ r · I d ( z ) . QUARE FUNCTIONS AND UNIFORM RECTIFIABILITY 27
Let F ⊂ F be the subfamily of F consisting of cubes with maximal length. In particularthe cubes in F are pairwise disjoint. Moreover it is easy to check that if I, J ∈ F and(5.30) 20 I ∩ J = ∅ , then ℓ ( I ) ≈ ℓ ( J ).We denote by G ( x, r ) the family of cubes I ∈ e D ( R n ) which satisfy • ℓ ( I ) ≤ , • I ∩ ( ∂B n (0 , ∪ ∂B n (0 , = ∅ , • (cid:0) x + Π − ( r · I ) (cid:1) ∩ supp( µ ) = ∅ , and • I is not contained in any cube from F .We denote by T ( x, r ) the family of cubes I ∈ e D ( R n ) which satisfy • ℓ ( I ) ≤ , • I ∩ ( ∂B n (0 , ∪ ∂B n (0 , = ∅ , • (cid:0) x + Π − ( r · I ) (cid:1) ∩ supp( µ ) = ∅ , and • I ∈ F .Now we write G r ( x ) = 1 r n X I ∈G ( x,r ) a I Z ψ I (cid:18) Π( y − x ) r (cid:19) dµ ( y )+ 1 r n X I ∈T ( x,r ) X J ∈ e D ( R n ): J ⊂ I a J Z ψ J (cid:18) Π( y − x ) r (cid:19) dµ ( y )=: G r, ( x ) + G r, ( x ) , (5.31)so that I ≤ X Q ∈D : Q ⊂ R,α (1000 Q ) ≤ δ Z Q Z ℓ ( Q ) ℓ ( Q ) | G r, ( x ) | drℓ ( Q ) dµ ( x )+ X Q ∈D : Q ⊂ R,α (1000 Q ) ≤ δ Z Q Z ℓ ( Q ) ℓ ( Q ) | G r, ( x ) | drℓ ( Q ) dµ ( x )=: I + I . (5.32)First we will deal with the term G r, ( x ). To this end we need several auxiliary lemmas. Lemma 5.7. If I ∈ G ( x, r ) , then there exists P := P ( I ) ∈ D with ℓ ( P ) ≈ rℓ ( I ) such that supp( µ ) ∩ (cid:0) x + Π − ( r · I ) (cid:1) ⊂ P. Proof.
Notice that, by definition, supp( µ ) ∩ ( x + Π − ( r · I )) = ∅ . Observe also that theconclusion of the lemma holds if ℓ ( r · I ) ≈ ℓ ( Q ) because α (1000 Q ) ≤ δ .So assume that ℓ ( r · I ) ≪ ℓ ( Q ) and consider z ∈ r · I . Since I ∈ G ( x, r ), I / ∈ F , and d is 3-Lipschitz, we have d ( z ) ≤ c rℓ ( I ) , for some absolute constant c . Take y ∈ x + Π − ( r · I ) such that ℓ ( y ) ≤ c rℓ ( I ) . Let ε = c rℓ ( I ). By definition, there exists some cube P ′ ∈ G such that ℓ ( P ′ ) + dist( y, P ′ ) ≤ ℓ ( y ) + ε ≤ c r ℓ ( I ) . Let
A >
10 be some big constant to be fixed below. Suppose that there are two cubes P , P ∈ D which satisfy the following properties(i) rℓ ( I ) ≤ ℓ ( P ) = ℓ ( P ) ≤ rℓ ( I ),(ii) dist( P , P ) ≥ Aℓ ( P ) , (iii) P i ∩ (cid:0) x + Π − ( r · I ) (cid:1) = ∅ for i = 1 , P , P ′ ) ≥ dist( P , P ′ ). Then from (ii) we infer thatdist( P , P ′ ) & A ℓ ( P ) . Let P ′′ ∈ D such that P ∪ P ′ ⊂ P ′′ with minimal side length, so that ℓ ( P ′′ ) ≈ ℓ ( P ) + ℓ ( P ′ ) + dist( P , P ′ ) . Since α (1000 Q ) ≤ δ and ℓ ( P ) , ℓ ( P ) , ℓ ( P ′ ) ≪ ℓ ( Q ), it followseasily that we must also have ℓ ( P ′′ ) ≪ ℓ ( Q ). It is not difficult to check that either β ( P ′′ ) ≫ δ, or ∡ ( L P ′′ , L Q ) ≫ δ. In either case one has X S ∈D : P ′′ ⊂ S ⊂ Q α ( S ) ≫ δ. We deduce that X S ∈D : P ′ ⊂ S ⊂ Q α (100 S ) ≫ δ, because P ′ ⊂ P ′′ . This contradicts the fact that P ′ ∈ G .We have shown that a pair of cubes P , P such as the ones above does not exist. Thus,if P ∈ D satisfies rℓ ( I ) ≤ ℓ ( P ) ≤ rℓ ( I ) , and P ∩ (cid:0) x + Π − ( r · I ) (cid:1) = ∅ , then any other cube P for which these properties also hold must be contained in the ball B ( x P , c A ℓ ( P )), where x P stands for the center of P and c is some absolute constant.Hence letting P = P ( I ) be some suitable ancestor of P , the lemma follows. (cid:3) Lemma 5.8.
Let I ∈ G ( x, r ) and let P = P ( I ) ∈ D be the cube from Lemma 5.7, so that supp( µ ) ∩ ( x + Π − ( r · I )) ⊂ P. We have (5.33) (cid:12)(cid:12)(cid:12)(cid:12)Z ψ I (cid:18) Π( y − x ) r (cid:19) dµ ( y ) (cid:12)(cid:12)(cid:12)(cid:12) . (cid:18) ℓ ( Q ) ℓ ( P ) (cid:19) n/ X S ∈D : P ⊂ S ⊂ Q α ( S ) + dist( x, L Q ) ℓ ( Q ) ℓ ( P ) n . QUARE FUNCTIONS AND UNIFORM RECTIFIABILITY 29
Proof.
Without loss of generality we assume that x = 0 and as before we let L = R n be the n -plane parallel to L Q containing 0. Let also y P ∈ B P ∩ supp( µ ) be such thatdist( y P , L P ) . α ( P ) ℓ ( P ). The existence of such point follows from [To1, Remark 3.3] andChebychev’s inequality. We also denote by e L P the n -plane parallel to L which contains y P . We set σ P = c P H n | L P and e σ P = c P H n | e L P . Let φ P be a smooth function such that χ B P ≤ φ P ≤ χ B P and k∇ φ P k ∞ . ℓ ( P ) − . Since α ( P ) is assumed to be very small, wehave Π − ( r · I ) ∩ e L P ⊂ B P . Then we write Z ψ I (cid:18) Π( y ) r (cid:19) dµ ( y ) = Z φ P ( y ) ψ I (cid:18) Π( y ) r (cid:19) dµ ( y )= Z φ P ( y ) ψ I (cid:18) Π( y ) r (cid:19) ( dµ ( y ) − dσ P ( y ))+ Z φ P ( y ) ψ I (cid:18) Π( y ) r (cid:19) ( dσ P ( y ) − d e σ P ( y )) + Z ψ I (cid:18) Π( y ) r (cid:19) d e σ P ( y )=: A + A + A . (5.34)Now we turn our attention to A : | A | = (cid:12)(cid:12)(cid:12)(cid:12)Z φ P ( y ) ψ I (cid:18) Π( y ) r (cid:19) ( dµ ( y ) − dσ P ( y )) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:13)(cid:13)(cid:13)(cid:13) ∇ (cid:18) φ P ψ I (cid:18) Π( · ) r (cid:19)(cid:19)(cid:13)(cid:13)(cid:13)(cid:13) ∞ α ( P ) ℓ ( P ) n +1 . (cid:18) ℓ ( P ) 1 ℓ ( I ) n/ + 1 ℓ ( I ) n/ r (cid:19) α ( P ) ℓ ( P ) n +1 ≈ (cid:18) ℓ ( Q ) ℓ ( P ) (cid:19) n/ α ( P ) ℓ ( P ) n , (5.35)where we used that ℓ ( P ) ≈ ℓ ( I ) ℓ ( Q ) and that k∇ Π k ∞ . B P since B P lies far fromthe subspace Π ⊥− ( { } ).We will now estimate the term A . We have | A | = (cid:12)(cid:12)(cid:12)(cid:12)Z φ P ( y ) ψ I (cid:18) Π( y ) r (cid:19) ( dσ P ( y ) − d e σ P ( y )) (cid:12)(cid:12)(cid:12)(cid:12) . As in [To1, Lemma 5.2], ∡ ( L P , e L P ) = ∡ ( L P , L Q ) . X S ∈D : P ⊂ S ⊂ Q α ( S ) . Therefore, dist H ( e L P ∩ B P , L P ∩ B P ) . X S ∈D : P ⊂ S ⊂ Q α ( S ) ℓ ( P ) , and, as in (5.35), | A | . (cid:13)(cid:13)(cid:13)(cid:13) ∇ (cid:18) φ P ψ I (cid:18) Π( · ) r (cid:19)(cid:19)(cid:13)(cid:13)(cid:13)(cid:13) ∞ ℓ ( P ) n dist H ( e L P ∩ B P , L P ∩ B P ) . (cid:18) ℓ ( Q ) ℓ ( P ) (cid:19) n/ X S ∈D : P ⊂ S ⊂ Q α ( S ) ℓ ( P ) n . (5.36)We now consider A . Let B be a ball centered in L such that supp ψ I (cid:0) · r (cid:1) ⊂ B anddiam( B ) . ℓ ( P ). For some constant c ∗ , with 0 ≤ c ∗ .
1, to be fixed below, we write (cid:12)(cid:12)(cid:12)(cid:12)Z ψ I (cid:18) Π( y ) r (cid:19) d e σ P ( y ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) c P Z ψ I (cid:16) yr (cid:17) d (Π ♯ H n | e L P )( y ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) c P Z ψ I (cid:16) yr (cid:17) d (Π ♯ H n | e L P )( y ) − c ∗ c P Z ψ I (cid:16) yr (cid:17) d H n | L ( y ) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) c ∗ c P Z ψ I (cid:16) yr (cid:17) d H n | L ( y ) (cid:12)(cid:12)(cid:12)(cid:12) . k∇ ψ I k ∞ ℓ ( Q ) dist B (Π ♯ H n | e L P , c ∗ H n | L ) , (5.37)where in the last inequality we took into account that c ∗ c P . R R n ψ I (cid:0) yr (cid:1) dy = 0.Notice that the map Π | e L P → L need not be affine and so the term dist B (Π ♯ H n | e L P , c ∗ H n | L )requires some careful analysis. Anyway, we claim that, for some appropriate constant c ∗ . B (Π ♯ H n | e L P , c ∗ H n | L ) . X S ∈D : P ⊂ S ⊂ Q α ( S ) + dist(0 , L Q ) ℓ ( Q ) ! ℓ ( P ) n +1 , which implies that | A | . (cid:18) ℓ ( Q ) ℓ ( P ) (cid:19) n/ X S ∈D : P ⊂ S ⊂ Q α ( S ) + dist(0 , L Q ) ℓ ( Q ) ! ℓ ( P ) n . Notice that the lemma is an immediate consequence of the estimates we have for A , A and A .To conclude, it remains to prove the claim (5.38). This task requires some preliminarycalculations and we defer it to Lemma 5.9. (cid:3) Our next objective consists in comparing the measures Π ♯ H n | e L P and H n | L from thepreceding lemma. To this end, we consider the map e Π := Π | e L P → L . Abusing notation,identifying both e L P and L with R n , we also denote by e Π the corresponding mapping in R n , that is e Π : R n → R n . Then, writing h = y VP , for y = ( y , . . . , y n , h ) we have e Π i ( y ) = y i s y + · · · + y n + | h | y + · · · + y n = y i s | h | y + · · · + y n , QUARE FUNCTIONS AND UNIFORM RECTIFIABILITY 31 for i = 1 , . . . , n . Hence, for i, j = 1 , . . . , n , ∂ j e Π i = δ ij | y || y H | − | h | y i y j | y || y H | = | y || y H | (cid:18) δ ij − | h | y i y j | y | | y H | (cid:19) , where δ ij denotes Kronecker’s delta. For y ∈ P , | ∂ j e Π i ( y ) − ∂ j e Π i ( y P ) | ≤ (cid:12)(cid:12)(cid:12)(cid:12) | y || y H | − | y P || y HP | (cid:12)(cid:12)(cid:12)(cid:12) + | h | (cid:12)(cid:12)(cid:12)(cid:12) y i y j | y | | y H | − y P i y P j | y P | | y P H | (cid:12)(cid:12)(cid:12)(cid:12) . Moreover, (cid:12)(cid:12)(cid:12)(cid:12) | y || y H | − | y P || y HP | (cid:12)(cid:12)(cid:12)(cid:12) ≈ (cid:12)(cid:12)(cid:12)(cid:12) | y | | y H | − | y P | | y HP | (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) | y | | y HP | − | y P | | y H | | y H | | y HP | (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) ( | y H | + | h | ) | y HP | − ( | y HP | + | h | ) | y H | | y H | | y HP | (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) | h | ( | y HP | − | y H | ) | y H | | y HP | (cid:12)(cid:12)(cid:12)(cid:12) ≈ | h | | y | || y HP | − | y H ||| y | . | h | ℓ ( P ) r , and in a similar manner we get | h | (cid:12)(cid:12)(cid:12)(cid:12) y i y j | y | | y H | − y P i y P j | y P | | y P H | (cid:12)(cid:12)(cid:12)(cid:12) . | h | ℓ ( P ) r . Hence(5.39) | ∂ j e Π i ( y ) − ∂ j e Π i ( y P ) | . | h | ℓ ( P ) r . Now we write | J e Π( y ) − J e Π( y P ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X σ sgn( σ ) n Y j =1 ∂ j e Π σ ( j ) ( y ) − X σ sgn( σ ) n Y j =1 ∂ j e Π σ ( j ) ( y P ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ c ( n ) sup i,j (cid:12)(cid:12) ∂ j e Π i ( y ) − ∂ j e Π i ( y P ) (cid:12)(cid:12) (cid:0) sup i,j | ∂ j e Π i ( y ) | n − + sup i,j | ∂ j e Π i ( y P ) | n − (cid:1) . sup i,j (cid:12)(cid:12) ∂ j e Π i ( y ) − ∂ j e Π i ( y P ) (cid:12)(cid:12) , (5.40)where the sum is computed over all permutations of { , . . . , n } and sgn( σ ) denotes thesignature of the permutation σ . Moreover, in the last inequality we used again that k∇ Π k ∞ . B P since B P lies far from the subspace Π ⊥− ( { } ).Therefore, by (5.40) and (5.39),(5.41) | J e Π( y ) − J e Π( y P ) | . | h | ℓ ( P ) r for y ∈ P .
Lemma 5.9.
Let B be a ball centered in Π( P ) with diam( B ) . ℓ ( P ) . Then dist B (Π ♯ H n | e L P , c ∗ H n | L ) . X S ∈D : P ⊂ S ⊂ Q α ( S ) + dist(0 , L Q ) ℓ ( Q ) ℓ ( P ) n +1 , where c ∗ = ( J e Π( y P )) − .Proof. Let f be 1-Lipschitz with supp f ⊂ B . Then, recalling that e σ P = c P H n | e L P , (cid:12)(cid:12)(cid:12)(cid:12)Z f d (Π ♯ H n | e L P ) − c ∗ Z f d H n | L (cid:12)(cid:12)(cid:12)(cid:12) ≈ (cid:12)(cid:12)(cid:12)(cid:12) c ∗ Z f (Π( y )) d H n | e L P − Z f ( y ) d H n | L (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) c ∗ Z R n f ( e Π( y )) dy − Z R n f ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z R n f ( e Π( y )) J e Π( y P ) dy − Z R n f ( e Π( y )) J e Π( y ) dy (cid:12)(cid:12)(cid:12)(cid:12) , where we changed variables in the last line. Now notice that supp f ◦ e Π ⊂ B ′ , where B ′ is a ball concentric with B such that diam( B ) . ℓ ( P ). In addition, since supp f ⊂ B and k∇ f k ∞ ≤ k f k ∞ . ℓ ( P ). Hence, by (5.41), (cid:12)(cid:12)(cid:12)(cid:12)Z f d (Π ♯ H n | e L P ) − c ∗ Z f d H n | L (cid:12)(cid:12)(cid:12)(cid:12) . Z R n | f ( e Π( y )) || J e Π( y P ) − J e Π( y ) | dy . | h | ℓ ( P ) r Z B ′ ℓ ( P ) dy . | h | ℓ ( P ) n +1 r . Moreover, by [To1, Remark 5.3] and the choice of y P , | h | = dist( y P , L ) ≤ dist( y P , L Q ) + dist( L , L Q ) . X S ∈D : P ⊂ S ⊂ Q α ( S ) ℓ ( S ) + dist(0 , L Q ) . Hence (cid:12)(cid:12)(cid:12)(cid:12)Z f d (Π ♯ H n | e L P ) − c ∗ Z f d H n | L (cid:12)(cid:12)(cid:12)(cid:12) . P S ∈D : P ⊂ S ⊂ Q α ( S ) ℓ ( Q ) + dist(0 , L Q ) ℓ ( Q ) ℓ ( P ) n +1 , and the lemma follows. (cid:3) We denote e G ( x, r ) := { P ( I ) } I ∈G ( x,r ) . We need the following auxiliary result.
Lemma 5.10.
For every a ≥ and every S ∈ D , X P ∈ e G ( x,r ): P ⊂ a S µ ( P ) . µ ( S ) , with the implicit constant depending on a .Proof. We assume x = 0 for simplicity. Notice that for every P ∈ e G (0 , r ) such that P ⊂ a S there exists some I ∈ G (0 , r ) such that rℓ ( I ) ≈ ℓ ( P ) and r · I ⊂ a ′ Π( B S ) where a ′ onlydepends on a . Therefore X P ∈ e G (0 ,r ); P ⊂ a S µ ( P ) . X { ℓ ( r · I ) n : I ∈ G (0 , r ); r · I ⊂ a ′ Π( B S ) } . X { ℓ ( r · I ) n : I ∈ e D ( R n ); r · I ⊂ a ′ Π( B S ); r · I ∩ ( ∂B n (0 , r ) ∪ ∂B n (0 , r )) = ∅ }≤ c ( a ) ℓ ( S ) n ≈ c ( a ) µ ( S ) . QUARE FUNCTIONS AND UNIFORM RECTIFIABILITY 33 (cid:3)
We can now estimate the term G r, ( x ) in (5.31). Lemma 5.11.
We have (5.42) | G r, ( x ) | . X P ∈ e G ( x,r ) (cid:18) α ( aP ) + d ( x, L Q ) ℓ ( Q ) (cid:19) µ ( P ) µ ( Q ) , for some absolute constant a ≥ .Proof. Using (5.33) and (c) from Lemma 5.1, | G r, (0) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r n X I ∈G (0 ,r ) a I Z ψ I (cid:18) Π( y ) r (cid:19) dµ ( y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . ℓ ( Q ) n X I ∈G (0 ,r ) X S ∈D : P ( I ) ⊂ S ⊂ Q α ( S ) + dist(0 , L Q ) ℓ ( Q ) ℓ ( P ( I )) n . (5.43)Notice that by the definition of G (0 , r ), for every I ∈ G (0 , r ) { P ∈ D : P = P ( I ) } . . Then(5.44) | G r, (0) | . X P ∈ e G (0 ,r ) X S ∈D : P ⊂ S ⊂ Q α ( S ) ℓ ( P ) n ℓ ( Q ) n + X P ∈ e G (0 ,r ) dist(0 , L Q ) ℓ ( Q ) ℓ ( P ) n ℓ ( Q ) n . If S ∈ D is such that P ⊂ S ⊂ Q , then there exists e S ∈ e G (0 , r ), with ℓ ( e S ) ≈ ℓ ( S ), suchthat S ⊂ a e S for some a ≥
1. In fact, since P ∈ e G (0 , r ) we can find I ′ ∈ G (0 , r ) with ℓ ( r · I ′ ) ≈ ℓ ( S ) such that Π( S ) ∩ r · I ′ = ∅ . Therefore we can take e S := P ( I ′ ).Hence for P ∈ e G (0 , r ),(5.45) X S ∈D : P ⊂ S ⊂ Q α ( S ) . X S ∈ e G (0 ,r ): P ⊂ a S ⊂ a Q α ( a S ) . Thus, using also Lemma 5.10, X P ∈ e G (0 ,r ) X S ∈D : P ⊂ S ⊂ Q α ( S ) ℓ ( P ) n ℓ ( Q ) n . X P ∈ e G (0 ,r ) X S ∈ e G (0 ,r ): P ⊂ a S ⊂ a Q α ( a S ) ℓ ( P ) n ℓ ( Q ) n ≈ X S ∈ e G (0 ,r ): S ⊂ a Q α ( a S ) X P ∈ e G (0 ,r ): P ⊂ a S µ ( P ) µ ( Q ) . X S ∈ e G (0 ,r ): S ⊂ a Q α ( a S ) µ ( S ) µ ( Q ) . (5.46)Together with (5.44), this yields (5.42). (cid:3) Now we will deal with the term I in (5.32). Lemma 5.12.
We have I . µ ( R ) . Proof. | G r, ( x ) | . X P ∈ e G ( x,r ) (cid:18) α ( aP ) + d ( x, L Q ) ℓ ( Q ) (cid:19) µ ( P ) µ ( Q ) X P ∈ e G ( x,r ) µ ( P ) µ ( Q ) . X P ∈ e G ( x,r ) (cid:18) α ( aP ) + d ( x, L Q ) ℓ ( Q ) (cid:19) µ ( P ) µ ( Q ) . Then I . X Q ∈D ( R ) ℓ ( Q ) n +1 Z Q Z ℓ ( Q ) ℓ ( Q ) X P ∈ e G ( x,r ) (cid:18) α ( aP ) + d ( x, L Q ) ℓ ( Q ) (cid:19) µ ( P ) drdµ ( x ) . X Q ∈D ( R ) ℓ ( Q ) n +1 Z Q Z ℓ ( Q ) ℓ ( Q ) X P ⊂ a ′′ Q : cB P ∩ ( ∂B ( x,r ) ∪ ∂B ( x, r )) = ∅ (cid:18) α ( aP ) + d ( x, L Q ) ℓ ( Q ) (cid:19) µ ( P ) drdµ ( x ) . By Fubini, Z ℓ ( Q ) ℓ ( Q ) X P ⊂ a ′′ Q : cB P ∩ ∂B ( x,r ) = ∅ (cid:18) α ( aP ) + d ( x, L Q ) ℓ ( Q ) (cid:19) µ ( P ) dr = X P ⊂ a ′′ Q (cid:18) α ( aP ) + d ( x, L Q ) ℓ ( Q ) (cid:19) µ ( P ) Z { r : cB P ∩ ∂B ( x,r ) = ∅ } dr . X P ⊂ a ′′ Q (cid:18) α ( aP ) + d ( x, L Q ) ℓ ( Q ) (cid:19) µ ( P ) ℓ ( P ) , where we used the fact that if r > cB P ∩ ∂B ( x, r ) = ∅ then | x P | − c ℓ ( P ) ≤ r ≤ | x P | + c ℓ ( P ) , where x P is the center of B P . Therefore, I . X Q ∈D ( R ) ℓ ( Q ) n Z Q X P ⊂ a ′′ Q (cid:18) α ( aP ) + d ( x, L Q ) ℓ ( Q ) (cid:19) ℓ ( P ) ℓ ( Q ) µ ( P ) dµ ( x ) . X Q ∈D ( R ) X P ⊂ a ′′ Q α ( aP ) ℓ ( P ) ℓ ( Q ) µ ( P )+ X Q ∈D ( R ) ℓ ( Q ) n Z Q d ( x, L Q ) ℓ ( Q ) dµ ( x ) X P ∈D : P ⊂ a ′′ Q ℓ ( P ) ℓ ( Q ) µ ( P ) . X P ∈D : P ⊂ a ′′ R α ( aP ) µ ( P ) X Q ∈D : a ′′ Q ⊃ P ℓ ( P ) ℓ ( Q ) + X Q ∈D ( R ) Z Q d ( x, L Q ) ℓ ( Q ) dµ ( x ) . µ ( R ) . (5.47) QUARE FUNCTIONS AND UNIFORM RECTIFIABILITY 35 (cid:3)
Finally we turn our attention to I . Recall that I = X Q ∈D : Q ⊂ R,α (1000 Q ) ≤ δ Z Q Z ℓ ( Q ) ℓ ( Q ) | G r, ( x ) | drℓ ( Q ) dµ ( x ) . For x ∈ supp( µ ) and r > f x,r ( y ) = X I ∈T ( x,r ) X J ∈ e D ( R n ): J ⊂ I a J ψ J (cid:18) Π( y − x ) r (cid:19) , so that G r, ( x ) = 1 r n Z f x,r ( y ) dµ ( y ) . Lemma 5.13.
The functions f x,r satisfy • supp f x,r ⊂ S I ∈T ( x,r ) P ( ˆ I ) , where ˆ I is the father of I , • k f x,r k ∞ . .Proof. We assume again that x = 0. Notice that supp f x,r ⊂ Π − ( r · I ) ∩ supp( µ ) andsince I ∈ F , we have ˆ I ∈ G ( x, r ). Therefore by Lemma 5.7, Π − ( r · I ) ∩ supp( µ ) ⊂ P ( ˆ I ).We will now show that k f x,r k ∞ .
1. Recalling (5.30) if
I, J ∈ F and 20 I ∩ J = ∅ ,then ℓ ( I ) ≈ ℓ ( J ). If I ∈ F \ T ( x, r ) or I ⊂ J for some J ∈ F \ T ( x, r ), then by Lemma5.1 a I = 0. Therefore, f x,r ( y ) = X I ∈F X J ∈ e D ( R n ): J ⊂ I a J ψ J (cid:18) Π( y ) r (cid:19) . We now consider the function e f ( z ) = X I ∈F X J ⊂ I a J ψ J ( z ) . The second assertion in the lemma follows after checking that k e f k ∞ .
1. To this end,recall that by (5.1), for any k ∈ Z , we have e h = P I ∈ e D ( R n ) a I ψ I . We can also write(5.48) e h ( z ) = X I ∈ e D k ( R n ) X J ⊂ I a J ψ J ( z ) + X I ∈D k ( R n ) β I φ I ( z ) , where β I = h e h, φ I i and the functions φ I satisfy • supp φ I ⊂ I , • k φ I k ∞ . ℓ ( I ) n/ , • k∇ φ I k ∞ . ℓ ( I ) n/ , • k φ I k = 1. See [Mal, Theorem 7.9]. We note that supp ψ I ⊂ I and supp φ I ⊂ I since we are takingDaubechies wavelets with 3 vanishing moments, see [Mal, p. 250].Now let z ∈ I for some I ∈ F with ℓ ( I ) = 2 − k . Notice that k β I φ I k ∞ . | β I | ℓ ( I ) − n/ ≤ Z | e h ( y ) φ I ( y ) | dy ℓ ( I ) − n/ . k φ I k ℓ ( I ) − n/ . ℓ ( I ) n/ k φ I k ℓ ( I ) − n/ = 1 . (5.49)By the finite superposition of supp φ I for I ∈ D k , (5.49) implies that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X I ∈D k ( R n ) β I φ I ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . . Therefore by (5.48) we deduce that(5.50) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X I ∈ e D k ( R n ) X J ⊂ I a J ψ J ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . . We will now prove that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e f ( z ) − X I ∈ e D k ( R n ) X J ⊂ I a J ψ J ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . . Together with (5.50), this shows that | e f ( z ) | . (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e f ( z ) − X I ∈ e D k ( R n ) X J ⊂ I a J ψ J ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X I ∈F :5 I ∩ I = ∅ X J ⊂ I a J ψ J ( z ) − X I ∈D k ( R n ):5 I ∩ I = ∅ X J ⊂ I a J ψ J ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . X J ∈ A △ A | a J ψ J ( z ) | , (5.51)where A = { J ∈ e D ( R n ) : J ⊂ I, for some I ∈ F such that 5 I ∩ I = ∅ } and A = { J ∈ e D ( R n ) : J ⊂ I, for some I ∈ D k ( R n ) such that 5 I ∩ I = ∅ } . It follows as in (5.49) that k a J ψ J k ∞ .
1. Therefore, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e f ( z ) − X I ∈ e D k ( R n ) X J ⊂ I a J ψ J ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . A . . This follows from the fact that if I ∈ F such that 5 I ∩ I = ∅ then ℓ ( I ) ≈ ℓ ( I ). (cid:3) Lemma 5.14.
We have I . µ ( R ) . QUARE FUNCTIONS AND UNIFORM RECTIFIABILITY 37
Proof.
Lemma 5.13 implies that | G r, ( x ) | . ℓ ( Q ) n Z | f x,r ( y ) | dµ ( y ) . ℓ ( Q ) n X I ∈T ( x,r ) µ ( P ( ˆ I )) . As noted earlier, for I ∈ T ( x, r ), the parent of I , denoted by ˆ I , belongs to G ( x, r ). Observealso that r diam( I ) ≤ z ∈ r · I d ( z ) , because T ( x, r ) ∈ F . So every z ′ ∈ r · I ⊂ r · ˆ I satisfies d ( z ′ ) ≥ r diam( I ) =2500 r diam( ˆ I ). This implies that d ( z ) & r ℓ ( I ) for all z ∈ r · ˆ I , because d ( · ) is 3-Lipschitz.As a consequence, by the definition of d ( · ), there exists some y ∈ P ( ˆ I ) such that ℓ ( y ) & r ℓ ( I ) ≈ ℓ ( P ( ˆ I )). Then it follows easily that there exists some descendant U of P ( ˆ I ) with ℓ ( U ) ≈ ℓ ( P ( ˆ I )) such that X S ∈D : U ⊂ S ⊂ Q α (100 S ) ≥ δ. This clearly implies that either X S ∈D : P (ˆ I ) ⊂ S ⊂ Q α (100 S ) ≥ δ , or X S ∈D : U ⊂ S ⊂ P (ˆ I ) α (100 S ) ≥ δ . Since ℓ ( U ) ≈ ℓ ( P ( ˆ I )), from the second condition one infers that α (100 P ( ˆ I )) ≥ cδ. Hencein either case, for some small constant c > X S ∈D : P ⊂ S ⊂ Q α (100 S ) ≥ cδ. Therefore, | G r, ( x ) | . δ ℓ ( Q ) n X I ∈T ( x,r ) µ ( P ( ˆ I )) X S ∈D : P ⊂ S ⊂ Q α (100 S ) . ℓ ( Q ) n X P ∈ e G ( x,r ) µ ( P ) X S ∈D : P ⊂ S ⊂ Q α (100 S ) . Notice that X P ∈ e G ( x,r ) ℓ ( P ) n ℓ ( Q ) n X S ∈D : P ⊂ S ⊂ Q α (100 S )is smaller, modulo the constants 1000 and 100, than the right side in (5.44). Therefore bythe same arguments we used for I we get I . µ ( R ) . (cid:3) From Lemmas 5.12 and 5.14 we deduce that I . µ ( R ) . Together with Lemma 5.5 thiscompletes the proof of Theorem 5.2. Proof of Proposition 1.3
We will only prove the equivalence (a) ⇔ (c), as (a) ⇔ (b) is very similar.By Theorem 4.1, it is clear that uniform n -rectifiability implies the boundedness of thesquare function in (c) for any positive integer k . As for the converse, next we show thatLemma 3.1 holds with e ∆ µ,ϕ replaced by e ∆ kµ,ϕ . Recall e ∆ kµ,ϕ ( x, t ) = Z ∂ kϕ ( x − y, t ) dµ ( y ) , where ∂ kϕ ( x, t ) = t k ∂ kt ϕ t ( x ) . Lemma 6.1.
Let k ≥ and let µ be an n -AD-regular measure such that ∈ supp( µ ) . Forall ε > there exists δ > such that if Z δ − δ Z x ∈ ¯ B (0 ,δ − ) | e ∆ kµ,ϕ ( x, r ) | dµ ( x ) dr ≤ δ, then dist B (0 , ( µ, e U ( ϕ, c )) < ε. Proof.
Suppose that there exists an ε >
0, and for each m ≥ n -AD-regularmeasure µ m such that 0 ∈ supp( µ m ),(6.1) Z m /m Z x ∈ ¯ B (0 ,m ) | e ∆ kµ m ,ϕ ( x, r ) | dµ m ( x ) dr ≤ m , and(6.2) dist B (0 , ( µ m , e U ( ϕ, c )) ≥ ε. By (1.1) we can replace { µ m } by a subsequence converging weak * (i.e. when testedagainst compactly supported continuous functions) to a measure µ and it is easy to checkthat 0 ∈ supp( µ ) and that µ is also n -dimensional AD-regular with constant c . We claimthat Z ∞ Z x ∈ R d | e ∆ kµ,ϕ ( x, r ) | dµ ( x ) dr = 0 . The proof of this statement is elementary and is almost the same as the analogous onein Lemma 3.1. We leave the details for the reader.Our next objective consists in showing that µ ∈ e U ( ϕ, c ). To this end, denote by G thesubset of those points x ∈ supp( µ ) such that Z ∞ | e ∆ kµ,ϕ ( x, r ) | dr = 0 . It is clear that G has full µ -measure. For x ∈ G and r >
0, consider the function f x ( r ) = ϕ r ∗ µ ( x ). Notice that f x : (0 , + ∞ ) → R is C ∞ and satisfies ∂ kr f x ( r ) = ( ∂ kr ϕ r ) ∗ µ ( x ) = r − k e ∆ kµ,ϕ ( x, r ) = 0for a.e. r >
0. Thus f x is a polynomial on r of degree at most k −
1, whose coefficientsmay depend on x . However, since µ is n -AD-regular, it follows easily that there existssome constant c such that | f x ( r ) | = | ϕ r ∗ µ ( x ) | ≤ c for all r > QUARE FUNCTIONS AND UNIFORM RECTIFIABILITY 39
Thus f x must be constant on r . So for all x ∈ G and 0 < R ≤ R , ϕ R ∗ µ ( x ) = ϕ R ∗ µ ( x ) . This is the same estimate we obtained in (3.6) in Lemma 3.1. So proceeding exactly inthe same way as there we deduce then that ϕ R ∗ µ ( x ) = ϕ R ∗ µ ( y ) for all x, y ∈ supp µ and all 0 < R ≤ R .That is, µ ∈ e U ( ϕ, c ). However, by condition (6.2), letting m → ∞ , we havedist B (0 , ( µ, e U ( ϕ, c )) ≥ ε, because dist B (0 , ( · , e U ( ϕ, c )) is continuous under the weak * topology. So µ e U ( ϕ, c ),which is a contradiction. (cid:3) Applying the previous lemma and arguing in the same way as in Section 3 one provesthe implication (c) ⇒ (a) of Proposition 1.3. References [ADT] J. Azzam, G. David and T. Toro,
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Vasileios Chousionis. Department of Mathematics and Statistics, P.O. Box 68, FI-00014University of Helsinki, Finland
E-mail address : [email protected] John Garnett. Department of Mathematics, University of California at Los Angeles.6363 Math Sciences Building, Los Angeles, CA 90095-1555
E-mail address : [email protected] Triet Le. Department of Mathematics, University of Pennsylvania. David RittenhouseLab. 209 South 33rd Street, Philadelphia, PA 19104.
E-mail address : [email protected] Xavier Tolsa. Instituci´o Catalana de Recerca i Estudis Avanc¸ats (ICREA) and Departa-ment de Matem`atiques, Universitat Aut`onoma de Barcelona, Catalonia
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