Infinitesimal splitting for spaces with thick curve families and Euclidean embeddings
IINFINITESIMAL SPLITTING FOR SPACES WITH THICK CURVE FAMILIES AND EUCLIDEANEMBEDDINGS
GUY C. DAVID AND SYLVESTER ERIKSSON-BIQUEA
BSTRACT . We study metric measure spaces that admit “thick” families of rectifiable curves or curve fragments, in the formof Alberti representations or curve families of positive modulus. We show that such spaces cannot be bi-Lipschitz embeddedinto any Euclidean space unless they admit some “infinitesimal splitting”: their tangent spaces are bi-Lipschitz equivalent toproduct spaces of the form Z × R k for some k ≥ . We also provide applications to conformal dimension and give newproofs of some previously known non-embedding results.
1. I
NTRODUCTION
Many natural problems in analysis on metric spaces involve studying spaces that support “thick” families of rec-tifiable curves (or curve fragments), in one sense or another. In this paper, we show that such spaces cannot bebi-Lipschitz embedded into any Euclidean space unless they admit some “infinitesimal splitting”: their tangent spacesare bi-Lipschitz equivalent to product spaces of the form Z × R k for some k ≥ . Our methods are relatively directand admit a number of consequences.1.1. Background.
In 1999, Cheeger [16] proved a deep extension of Rademacher’s theorem (Lipschitz functions aredifferentiable almost everywhere) to certain abstract metric measure spaces. These are the so-called
PI spaces , thosethat are doubling and support a Poincaré inequality in the sense of [29]. As one of many consequences, he showed thatif a PI space admits a bi-Lipschitz embedding into some Euclidean space, then its tangent spaces must be bi-Lipschitzequivalent to Euclidean spaces almost everywhere:
Theorem 1.1 (Cheeger [16], Theorem 14.1) . Let ( X, µ ) be a PI space. Suppose that X admits a bi-Lipschitz em-bedding into some R n . Then for µ -almost-every x ∈ X , there is an integer k ≥ such that every tangent space ( Y, y ) ∈ Tan(
X, x ) is bi-Lipschitz equivalent to R k . (Here, the notion of “tangent” is in the pointed Gromov-Hausdorff sense; see section 2.2.)In other words, to admit a bi-Lipschitz embedding into a Euclidean space, a PI space must itself be infinitesimallyEuclidean. (Extensions of this are known, see [16, Theorem 14.2] and more recent results in [17, 18, 52, 21, 19, 23].)Since we know of many abstract PI spaces that are not infinitesimally Euclidean, this consequence of Cheeger’sresult can be viewed as a generalized non-bi-Lipschitz embedding theorem, i.e., a checkable criterion for a space toadmit no bi-Lipschitz embedding into any Euclidean space.A Poincaré inequality is sufficient but not necessary to prove a Rademacher-type theorem in metric spaces, andhence a non-embeddability criterion like Theorem 1.1. Indeed, a number of weaker sufficient conditions implyingCheeger’s Rademacher theorem have been found since its discovery [34, 8, 52]. Most importantly for our purposes, Date : June 19, 2020.2010
Mathematics Subject Classification. a r X i v : . [ m a t h . M G ] J un GUY C. DAVID AND SYLVESTER ERIKSSON-BIQUE
Bate [8], building on work of Alberti [1] and Alberti-Csörnyei-Preiss [2], showed that Cheeger’s differentiable struc-ture is equivalent to the presence of a universal family of Alberti representations . An Alberti representation of ameasure is a decomposition into -rectifiable measures supported on fragments of rectifiable curves (see Definition2.6). For example, Fubini’s theorem gives simple Alberti representations of Lebesgue measure on [0 , .Essentially, Bate shows that if a space supports a “large enough” family of independent Albert representations,then it supports a Rademacher theorem for Lipschitz functions, from which one can deduce an analog of the non-embeddability criterion Theorem 1.1.Given the above, it is natural to ask whether there are conditions, weaker than any of those studied above, that arenot strong enough to yield a Rademacher-type theorem but that still prevent bi-Lipschitz embeddings.In this paper, we answer this question by studying spaces that support a single Alberti representation (or moregenerally k independent independent Alberti representations), but not necessarily enough to form a “universal” familyin Bate’s sense, and thus not necessarily enough to yield a differentiable structure for Lipschitz functions.Nonetheless, we show that such smaller families of Alberti representations still strongly constrain the ability of thespace to bi-Lipschitz embed into any Euclidean space. The following is our main theorem. Theorem 1.2.
Let X ⊆ R n be a closed set supporting a doubling Radon measure µ . Suppose that a non-trivialRadon measure µ (cid:28) µ supports k independent Alberti representations, for some k ≥ .Then for µ -almost-every x ∈ X , there is a k -dimensional vector subspace V ⊆ R n with the following property:Every intrinsic tangent Y ∈ Tan R n ( X, x ) of X at x is a product Z × V , for some closed set Z ⊆ V ⊥ ⊆ R n . For a vector subspace V ⊂ R n , its orthogonal complement is denoted by V ⊥ . An “intrinsic tangent” of a subset X ⊆ R n is simply a limit of rescalings of X centered at a fixed basepoint. For a precise definition, see section 2.2below.It is easy to recast Theorem 1.2 as a result that constrains bi-Lipschitz embeddings of metric spaces: Corollary 1.3.
Let X be a complete metric space supporting a doubling Radon measure µ . Suppose that a non-trivialRadon measure µ (cid:28) µ supports k independent Alberti representations, for some k ≥ .If X admits a bi-Lipschitz embedding into some Euclidean space, then for µ -almost-every x ∈ X and every tangent ( Y, y ) ∈ Tan(
X, x ) , Y is bi-Lipschitz equivalent to a product Z × R k , for some complete metric space Z . Corollary 1.3 gives a simple non-bi-Lipschitz embedding criterion that applies to a wider class of examples thanTheorem 1.1.Theorem 1.2 and Corollary 1.3 are proven without recourse to any more general metric measure Rademachertheorem. Rather, their proofs rely only on one of the preliminary results of Bate’s paper [8] (that Alberti representationsinduce “partial derivatives” almost everywhere) – see Proposition 2.13 – and an adaptation of a principle of Preiss [50]about the structure of the space of tangent objects, Proposition 2.5.
Remark 1.4.
We note that there are many other conditions, independent from any of those discussed above, thatprevent or constrain bi-Lipschitz embeddability. In particular, the results of [38] and [47] rest also, remarkably, onstudying a single family of curves that is “thick” in some quantitative sense (although different than the senses usedhere). This is part of a larger program to characterize metric spaces embedding into the so-called RNP Banach spaces,and conversely to characterize RNP-Banach spaces by the spaces embedding into them; see [46].1.2.
Corollaries of the main results.
We view Corollary 1.3 as a generalized non-embedding result. Since “thick"families of curves arise in many settings, it has a number of specific consequences.1.2.1.
Modulus.
From our perspective, the most important consequences of Corollary 1.3 involve its relationship witha well-known way of measuring the “thickness” of a family of curves Γ in a metric measure space ( X, µ ) . This is the p -modulus of the family ( p ≥ ), which we denote Mod p (Γ , µ ) . This notion plays a central role in the modern theoryof analysis on metric spaces [26, 30]. (We give a precise definition in Section 4.) NFINITESIMAL SPLITTING FOR SPACES WITH THICK CURVE FAMILIES AND EUCLIDEAN EMBEDDINGS 3
In Proposition 4.5 and Corollary 4.9 below, we show that path families of positive modulus induce non-trivialAlberti representations. This idea is essentially already contained in work of the second-named author and his col-laborators [24] and the proof below reworks that argument in a slightly different context. This is also closely relatedto the results of [4] and [25]. The formulation in Proposition 4.5 is slightly different and applies more directly to oursetting. (To link modulus and Alberti representations, we also use ideas of Keith [33] and Bate [8, Corollary 5.8].)As a consequence of Proposition 4.5 and Corollary 1.3, we obtain the following result for spaces that contain curvefamilies of positive modulus.
Corollary 1.5.
Let X be a complete metric space admitting a Radon measure µ that is absolutely continuous withrespect to a doubling measure µ . Suppose that X contains family Γ ⊆ Curv( X ) of non-constant curves with Mod p (Γ , µ ) > for some p ∈ [1 , ∞ ) .If X admits a bi-Lipschitz embedding into some Euclidean space, then there exists a non-trivial Radon measure µ (cid:48) (cid:28) µ , such that, for µ (cid:48) -almost every x ∈ X , every tangent Y of X at x is bi-Lipschitz to a product Z × R , for somecomplete metric space Z . For many metric measure spaces, it is known, or not difficult to check, that they contain a path family of positivemodulus and do not have tangents that split as Z × R . Therefore such metric spaces cannot bi-Lipschitz embed intoany Euclidean space. We give some applications of this argument below.1.2.2. Conformal dimension.
An important problem in metric geometry is to understand the conformal dimension of a metric space, a quasisymmetric invariant first introduced by Pansu [48], and much used since [41]. There are anumber of variations of this quantity, but we focus on the
Ahlfors regular conformal dimension . This is a variant firstnamed by Bonk-Kleiner in [11], where they attribute the idea to Bourdon-Pajot [13].We recall that a metric space X is Ahlfors Q -regular if there is a constant C ≥ such that C − r Q ≤ H Q ( B ( x, r )) ≤ Cr Q for all r ≤ diam( X ) , where H Q denotes the Q -dimensional Hausdorff measure.The Ahlfors regular conformal dimension of X measures the infimal dimension Q of all Ahlfors regular quasisym-metric deformations of X : Definition 1.6.
The Ahlfors regular conformal dimension of a metric space X is(1.1) cdim AR ( X ) = inf { Q : Y is Ahlfors Q -regular and quasisymmetric to X } . We refer the reader to [26, Ch. 10] for a precise definition of quasisymmetric mappings, and to [11, 41] for morebackground on the Ahlfors regular conformal dimension, which we now discuss briefly.By definition, the Ahlfors regular conformal dimension and its variations are quasisymmetric invariants. Theyhave thus played an important role in geometric group theory and quasiconformal geometry, and their properties areconnected to many deep questions. We refer to [41] for a book-length account of many of these connections.In particular, it is a difficult problem to understand for which metric spaces the conformal dimension is actuallyachieved as a minimum. This problem is closely related to an approach to Cannon’s conjecture initiated by Bonk andKleiner [10, 11].As an example, it known that cdim AR ( S ) is strictly less than the Hausdorff dimension of the standard Sierpi´nskicarpet S [35], but not whether the infimum in (1.1) is achieved by some Ahlfors regular space Y when X = S . See[41, 11] for additional details. (The exact value of cdim AR ( S ) is also a well-known open problem.)Complicating the problem further, even if the conformal dimension of a subset X ⊆ R N is achieved by a space Y quasisymmetric to X , there is no reason that Y should be a subset of Euclidean space. Below, we show that thisnon-embedding phenomenon should be expected quite generally.We need the following important result of Keith and Laakso. GUY C. DAVID AND SYLVESTER ERIKSSON-BIQUE
Theorem 1.7 (Keith-Laakso [35], Corollary 1.0.2) . Let Q ≥ and let X be a complete, Ahlfors Q -regular metricspace.Then cdim AR ( X ) = Q if and only if there is a weak tangent of X that contains a family of non-constant curveswith positive p -modulus, for some p ≥ . (As remarked on [35, p. 1279], this version follows from the version stated there. The notion of a “weak tangent”is defined in section 2.2.)As a consequence of Corollary 1.5 and Theorem 1.7, we obtain: Corollary 1.8.
Let Q ≥ and let X be a complete, Ahlfors Q -regular metric space, where Q = cdim AR ( X ) . If X admits a bi-Lipschitz embedding into some Euclidean space R N , then there is a complete metric space Z and a weaktangent of X that is bi-Lipschitz equivalent to Z × R . Corollary 1.8 shows that we should not expect minimizers for conformal dimension to appear within Euclideanspace except under quite special circumstances, i.e., in the presence of some form of splitting. We illustrate a moreconcrete special case here.For this we introduce the terminology of linear connectedness: A metric space X is linearly connected (withconstant C ≥ ) if every pair of points x, y ∈ X can be joined by a compact, connected subset E ⊆ X with diam( E ) ≤ Cd ( x, y ) . In particular, quasiconvex spaces (in which every pair of points can be joined by a curve withlength comparable to the distance between the points) are linearly connected. Corollary 1.9.
Let X be a linearly connected metric space. Suppose that < cdim AR ( X ) < and that cdim AR ( X ) is achieved by a space Y . (In other words, Y is quasisymmetric to X and Ahlfors Q -regular with Q = cdim AR ( X ) =cdim AR ( Y ) .)Then Y admits no bi-Lipschitz embedding into any Euclidean space. As an example, consider the classical Sierpi´nski carpets S p for odd integers p > (the most famous example being S = S ). These are plane fractals formed by dividing the unit square into p − × p − squares and removing the middlesquare, then iterating this construction on the remaining squares. See [12] for details on this notation.The spaces S p are all linearly connected and have cdim AR ( S p ) ∈ (1 , (see [12, p. 595]). It is an open questionwhether the Ahlfors regular conformal dimensions of these spaces are actually achieved. Corollary 1.9 shows that theycannot be achieved by subsets of any Euclidean space.We note that the linear connectedness condition cannot be removed from Corollary 1.9: The product C × [0 , ⊆ R of the standard Cantor set with the unit interval is Ahlfors Q -regular (for Q = 1 + log(2)log(3) ∈ (1 , ) and known to beminimal for conformal dimension (by a theorem of Tyson [55]). However, it sits isometrically in R .One may even make this example connected by taking its union with [0 , ×{ } , showing that “linearly connected”cannot be replaced by “connected” in Corollary 1.9. This example is also easily seen to have a weak tangent that isbi-Lipschitz equivalent to a product Z × R .1.2.3. The slit carpet.
The slit carpet M is a metric space homeomorphic to the standard Sierpi´nski carpet with anumber of interesting properties. It was first proposed by Bonk and Kleiner and first studied in print by Merenkov[44]. Following [44], we define the space as follows, mostly using notation from [22]. We will be rather brief here,referring the reader to [44] or [22] for more details.Let Q = [0 , denote the unit square in R . For each dyadic subsquare Q ⊆ Q , let s Q denote a central vertical“slit” in Q of half the side length. More specifically, if Q = [ a − k , ( a + 1)2 − k ] × [ b − k , ( b + 1)2 − k ] , then s Q = (cid:2) ((2 a + 1)2 − k − , (4 b + 1)2 − k − ) , ((2 a + 1)2 − n − , (4 b + 3)2 − k − ) (cid:3) ⊆ Q. NFINITESIMAL SPLITTING FOR SPACES WITH THICK CURVE FAMILIES AND EUCLIDEAN EMBEDDINGS 5 ( A ) M ( B ) M ( C ) M F IGURE
1. The first three M i .Define now M = Q = [0 , , and inductively set M k +1 = M k \ (cid:91) Q dyadic, side ( Q )=2 − ( k +1) s Q . (See Figure 1, borrowed from [22].)We then define M k as the completion of M k with respect to the shortest path metric d k on M k , continuing to callthis new complete metric on M k by d k . In other words, we “cut along” each slit in a square of scale k or lower. Notethat the d k -diameter of each M k is bounded by .Merenkov observes in [44] that for each k ≤ j , there is a -Lipschitz mapping π j,k : M j → M k obtained byidentifying opposing points on slits of levels greater than j corresponding to the same point in M k . These mapscompose in the obvious way. We then define the Merenkov slit carpet M as the inverse limit of the system M π , ←−− M π , ←−− M π , ←−− . . . , equipped with the metric d ( x, y ) = lim k →∞ d k ( x k , y k ) for x = ( x k ) and y = ( y k ) such that π k ( x k ) = x k − , and similarly for y k . Note that this is the limit of a bounded,increasing sequence.In [22], the present authors answered a 1997 question of Heinonen and Semmes [28, Question 8] by proving thefollowing. Corollary 1.10 (Originally proven in Theorem 1.2 of [22]) . The slit carpet M does not admit a bi-Lipschitz embeddinginto any Euclidean space. (Actually, the result in [22] is more general, but the above corollary already answers [28, Question 8].)We give an alternative proof of this fact below, based on Corollary 1.5. This proof is based on the idea that thevertical lines in Figure 1 form a positive modulus family of curves, while at the same time the slit carpet admits noinfitesimal splitting. As in Corollaries 1.9 above and 1.11 below, the main work is to verify the latter claim.1.2.4. The Heisenberg group.
The most well-known example of a doubling metric space that does not bi-Lipschitzembed into any Euclidean space is the first Heisenberg group H . This was first observed by Semmes [53] as aconsequence of Pansu’s version of Rademacher’s theorem in Carnot groups [49]. There are also a number of otherproofs that H admits no bi-Lipschitz embedding into any Euclidean space [31].We give a new short proof of this non-embedding. Our proof is independent of Pansu’s theorem (or the generalmetric measure space Rademacher theorem of Cheeger [16]), and relies only on Theorem 1.2 and some basic propertiesof the Heisenberg group. (Our proof does involve a “blowup” argument via Theorem 1.2, so it has that in commonwith the Pansu-Semmes approach.) GUY C. DAVID AND SYLVESTER ERIKSSON-BIQUE
Corollary 1.11 (Originally due to Semmes and Pansu) . The Heisenberg group H (with its Korányi or Carnot-Carathéodory metric) admits no bi-Lipschitz embedding into any Euclidean space. Final introductory remarks and outline.
A few more remarks are in order concerning our main results.First of all, Theorem 1.2, Corollary 1.3, Corollary 1.5, and Corollary 1.8 are all completely false if one does notassume that the space X lies inside, or admits a bi-Lipschitz embedding into, some Euclidean space. To be concrete,the Heisenberg group H supports a doubling (even Ahlfors -regular) measure supporting two independent Albertirepresentations, it supports curve families of positive modulus, and it is minimal for conformal dimension. (See [26,Theorems 9.6, 9.27, 15.10] and subsection 5.3.) However, no weak tangent of H is bi-Lipschitz to some Z × R ; this isnot difficult to prove given [6, Theorem 7.2]. (One could actually prove Corollary 1.11 along these lines, but we givea different argument that avoids the tools of [6, 49].)We note that Theorem 1.2 is related to the main result of [3], which gives a “partial differentiable structure” formeasures on Euclidean space that support independent Alberti representations. Here our focus is on the geometrictangent structure of subsets, rather than on differentiability properties of mappings, and our methods are different.There are certainly further questions one could ask in our setting about the tangents of the measure µ in Theorem1.2, rather than the tangents of the support X . Related results for abstract metric measure spaces appear in [19]. Onecould also ask about embeddings into infinite-dimensional Banach spaces. In the interest of keeping the present paperreasonably direct, we defer these questions to future work.Using results of Schioppa [52, Theorem 3.24 and Corollary 3.93], Theorem 1.2 and Corollary 1.3 can be directlyrecast in terms of so-called Weaver derivations. In other words, if X , µ , and µ are as in Theorem 1.2, but we assumethat µ supports k independent Weaver derivations rather than k independent Alberti representations, the conclusionstill holds. We refer the reader to [57] or [27, Section 13] for more on Weaver derivations, and [52] for more on theconnection between Weaver derivations and Alberti representations.1.3.1. Outline of the paper.
In Section 2, we give basic definitions and preliminary results. Within that section,subsection 2.2 defines the different notions of tangents and our version of Preiss’s principle, Proposition 2.5, is stated,though its proof is deferred to the appendix (Section 6). Subsection 2.5 defines Alberti representations and states theresult of Bate, Proposition 2.13, that we will need, in addition to some other preliminary facts.Theorem 1.2 is then proven in Section 3. Section 4 contains Proposition 4.5 and Corollary 4.9, which relate thenotions of modulus and Alberti representations. All the corollaries listed in subsection 1.2 are then proven in Section5. 2. N
OTATION AND PRELIMINARIES
Metric spaces and measures.
We write ( X, d ) for a metric space, or just X if the metric is understood. We usestandard notation; in particular B ( x, r ) and B ( x, r ) denote the open and closed balls, respectively, of radius r centeredat x ∈ X . If c > and X = ( X, d ) is a metric space, then cX denotes the metric space ( X, cd ) . We write ( X, µ ) fora metric measure space with a given Radon measure µ that is finite and positive on all balls, and where the metric d isimplied.As usual, a map f : X → Y between two metric spaces is Lipschitz if there is a constant L ≥ such that d ( f ( x ) , f ( y )) ≤ Ld ( x, y ) for all x, y ∈ X. In this case, we may also call f L -Lipschitz to emphasize the constant. The infimum of all L such that f is L -Lipschitzis denoted LIP ( f ) .A map f : X → Y between two metric spaces is bi-Lipschitz (or L -bi-Lipschitz ) if there is a constant L ≥ suchthat L − d ( x, y ) ≤ d ( f ( x ) , f ( y )) ≤ Ld ( x, y ) for all x, y ∈ X. Two metric spaces are bi-Lipschitz equivalent if there is a bi-Lipschitz surjection from one onto the other.
NFINITESIMAL SPLITTING FOR SPACES WITH THICK CURVE FAMILIES AND EUCLIDEAN EMBEDDINGS 7 A pointed metric space is a pair ( X, x ) , where X is a metric space and x ∈ X is a point (the “basepoint”). Twopointed metric spaces are pointedly isometric if there is an isometry between them that preserves basepoints.Corollary 1.3 uses the notion of a product X × Y of two metric spaces ( X, d X ) and ( Y, d Y ) . There are manyequivalent ways to metrize this product. For concreteness, we take d X × Y (( x, y ) , ( x (cid:48) , y (cid:48) )) = ( d X ( x, x (cid:48) ) + d Y ( y, y (cid:48) ) ) / . Any other natural choice would yield a bi-Lipschitz equivalent metric on X × Y .For a metric space X , we let M ( X ) be the space of finite Borel measures on X . A non-trivial measure µ ∈ M ( X ) is doubling if there is a constant C ≥ such that µ ( B ( x, r )) ≤ Cµ ( B ( x, r )) for all x ∈ X, r > . A metric space that supports a doubling measure must be a doubling metric space : every ball of radius r can be coveredby N balls of radius r/ , where N is a fixed constant. (See [26] for more on doubling metrics and measures.)A complete doubling metric space is proper : every closed ball is compact.Lastly, we write H Q for the Q -dimensional Hausdorff measure on a metric space X (with X understood fromcontext), and dim H ( X ) for the Hausdorff dimension of X . See [26, Section 8.3] for definitions.2.2. Tangents of metric spaces and sets.
The next few definitions will use the notion of pointed Gromov-Hausdorffconvergence of a sequence of pointed metric spaces. See [14, Section 8]. Versions are also given in [20, 33, 35, 21],among other places.
Definition 2.1.
Let X be a metric space, and let λ i ( i ∈ N ) be a sequence of positive real numbers. • If x ∈ X , λ i → , and the sequence of pointed metric spaces ( λ − i X, x ) , converges in the pointed Gromov-Hausdorff sense to a complete pointed metric space ( Y, y ) , then ( Y, y ) iscalled a tangent of X at x . The collection of all tangents of X at x is written Tan(
X, x ) . • If { x i } ⊆ X , < λ i < diam( X ) for each i , and the sequence of pointed metric spaces ( λ − i X, x i ) , converges in the pointed Gromov-Hausdorff sense to a complete pointed metric space ( W, w ) , then ( W, w ) iscalled a weak tangent of X . The collection of all weak tangents of X is written WTan ( X ) .Technically speaking, elements of Tan(
X, x ) are not pointed metric spaces but rather pointed isometry classes,since the pointed Gromov-Hausdorff topology does not distinguish between isometric metric spaces; similarly, ele-ments of WTan ( X ) are isometry classes. We tend to elide this distinction for notational convenience. Of course, if ( Y, y ) ∈ Tan(
X, x ) then Y ∈ WTan ( X ) .If a space already sits inside an ambient Euclidean space, then one can more naturally take tangents by rescalinginside the Euclidean space and taking a limit in the pointed Hausdorff sense: If { A j } are subsets of R n , then we saythat { A j } converges to a closed set A ⊆ R n in the pointed Hausdorff sense if lim j →∞ d R ( A j , A ) = 0 for all R > , where d R ( A, B ) = max { sup { dist( a, B ) : a ∈ A ∩ B (0 , R ) } , sup { dist( b, A ) : b ∈ B ∩ B (0 , R ) }} . Of course, one could take a sum rather than a max above (as done in [20, Chapter 8]) and this would only change thedefinition by at most a factor of . GUY C. DAVID AND SYLVESTER ERIKSSON-BIQUE
Definition 2.2.
Let A be a subset of R n and a ∈ A . An intrinsic tangent of A at a is any closed set which is a pointedHausdorff limit of sets of the form λ − j ( A − a ) , where { λ j } is a sequence of positive real numbers tending to as j → ∞ .The collection of all intrinsic tangents of A at a is written Tan R n ( A, a ) .An important difference between tangents and intrinsic tangents is that Tan R n ( A, a ) may contain distinct elementsthat are isometric.If one has a Lipschitz function defined on a subset of R n , there is also a way to pass to a tangent (or “blowup”) ofthe function simultaneously with the set. This is given in [20, Chapter 8]. Definition 2.3.
Let A be a subset of R n and a ∈ A . Let f : A → R m be a Lipschitz function.Suppose that { λ j } is a sequence of positive real numbers tending to as j → ∞ and that λ − j ( A − a ) converge to the closed set ˆ A in the pointed Hausdorff sense . Moreover, suppose that ˆ f : ˆ A → R m is such that, whenever x j ∈ A and λ − j ( x j − a ) → x ∈ ˆ A, we then have λ − j ( f ( x j ) − f ( a )) → ˆ f ( x ) . We then call the pair ( ˆ A, ˆ f ) an intrinsic tangent of ( A, f ) at a and write ( ˆ A, ˆ f ) ∈ Tan R n ( A, f, a ) . The following facts are all standard, well-known consequences of the compactness theorems for pointed Hausdorffand pointed Gromov-Hausdorff convergence. See, e.g., [20, Lemmas 8.6 and 8.13]. We include this lemma simply asa summary of the basic facts that we will use.
Lemma 2.4.
Let X be a complete doubling metric space and x ∈ X . Let A be a closed subset of R n and a ∈ A . Let f : A → B ⊆ R m be Lipschitz. Let { λ j } be any sequence of positive real numbers tending to .(i) There is a subsequence λ j k such that ( λ − j k X, x ) converges in the pointed Gromov-Hausdorff sense to a doublingpointed metric space in Tan(
X, x ) . In particular, Tan(
X, x ) (cid:54) = ∅ .(ii) There is a subsequence λ j k such that λ − j k ( A − a ) converges in the pointed Hausdorff sense to an element ˆ A of Tan R n ( A, a ) . In particular, Tan R n ( A, a ) (cid:54) = ∅ .(iii) In the subsequence from (ii), we may also obtain that • the functions λ − j k ( f ( · ) − f ( a )) converge to a Lipschitz function ˆ f : ˆ A → R m in the sense above, to yield ( ˆ A, ˆ f ) ∈ Tan R n ( A, a, f ) , • the sequence λ − j k ( B − f ( a )) converges in the pointed Hausdorff sense to an element ˆ B of Tan R m ( B, f ( a )) ,and • ˆ f ( ˆ A ) ⊆ ˆ B .(iv) If f is L -bi-Lipschitz, then so is ˆ f .(v) If X and A are Ahlfors Q -regular, then so are every element of WTan ( X ) and Tan R n ( A, a ) .(vi) If µ is a doubling measure on A and a is a point of density of a subset A (cid:48) ⊆ A , then Tan R n ( A (cid:48) , f, a ) =Tan( A, f, a ) .(vii) If Y ∈ WTan ( X ) , then WTan ( Y ) ⊆ WTan ( X ) .Proof. For (i), (ii), and (iii), see [20, Lemmas 8.6 and 8.13]. For (iv), see [20, Lemma 8.20]. For (v), see [20, Lemma8.28]. For (vi) concerning the tangent spaces, see [20, Lemma 9.6] or [39, Proposition 3.1]; the extension to thetangent mappings is simple, as remarked in [21]. For (vii), see [20, Lemma 9.5]. (cid:3)
NFINITESIMAL SPLITTING FOR SPACES WITH THICK CURVE FAMILIES AND EUCLIDEAN EMBEDDINGS 9
We will need one more fact about tangents, a principle that appears in many different forms and goes back to Preiss[50]. Versions appear in, e.g., [42, 5, 39, 21]. Informally, this is the principle that “tangents with moved basepointsare still tangents”.
Proposition 2.5.
Let A ⊆ R n be a closed set supporting a doubling measure µ . Let f : A → R m be a Lipschitzmapping. Then for µ -a.e. a ∈ A , the following holds:For all ( ˆ A, ˆ f ) ∈ Tan R n ( A, a, f ) and all b ∈ ˆ A , we have ( ˆ A − b, ˆ f ( · + b ) − ˆ f ( b )) ∈ Tan R n ( A, a, f ) . The proof of Proposition 2.5 is a minor modification of facts in the literature, and so postponed until the Appendix(Section 6).2.3.
Curves and fragments.
The key objects in this paper are families of curves (or curve fragments) in metricspaces. We introduce some notation to discuss these objects. Our definitions and notation follow those in [52] for themost part, with some minor changes.Fix a separable, locally compact metric space X . A fragment in X is a bi-Lipschitz map γ : C → X , where C ⊆ R is compact and the one-dimensional Lebesgue measure L ( C ) is positive. We write Frag( X ) for the collection offragments in X . If γ ∈ Frag( X ) , then the domain C of γ is denoted dom( γ ) and the image in X is denoted im( γ ) .If f : X → R m is any function, then we define ( f ◦ γ ) (cid:48) ( t ) = lim t (cid:48) → t,t (cid:48) ∈ C f ( γ ( t (cid:48) )) − f ( γ ( t )) t (cid:48) − t , when the limit existsand t ∈ C = dom( γ ) is a density point. If X ⊂ R n , then we simply write γ (cid:48) ( t ) , when f = id is the identity map. A density point of a compact set C ⊂ R is a t ∈ C , where lim h → L ( C ∩ ( t − h,t + h ))2 h = 1 .In Section 4, we will also consider Curv( X ) , the collection of all non-constant, Lipschitz maps γ : I → X , where I is a compact interval in R of positive length. Thus, elements of Curv( X ) represent honest curves in X . We willalso use the notation dom( γ ) to denote the domain of an element γ ∈ Curv( X ) .Note that neither Curv( X ) nor Frag( X ) is a subpsace of the other. We now discuss the appropriate topologies on Frag( X ) and Curv( X ) , borrowing from [52, Section 2].The spaces Frag( X ) and Curv( X ) both admit embeddings into the space Haus ( R × X ) of non-empty compactsubsets of R × X , by γ (cid:55)→ { ( t, γ ( t )) : t ∈ dom( γ ) } . The space Haus ( R × X ) is given the Hausdorff metric and the induced topology. If X is complete, then so is Haus ( R × X ) .Therefore, we topologize Frag( X ) and Curv( X ) as subspaces of Haus ( R × X ) . We note that these spaces are σ -compact if X is proper.2.4. Line integrals and metric derivatives.
Let X be a metric space and γ ∈ Curv( X ) . We denote by len( γ ) thelength of γ , as in [26, Chapter 7]. If g : X → R is a Borel function, then ´ γ g ds is defined as ˆ len( γ )0 g (˜ γ ( t )) dt, where ˜ γ is the arc length parametrization of γ ; see [26, Ch.7].Following [7, Definition 4.1.2], the metric derivative of γ at a point t ∈ dom( γ ) is d γ ( t ) := lim h → d ( γ ( t + h ) , γ ( t )) | h | , whenever the limit exists. By [7, Theorem 4.1.6], d γ ( t ) does exist for a.e. t ∈ dom( γ ) , and len( γ ) = ˆ dom( γ ) d γ ( t ) dt. It follows that the arc length parametrization ˜ γ satisfies d ˜ γ ( t ) = 1 for a.e. t ∈ dom(˜ γ ) .2.5. Alberti representations.
Fix a complete, locally compact, separable metric space X . Recall that M ( X ) denotesthe space of Radon measures on X . We equip M ( X ) with the weak ∗ topology arising from viewing M ( X ) as the dualspace of C c ( X ) , the space of compactly supported continuous functions on X . See [52, Assumption 2.3] for details.Inside M ( X ) , we consider the subspace P ( X ) consisting of probability measures. Note that elements of P ( X ) areBorel.Fix a metric space X and a measure µ ∈ M ( X ) . The following definition is due to Bate [8], based on earlier workof Alberti [1]. In [52], the definition was clarified and modified slightly, and this is the definition we present below. Definition 2.6. An Alberti representation A of µ is a pair ( P, ν ) where(i) P is a Radon probability measure on Frag( X ) ,(ii) ν : Frag( X ) → M ( X ) is a Borel map with ν γ (cid:28) H | im( γ ) for each γ ∈ Frag( X ) ,(iii) the measure µ can be represented as µ ( A ) = ˆ Frag( X ) ν γ ( A ) dP ( γ ) , for each A ⊆ X Borel,(iv) and, for each Borel A ⊆ X and compact interval I ⊆ R , the map γ (cid:55)→ ν γ ( A ∩ γ (dom( γ ) ∩ I )) is Borel.Note that, given the topologies defined above, the statement that the map ν : Frag( X ) → M ( X ) is Borel meansthat γ (cid:55)→ ˆ X g ( x ) dν γ ( x ) is a Borel map from Frag( X ) to R for each g ∈ C c ( X ) .A cone in R n is a set of the form Cone( w, t ) := { v ∈ R n : v (cid:54) = 0 and v · w ≥ t | v |} , for some w ∈ S n − and t ∈ R . Note that, for any w ∈ S n − and t ≤ − , Cone( w, t ) = R n \ { } . Remark 2.7.
Our definition of a cone departs slightly from those in [8] and [52]. In particular, our cones may haveopening angle larger than π . It is clear that any cone of the types in [8] and [52] is a subset of a cone like one above. Definition 2.8.
Fix a metric space X , a Lipschitz map φ : X → R n , and a cone C ⊆ R n .A fragment γ ∈ Frag( X ) is said to be in the φ -direction of C if ( φ ◦ γ ) (cid:48) ( t ) ∈ C for a.e. t ∈ dom( γ ) .An Alberti representation A = ( P, ν ) of a measure µ ∈ M ( X ) is said to be in the φ -direction of C if P -a.e. γ ∈ Frag( X ) is in the φ -direction of C . Definition 2.9.
Cones C , . . . , C k in R n are called independent if each collection { v , . . . , v k : v i ∈ C i } is linearly independent.A collection A , . . . , A k of Alberti representations of a measure µ ∈ M ( X ) is called φ -independent , for a Lipschitz φ : X → R m , if there are independent cones C , . . . , C k in R m such that each A i is in the φ -direction of C i .We call a collection A , . . . , A k of Alberti representations independent if they are φ -independent for some Lipschitzmap φ as above.A few remarks concerning this definition are in order. Remark 2.10. If X ⊆ R n and µ ∈ M ( X ) supports k φ -independent Alberti representations, for some φ : X → R m ,then the map φ may be extended to a Lipschitz map φ : R n → R m without altering the notion of φ -independence. NFINITESIMAL SPLITTING FOR SPACES WITH THICK CURVE FAMILIES AND EUCLIDEAN EMBEDDINGS 11
Remark 2.11.
Traditionally (i.e., in [8]), it is assumed that m = k in Definition 2.9, but we see no need to assumethis, and in fact it will be occasionally convenient not to. Remark 2.12.
In the case k = 1 of Definition 2.9, one may take all of R m \ { } as a single independent cone. Thus,a single Alberti representation ( P, ν ) is independent if and only if there is a Lipschitz map φ : X → R m such that ( φ ◦ γ ) (cid:48) ( t ) (cid:54) = 0 for P -a.e. γ ∈ Frag( X ) and a.e. t ∈ dom( γ ) .In particular, if X ⊆ R n , then a single non-trivial Alberti representation is automatically independent. Indeed, take φ to be the identity map. Since every γ ∈ Frag( X ) is bi-Lipschitz, we have that ( φ ◦ γ ) (cid:48) ( t ) = γ (cid:48) ( t ) (cid:54) = 0 for all γ ∈ Frag( X ) and a.e. t ∈ dom( γ ) .A last key fact for us will be the following result from [8]. Essentially, one would like to know that a phenomenonwhich happens at almost every point along each curve in an Alberti representation actually happens almost everywherein X . This is what the following result provides. (See also the more general [8, Proposition 2.9 ].) Proposition 2.13 (Corollary 2.13 of [8]) . Let X be a complete metric space with a Radon measure µ . Let φ : X → R m be Lipschitz such that µ has k φ -independent Alberti representations. Let f : X → R n be Lipschitz.Then for µ -a.e. x ∈ X , the following hold:(i) There are γ , . . . , γ k ∈ Frag( X ) such that γ i (0) = x and γ − ( x ) is a density point of dom( γ ) .(ii) The derivatives ( φ ◦ γ i ) (cid:48) (0) exist and form a linearly independent set in R m .(iii) The derivatives ( f ◦ γ i ) (cid:48) (0) exist. We briefly note that Bate assumes that k = m and n = 1 in the cited result, but the proof using [8, Lemma 2.8 andProposition 2.9] works in this generality.2.6. Connecting to other measures defined on curve families.
In Section 4, we will need to connect Alberti repre-sentations to a related type of measure defined on
Curv( X ) . Proposition 2.14.
Let X be a proper metric space, φ : X → R n bi-Lipschitz, and C ⊂ R n a cone. Suppose that P is a Radon measure on Curv( X ) , so that for ν -almost every γ , ( φ ◦ γ ) (cid:48) ( t ) ∈ C or d γ ( t ) = 0 for almost every t ∈ dom( γ ) .If the Borel measure defined by µ ( A ) = ˆ Curv( X ) ˆ γ A ds dP is locally finite (hence Radon), then it admits an Alberti representation in the φ -direction of C . Remark 2.15.
The literature is rife with different versions of Alberti representations, see [3, 8, 52, 19] for some ofthem. Modifications of this argument can be used to show that, roughly speaking, if one has a representation in one ofthese senses, then one has also a representation in any other sense.We briefly remark that the cones considered in [8] are slightly different from ours, but the proof applies for bothnotions of cone.
Proof.
By [8, Corollary 5.8] (see the “in particular...” statement), we can decompose X = A ∪ N , where µ | A admitsan Alberti representation in the φ -direction of C , and H (im( γ ) ∩ N ) = 0 for every γ ∈ Frag( X ) in the φ -direction of C . While the notation may suggest so, this measure need not be a probability measure.
If we can show that µ ( N ) = 0 , then µ restricted to the full-measure set A supports an Alberti representation in the φ -direction of C , and this completes the proof. To establish this, we will show that for P -almost every curve γ wehave ˆ γ N ds = 0 . First, for P -almost every curve γ : I → X , and almost every t ∈ I we have ( φ ◦ γ ) (cid:48) ( t ) ∈ C or d γ ( t ) = 0 . Let γ be any curve with such properties, and let ˜ γ : ˜ I → X be its arc length reparametrization. Then ( φ ◦ ˜ γ ) (cid:48) ( t ) ∈ C foralmost every t ∈ I .By [37, Lemma 4], we can find compact sets K j such that ˜ I = (cid:83) j K j ∪ S , | S | = 0 , and ˜ γ j := ˜ γ | K j is bi-Lipschitz.It follows that ˜ γ j ∈ Frag( X ) and in the φ -direction of C , and hence H (im(˜ γ j ) ∩ N ) = 0 for each j .Thus, ˆ γ N ds = ˆ ˜ I N (˜ γ ( t )) dt = (cid:88) j ˆ K j N (˜ γ ( t )) dt = 0 , which completes the proof. (cid:3)
3. P
ROOF OF T HEOREM
Lemma 3.1.
Let a closed set X ⊆ R n support a doubling measure µ . Let µ (cid:28) µ support k ψ -independent Albertirepresentations, for some Lipschitz ψ : R n → R m .Then for µ -a.e. x ∈ X , there are linearly independent vectors v , . . . , v k such that the following holds: For every Y ∈ Tan R n ( X, x ) , every y ∈ Y , and every i ∈ { , . . . , k } , there is a line through y in direction v i that is containedin Y .Proof. We apply Proposition 2.13, in the case φ = ψ and f is the inclusion X → R n .This tells us that, at µ -a.e. x ∈ X , there are γ , . . . , γ k ∈ Frag( X ) such that the following hold:(i) For each ≤ i ≤ k , we have γ i (0) = x with a density point of dom( γ ) .(ii) The vectors ( ψ ◦ γ ) (cid:48) (0) , . . . , ( ψ ◦ γ k ) (cid:48) (0) are linearly independent in R m .(iii) For each ≤ i ≤ k , γ (cid:48) i (0) exists.Fix an x ∈ X where the above hold and where the conclusion of Proposition 2.5 holds. Let w i = ( ψ ◦ γ i ) (cid:48) (0) ∈ R k . Let v i = γ (cid:48) i (0) ∈ R n . Note that v i (cid:54) = 0 as γ i is bi-Lipschitz, and w i (cid:54) = 0 by (ii).Consider an arbitrary tangent ( Y, ˆ ψ ) ∈ Tan R n ( X, ψ, x ) , subject to the sequence of scales λ k → .By Lemma 2.4 (items (ii), (iii), and (vi)), we may pass to a subsequence of { λ j } subject to which the followingtangents also exist (for each ≤ i ≤ k ): ( R , L i ) ∈ Tan R (dom( γ i ) , γ i , with L i ( R ) ⊆ Y, and ( R , ˆ ψ ◦ L i ) ∈ Tan R (dom( γ i ) , ψ ◦ γ i , . NFINITESIMAL SPLITTING FOR SPACES WITH THICK CURVE FAMILIES AND EUCLIDEAN EMBEDDINGS 13
Moreover, since γ i and ψ ◦ γ i are differentiable at , their tangent maps L i and ˆ ψ ◦ L i are linear. In particular, recalling γ (cid:48) i (0) = v i and ( ψ ◦ γ i ) (cid:48) (0) = w i , we have the following properties of L i :(3.1) L i ( t ) = tv i ∈ Y and ψ ( L i ( t )) = tw i for all t ∈ R . By definition, we also have ∈ Y and ˆ ψ (0) = 0 .To summarize, the above argument shows that for every element ( Y, ˆ ψ ) ∈ Tan(
X, x, ψ ) , there is a line L i through with the properties in (3.1).Consider again an arbitrary ( Y, ˆ ψ ) ∈ Tan(
X, x, ψ ) . Proposition 2.5 therefore says that for every y ∈ Y , the pair ( Y − y, ˆ ψ ( · + y ) − ˆ ψ ( y )) is also an element of Tan R n ( X, ψ, x ) .This implies that for every y ∈ Y and i ∈ { , . . . , k } , there is a function L yi : R → Y such that L yi ( t ) = y + tv i and ψ ( L i ( t )) = ˆ ψ ( y ) + tw i for all t ∈ R . In other words, L yi is the parametrization of a line through y in direction v i (contained in Y ), whose composition with ˆ ψ parametrizes a line through ˆ ψ ( y ) in direction w i .It remains to show that the vectors v i are linearly independent. Suppose to the contrary that there was a non-triviallinear combination k (cid:88) i =1 a i v i = 0 . Let y = 0 ∈ Y and p = 0 . For i = 1 , . . . , k + 1 , inductively set y i = L y i − i ( a i ) = y i − + a i v i and p i = ˆ ψ ( y i ) . Note that p i = p i − + a i w i for each i = 1 , . . . , k by the properties of L yi above.Then y k = 0 . This implies that p k = ˆ ψ ( y k ) = 0 . On the other hand p k = k (cid:88) i =1 a i w i . This contradicts the linear independence of the vectors w i . (cid:3) Lemma 3.2.
Let Y ⊆ R n be a closed set. Let v , . . . , v k be linearly independent in R n . Suppose that, for each y ∈ Y ,there are k lines L i = { y + tv i : t ∈ R } that pass through y and are contained in Y .Then Y = Z × V , where V = span ( { v , . . . , v k } ) and Z ⊆ V ⊥ is a closed set.Proof. Let V = span ( { v , . . . , v k } ) , and let Z = proj V ⊥ ( Y ) , the projection of Y to the orthogonal complement of V .We now claim that Y = Z × V . Certainly Y ⊆ Z × V , by definition of orthogonal projection. For the otherdirection, suppose p ∈ Z × V . Then p = proj V ⊥ ( y ) + a v + · · · + a k v k , where y ∈ Y and a i ∈ R . Let z = proj V ⊥ ( y ) ∈ Z . The point y = y is in Y . For i = 1 , . . . , k , we inductively set y i = y i − + a i v i . By assumption, each point y i isin Y . Note that the last point y k is equal to p . Hence p ∈ Y , which proves that Z × V ⊆ Y .Lastly, we argue that Z is closed. Indeed, if z n is a sequence in Z converging to z ∈ R n , then the points ( z n , ∈ Y ⊆ V ⊥ × V = R n converge to ( z, ∈ Y , since Y is closed. It follows that z ∈ Z . (cid:3) Proof of Theorem 1.2.
Let X ⊆ R n be a closed set admitting a doubling Radon measure µ . Let µ be a measureabsolutely continuous to µ that admits k φ -independent Alberti representations.Let x be a point at which the conclusion of Lemma 3.1 holds (a set of points that has full µ -measure). Let v , . . . , v k be the associated linearly independent vectors in R n .Let Y ∈ Tan R n ( X, x ) . Then each point y ∈ Y admits k lines L , . . . , L k through y , in directions v i , that arecontained in Y .By Lemma 3.2, this implies that Y = Z × V for a k -dimensional subspace V = span ( { v , . . . , v k } ) and someclosed set Z ⊆ V ⊥ ⊆ R n . This completes the proof. (cid:3)
4. M
ODULUS AND A LBERTI REPRESENTATIONS
In this section, we relate Alberti representations to the more classical notion of the modulus of a family of curves.The main result in this section is Proposition 4.5, which may be of independent interest. (See Remark 4.6 for more onthe provenance of this result.)We first recall the definition of the modulus of a family of curves. It is worth noting, that for us
Curv( X ) consistsonly of curves with Lipschitz parametrizations, while traditionally Modulus is defined for an a priori larger classof collections of γ : I → X , which are merely continuous. However generality is not lost, as the modulus of non-rectifiable curves vanishes by convention, and rectifiable curves can be reparametrized as Lipschitz curves withoutaffecting the modulus. Definition 4.1.
Let X be a metric space with a Radon measure µ , let Γ ⊆ Curv( X ) , and let p ≥ .A Borel measurable function ρ : X → [0 , ∞ ] is called admissible for Γ if ´ γ ρ ds ≥ for each γ ∈ Γ . We set A (Γ) to be the collection of all admissible functions for Γ .The p -modulus of Γ , with respect to the measure µ , is denoted(4.1) Mod p (Γ , µ ) = inf ρ ∈ A (Γ) ˆ X ρ p dµ. For our duality argument, we will need to work with continuous functions ρ . Thus, we define Mod cp (Γ , µ ) byreplacing the infimum in (4.1) with the infimum over all admissible ρ : X → [0 , ∞ ) that are in addition continuouswith compact support. In general, Mod cp (Γ , µ ) may be larger that Mod p (Γ , µ ) , and our first goal is to identify anassumption under which they are equal.We will need the following basic continuity fact both for the equality of Mod p and Mod cp and for the dualityargument below. It is a version of [33, Proposition 4] in our topology. Lemma 4.2.
Let ρ n : X → R be an increasing sequence of lower semi-continuous functions convering to ρ : X → R .If γ n : [ a, b ] → X converge uniformly to γ : [ a, b ] → X , or if γ n ∈ Curv( X ) converge to γ ∈ Curv( X ) , then lim inf n →∞ ˆ γ n ρ n ds ≥ ˆ γ ρ ds. Further, the map γ (cid:55)→ ´ γ ρ ds is lower semi-continuous on Curv( X ) . NFINITESIMAL SPLITTING FOR SPACES WITH THICK CURVE FAMILIES AND EUCLIDEAN EMBEDDINGS 15
Proof.
First, we reduce the case of γ n ∈ Curv( X ) to the case of uniform convergence with a common domain.Suppose that γ n converge to γ in Curv( X ) . Then their graphs Γ( γ n ) converge to Γ( γ ) in the Hausdorff metric onsubsets X × R . This claim still holds if we reparametrize each curve by an increasing affine map to have domain [0 , .After such reparametrization, by [56, Theorem 1], we see that these reparametrizations converge uniformly.Without loss of generality, we thus assume that a = 0 and b = 1 , and that the parametrized curves convergeuniformly. Note that for each fixed lower semicontinuous g : X → R , the map γ (cid:55)→ ´ γ g ds is lower semicontinuouson the space of curves γ : [0 , → X with the topology of uniform convergence. See, for example, [32, Lemma 2.2].Now, fix N ∈ N . Then, for n ≥ N we get that ˆ γ n ρ n ds ≥ ˆ γ n ρ N ds. Taking a limit inferior on both sides and using the lower semi-continuity noted above, we get lim inf n →∞ ˆ γ n ρ n ds ≥ ˆ γ ρ N ds. Finally, sending N → ∞ and using dominated convergence completes the claim.The latter claim on lower semi-continuity is a restatement of the first claim, by setting ρ n = ρ and assuming that γ n ∈ Curv( X ) converge to γ ∈ Curv( X ) . (cid:3) We will need the following Lemma. The proof is almost identical to the ones in [29, 33, 25], however, for complete-ness and since there is significant variation in the literature on terminology, we recall the main steps of the argument.We will follow the scheme of the proof of [33, Proposition 6], highlighting the main differences along the way, andthe reader may consult it for additional details.
Lemma 4.3. If Γ ⊂ Curv( X ) is compact, then Mod p (Γ , µ ) = Mod cp (Γ , µ ) Proof.
Since continuous admissible functions are also Borel admissible, then
Mod p (Γ , µ ) ≤ Mod cp (Γ , µ ) . We pro-ceed to show the reverse inequality by an approximation argument. If Mod p (Γ , µ ) = ∞ , there is nothing to prove.Otherwise, take any ρ admissible for Mod p (Γ , µ ) with finite L p -norm. Fix (cid:15) > . By the Vitali-Caratheodory theorem,we can approximate any Borel ρ from above by a lower semi-continuous function (cid:101) ρ with (cid:107) (cid:101) ρ (cid:107) pp ≤ (cid:107) ρ (cid:107) pp + (cid:15) .Since Γ is compact, we must have a some bounded ball B ( x, R ) ⊂ X which contains all of the curves. In contrastto [33, Proposition 6], we do not need to adjust (cid:101) ρ further.Next, we approximate from below. Let (cid:102) ρ n (cid:37) (cid:101) ρ be a sequence of continuous functions with compact supportforming an increasing sequence and converging to ρ .As in [33, Proposition 6], it suffices to prove that(4.2) ≤ lim sup n →∞ inf γ ∈ Γ ˆ γ (cid:102) ρ n ds. Indeed, in this case the function (1 − (cid:15) ) − (cid:101) ρ n would be admissible for Γ for sufficiently large n , forcing Mod cp (Γ , µ ) ≤ (1 − (cid:15) ) − p ( (cid:107) ρ (cid:107) pp + (cid:15) ) . Since this holds for all Borel admissible ρ and (cid:15) > , the desired inequality immediately follows.Choose curves γ n ∈ Γ so that lim inf n →∞ ˆ γ n (cid:102) ρ n ds ≤ lim sup n →∞ inf γ ∈ Γ ˆ γ (cid:102) ρ n ds. Since Γ is compact, there exists a subsequence of γ n which converges in Curv( X ) to some γ , which we continueto label γ n . (Unlike [33, Proposition 6], we do not need to reparametrize, or estimate lengths, as we are assumingcompactness.) By using Lemma 4.2, the fact that ρ ≤ (cid:101) ρ , and admissibility, we obtain (4.2): ≤ ˆ γ ρ ds ≤ ˆ γ (cid:101) ρ ds ≤ lim inf n →∞ ˆ γ n (cid:102) ρ n ds ≤ lim sup n →∞ inf γ ∈ Γ ˆ γ (cid:102) ρ n ds. (cid:3) A few standard, well-known facts about modulus will be used repeatedly: (See [26, Ch. 7].)(4.3) If Γ (cid:48) ⊆ Γ then Mod p (Γ (cid:48) , µ ) ≤ Mod p (Γ , µ ) . (4.4) Mod p ( ∪ ∞ i =1 Γ i , µ ) ≤ ∞ (cid:88) i =1 Mod p (Γ i , µ ) . We will also need the following notion, that translates a measure on a curve family into a measure on a space.
Definition 4.4.
Let P be a Radon probability measure on Curv( X ) . It then defines an induced Radon measure η P on X by η P ( E ) = ˆ Curv( X ) ˆ γ E ds dP ( γ ) . Our main result in this section is the following.
Proposition 4.5.
Let X be a proper metric space with a Radon measure µ . Let p ∈ [1 , ∞ ) , and let q ∈ (1 , ∞ ] be thedual exponent, so that p + q = 1 . Let Γ ⊆ Curv( X ) be compact.If Mod p (Γ , µ ) ∈ (0 , ∞ ) , then there is a Radon probability measure P on Curv( X ) , so that η P = f µ, with f ∈ L q ( µ ) and (4.5) || f || L q = Mod p (Γ , µ ) − p . Remark 4.6.
The proof of Proposition 4.5 that we give below is essentially already contained in work of the second-named author and his collaborators in [24, Section 3], though in a less general context.This result can also be deduced from [4] if p > , where it was shown that modulus is dual to probability measuresin a much more general context (for p > ).For completeness, and since the proof is short, we include an argument for all p ≥ in the case where Γ is acompact family of curves.Recall that C c ( X ) denotes the space of continuous functions with compact support, with the uniform topology.The main technical method in the proof of Proposition 4.5 is to express the Lagrangian in a given form and applythe following minimax principle on it. See also [51, Section 9] for another version. Theorem 4.7 ( Sion’s minimax theorem, Corollary 3.3 in [54] ) . Suppose that(i) G is a convex subset of some topological vector space,(ii) K is a compact convex subset of some topological vector space, and(iii) F : G × K → R satisfies(a) F ( · , y ) is convex and lower semi-continuous on G for every y ∈ K ,(b) F ( x, · ) is concave and upper semi-continuous on K for every x ∈ G .Then we have the equality sup y ∈ K inf x ∈ G F ( x, y ) = inf x ∈ G sup y ∈ K F ( x, y ) . The formulation here is slightly simplified.
NFINITESIMAL SPLITTING FOR SPACES WITH THICK CURVE FAMILIES AND EUCLIDEAN EMBEDDINGS 17
Proof of Proposition 4.5.
Note that, since Γ is compact, all curves of Γ must lie in a bounded subset of X , which mustbe compact and of finite µ -measure. Thus, it suffices to assume that X is compact and µ ( X ) < ∞ in the proof.Let K be the set of Radon probability measures on Γ . Then K is a compact, convex subset of the space of Radonmeasures on Curv( X ) , equipped with the topology of weak* convergence. Let G = { g : X → [0 , } ∩ C c ( X ) .Consider the functional Φ : G × K → R Φ( g, P ) = (cid:107) g (cid:107) L p ( µ ) − Mod p (Γ , µ ) p ˆ X g dη P . Lemma 4.2 gives the upper semi-continuity of Φ( g, · ) . The other conditions for Φ = F in Theorem 4.7 are verified asin [24, Theorem 3.7] .Let g ∈ G be any fixed function. Since g has compact support, it has finite L p ( µ ) -norm. Fix (cid:15) > . The function g (cid:15) = Mod p (Γ , µ ) p g (1 + (cid:15) ) || g || L p cannot be admissible, and thus there is a curve γ (cid:15) ∈ Γ with ˆ γ (cid:15) g (cid:15) ds ≤ . Then, ˆ γ (cid:15) g ds ≤ (1 + (cid:15) ) || g || L p Mod p (Γ , µ ) − p . Let P γ (cid:15) = δ γ (cid:15) , the Dirac measure supported on γ (cid:15) ∈ Curv( X ) . Then Φ( g, P γ (cid:15) ) ≥ − (cid:15) || g || L p ( µ ) . Therefore, sending (cid:15) → , we obtain inf g ∈ G sup P ∈ K Φ( g, P ) ≥ . By Theorem 4.7, we get sup P ∈ K inf g ∈ G Φ( g, P ) ≥ . Thus, there is a sequence of measures P (cid:15) ∈ K so that Φ( g, P (cid:15) ) ≥ − (cid:15), for each g ∈ G . By upper semi-continuity in P (cid:15) , and weak compactness, we can extract a weak limit P for which itholds that Φ( g, P ) ≥ , for each g ∈ G . In particular(4.6) ˆ X g dη P ≤ Mod p (Γ , µ ) − p || g || L p ( µ ) . Thus, the functional g → ˆ X g dη P extends to a L p bounded linear functional. Then, by the Riesz representation theorem we have η P = f µ, with the bound || f || L q ≤ Mod p (Γ , µ ) − p following from Estimate (4.6). On the other hand, by Lemma 4.3 we can find a sequence of continuous admissible ρ i so that lim i →∞ ˆ ρ pi dµ = Mod p (Γ , µ ) . By a standard approximation argument of continuous functions by simple functions, and the fact that ρ i is admissi-ble, we obtain that ˆ X ρ i dη P = ˆ Curv( X ) ˆ γ ρ i ds dP ≥ . Therefore, we get ≤ ˆ X ρ i dη P ≤ Mod p (Γ , µ ) − p || ρ i || L p . Sending i → ∞ , we get(4.7) ≤ lim i →∞ ˆ X ρ i dη P ≤ Mod p (Γ , µ ) − p Mod p (Γ , µ ) p = 1 . Thus, since we get equality in the limit,
Mod p (Γ , µ ) − p equals the norm of the functional g → ´ g dη P , and thus || f || L q = Mod p (Γ , µ ) − p . (cid:3) Remark 4.8.
Further details could be obtained. If p > , then the sequence in the last paragraph L p -converges ρ i → ρ ∗ , where ρ ∗ is an admissible function for Γ \ Γ (cid:48) where Γ (cid:48) has modulus zero, see e.g. [58]. Then, ρ ∗ pluggedinto Estimate 4.7 (without the limit), yields(4.8) ( ρ ∗ ) p = Mod p (Γ , µ ) p + qp f q almost everywhere. Further, ´ γ ρ ∗ ds = 1 for P -almost every γ , where P is the probability measure coming from thestatement.As an immediate corollary of Propositions 4.5 and 2.14, we explicitly point out the connection between modulusand Alberti representations: Corollary 4.9.
Let X be a proper metric space with a Radon measure µ . Suppose that x ∈ X admits a family Γ ⊆ Curv( X ) with positive modulus, that is Mod p (Γ , µ ) > for some p ≥ . Then there is a non-zero measure on X , absolutely continuous to µ , which admits an Alberti representation. Fur-thermore, if φ : X → R n is bi-Lipschitz, then the Alberti representation can be chosen φ -independent (i.e., in the φ -direction of some cone C ). As a reminder,
Curv( X ) by definition does not contain any constant curves. Indeed, the purpose of this restrictionis to prevent the following: A family of curves containing a constant curve would allow for no admissible functionsand thus have modulus ∞ . Such a family could not be used to construct a non-trivial Alberti representation, e.g., if itcontained no non-constant curves. Proof.
Fix x ∈ X . For each n ∈ N , define Γ n = { γ ∈ Γ : im( γ ) ⊆ B ( x , n ) , diam(im( γ )) ≥ n , LIP( γ ) ≤ n, dom( γ ) ⊆ [ − n, n ] } . Since Γ ⊆ ∪ ∞ n =1 Γ n , (4.4) tells us that Mod p (Γ n , µ ) > for some n , which we now fix.Consider the closure Γ n ⊂ Curv( X ) , which is compact by Arzelà-Ascoli. By (4.3), Mod p (Γ n , µ ) ≥ Mod p (Γ n , µ ) > . NFINITESIMAL SPLITTING FOR SPACES WITH THICK CURVE FAMILIES AND EUCLIDEAN EMBEDDINGS 19
Moreover, since each curve γ ∈ Γ n has diam(im( γ )) ≥ n and is contained in B ( x , n ) , we have Mod p (Γ n , µ ) ≤ n p µ ( B ( x , n )) < ∞ . By Proposition 4.5 we obtain a measure P on Curv( X ) , so that the corresponding measure η P is absolutelycontinuous with respect to µ . Then, applying Proposition 2.14 with cone C = R n \ { } , we obtain a non-trivialAlberti representation for µ that is φ -independent, i.e., in the φ -direction of C . (cid:3)
5. P
ROOFS OF THE COROLLARIES
In this section, we prove all the corollaries of our main result stated in the introduction.Before beginning the proofs, the following basic lemma allows us to reduce problems of bi-Lipschitz embeddingto Theorem 1.2.
Lemma 5.1.
Let X be a metric space with a doubling measure µ . Suppose that µ (cid:28) µ supports k φ -independentAlberti representations, for some φ : X → R m .Let f : X → Y be a bi-Lipschitz homeomorphism.Then(i) ˆ µ := f ∗ ( µ ) is a doubling measure supported on Y .(ii) ˆ µ := f ∗ ( µ ) (cid:28) ν .(iii) ˆ µ supports k φ ◦ f − -independent Alberti representations.Proof. The first two statements are immediate from the definitions.For the third statement, let A i = ( P i , ν i ) be independent Alberti representations for µ , for i = 1 , . . . , k .Note that f : X → Y and f − : Y → X induce continuous maps F : Frag( X ) → Frag( Y ) and F − : Frag( Y ) → Frag( X ) , by post-composition.Let ˆ ν : Frag( Y ) → M ( Y ) be defined by ˆ ν γ = f ∗ ( ν F − ( γ ) ) for each γ ∈ Frag( Y ) . It is immediate that ˆ ν γ (cid:28) H | im( γ ) for each γ ∈ Frag( Y ) , since bi-Lipschitz maps preserve sets of zero H -measure.We therefore define the Alberti representations ˆ A i = ( F ∗ ( P i ) , f ∗ ν i ) for i = 1 , . . . , k .It is easy to check that these satisfy conditions (i), (ii), and (iv) of Definition 2.6. For condition (iii), observe that if A ⊆ Y is Borel and i ∈ { , . . . , k } , then ˆ µ ( A ) = µ ( f − ( A ))= ˆ Frag( X ) ν iγ ( f − ( A )) dP i ( γ )= ˆ Frag( X ) ν iF ( γ ) ( A ) dP i ( γ )= ˆ Frag( Y ) ˆ ν iα ( A ) dP i ( α ) , as desired.Lastly, we check the independence of the new Alberti representations on Y . Let A = ( P, ν ) denote any one ofthe k original Alberti representations A i above. Then there is a cone C ⊆ R k such that ( φ ◦ γ ) (cid:48) ( t ) ∈ C for P -a.e. γ ∈ Frag( X ) and a.e. t ∈ dom( γ ) . Let G ⊆ Frag( X ) be the full P -measure set on which this holds.Let ˆ G = F ( G ) ⊆ Frag( Y ) , a set of full ˆ P -measure in Frag( Y ) . Consider any α ∈ ˆ G ⊆ Frag( Y ) . Then thefragment γ defined by t (cid:55)→ f − ( α ( t )) is in G . Therefore, for a.e. t ∈ dom( α ) = dom( γ ) , ( φ ◦ f − ◦ α ) (cid:48) ( t ) = ( φ ◦ γ ) (cid:48) ( t ) is in C .Thus, each new Albert representation ˆ A i is in the ( φ ◦ f − ) -direction of the same cone of which A i was in the φ -direction. Thus, the representations ˆ A i are ( φ ◦ f − ) -independent. (cid:3) As a consequence, we can now prove Corollary 1.3.
Proof of Corollary 1.3.
Let X be a complete metric space admitting a doubling Radon measure µ . Suppose that ameasure µ (cid:28) µ supports k independent Alberti representations, for some k ≥ .Suppose that f : X → R n is a bi-Lipschitz embedding. Let X (cid:48) = f ( X ) , µ (cid:48) = f ∗ ( µ ) , and µ (cid:48) = f ∗ ( µ ) .It follows from Lemma 5.1 that µ (cid:48) admits k independent Alberti representations. Therefore at µ (cid:48) -a.e. point x (cid:48) ∈ X (cid:48) ,every tangent Y (cid:48) ∈ Tan R n ( X (cid:48) , x (cid:48) ) is isometric to Z × R k , for some closed set Z ⊆ R n − k .The set of all preimages under f of such points x (cid:48) ∈ X (cid:48) forms a set of full µ -measure in X . At such a point x = f − ( x (cid:48) ) ∈ X , each tangent ( Y, y ) ∈ Tan(
X, x ) is bi-Lipschitz equivalent to an element of Tan R n ( X (cid:48) , x (cid:48) ) , witha bi-Lipschitz map given by a tangent map of f . Thus, any such tangent Y is bi-Lipschitz equivalent to a product Z × R k , for some complete metric space Z . This proves the corollary. (cid:3) Modulus and conformal dimension.
Here we prove Corollaries 1.5, 1.8, and 1.9.
Proof of Corollary 1.5 .
Let X be a complete metric space admitting a Radon measure µ that is absolutely contin-uous with respect to a doubling measure µ . Suppose that X contains a family of (non-constant) curves Γ so that Mod p (Γ , µ ) > for some p ∈ [1 , ∞ ) , and that X admits a bi-Lipschitz embedding φ into some R n .By Corollary 4.9, there is a non-trivial measure on X that is absolutely continuous to µ , hence to µ , and supportsa φ -independent Alberti representation. The corollary then follows from Corollary 1.3. (cid:3) Proof of Corollary 1.8.
Let X satisfy the assumptions of the corollary. Thus, X is Ahlfors Q -regular with Q =cdim AR ( X ) and X admits a bi-Lipschitz embedding into some R n .By the Keith-Laakso Theorem 1.7, there is a weak tangent W of X that contains a family of non-constant curveswith positive modulus. By Lemma 2.4, the space W also admits a bi-Lipschitz embedding into R n . By Corollary 1.5,there is a tangent Y of W that is bi-Lipschitz equivalent to Z × R for some complete metric space Z .As Y is also a weak tangent of the original space X (see Lemma 2.4(vii)), this completes the proof. (cid:3) Proof of Corollary 1.9.
Let X be linearly connected and let Y be quasisymmetric to X and Ahlfors Q -regular, where Q = cdim AR ( X ) = cdim AR ( Y ) ∈ (1 , . Suppose that Y did admit a bi-Lipschitz embedding into some Euclidean space, R N .By Corollary 1.8, Y would then admit a weak tangent W that is bi-Lipschitz equivalent to a product Z × R , forsome complete metric space Z . Let φ : Z × R → W be bi-Lipschitz. Let π : Z × R → Z be the projection to the Z factor. NFINITESIMAL SPLITTING FOR SPACES WITH THICK CURVE FAMILIES AND EUCLIDEAN EMBEDDINGS 21
The linear connectedness condition is preserved under both quasisymmetry and passage to weak tangents. (Theformer assertion is immediate from the definitions, and the latter is contained in the proof of [36, Proposition 5.4].)Thus, W is linearly connected. By Lemma 2.4(v), W is also Ahlfors Q -regular.Next, we observe that Z must contain at least two points: if not, then W would be bi-Lipschitz equivalent to Z × R ∼ = R , which would contradict the fact that W is Ahlfors Q -regular for Q > .We now observe that Z must contain a compact, connected set K with at least two points. To see this, fix distinctpoints z, z (cid:48) ∈ Z . Since W is linearly connected, there is a compact, connected set J in W that contains φ ( z, and φ ( z (cid:48) , . The set K = π ( φ − ( J )) is then a continuum containing z and z (cid:48) .Thus, Z contains a non-trivial continuum K and so W contains a bi-Lipschitz image of the space K × [0 , . Wenow argue that(5.1) dim H ( K × [0 , ≥ . That the Hausdorff dimension of a product is at least the sum of the dimensions of the factors is standard for compactsubsets of Euclidean space (see, e.g., [9, Theorem 3.2.1]); we give a brief argument in our setting here:Since K is compact and connected, H ( K ) > . By Frostman’s Lemma [43, Theorem 8.17], K supports a Radonmeasure µ satisfying µ ( B ( x, r )) ≤ r for all x ∈ K and < r ≤ diam( K ) . If L denotes Lebesgue measure on [0 , , then the measure µ × L on K × [0 , satisfies ( µ × L )( B ( p, r )) ≤ r for all p ∈ K × [0 , and < r ≤ diam( K × [0 , . By the “mass distribution principle” (see [26, p. 61]), we obtain (5.1).We therefore arrive at > Q = dim H ( W ) ≥ dim H ( K × [0 , ≥ , a contradiction. (cid:3) Slit carpet.
Here, we prove Corollary 1.10. This will follow from Corollary 1.5 and some facts about the slitcarpet.We first summarize some results of Merenkov [44].
Proposition 5.2 (Lemma 2.1, Proposition 2.4, and Lemma 4.2 of [44]) . The slit carpet M has the following properties:(i) It is homeomorphic to the standard Sierpi´nski carpet. In particular, it is compact and has topological dimension . (ii) It is geodesic and Ahlfors -regular.(iii) It admits a family of non-constant curves with positive -modulus (with respect to the measure H ).Proof of Corollary 1.10. By Proposition 5.2, we may fix a point p ∈ M where the conclusion of Corollary 1.5 holds.The self-similarity of M easily implies the following: There is a constant c > such that, for each r > , there is apoint q r and a compact set K r ⊆ B ( p, r ) such that B ( q r , cr ) ⊆ K r ⊆ B ( p, r ) , and K r is isometric to t M for some t ∈ (2 cr, r ) . Here, “topological dimension” can refer to Lebesgue covering dimension or (small) inductive dimension, which are equivalent for compact metricspaces [45]. All we will need to know is that M does not contain a topologically embedded copy of any open subset of R . Let ( Y, y ) ∈ Tan( M , p ) be obtained by rescaling along a sequence λ i → . Let q i = q λ i and K i = K λ i . Passingto a (subsequential) limit, the above properties imply (see, e.g., [20, Lemma 8.31]) that there is a point q ∈ Y and acompact set K ⊆ Y such that B ( q, c ) ⊆ K ⊆ B ( y, and K is bi-Lipschitz equivalent to M . By Corollary 1.5, Y is bi-Lipschitz equivalent to Z × R for some complete (and necessarily doubling) metric space Z . As a tangent of M , Y is quasiconvex and therefore so is Z .It follows that Z contains a non-trivial topological arc through each point. Hence, there is a homeomorphic imageof [0 , contained in B ( q, c ) . This implies that B ( q, c ) ⊆ K must have topological dimension . However, thiscontradicts the fact that K is bi-Lipschitz equivalent to M , which has topological dimension . (cid:3) Heisenberg group.
Here we give a brief introduction to the Heisenberg group and prove Corollary 1.11.5.3.1.
Preliminaries on the Heisenberg group.
We now fix some notation and definitions. We will be very brief,referring the reader to [40, 15] for details.The group H is R endowed with the non-abelian group law ( x , y , z ) · ( x , y , z ) = ( x + x , y + y , z + z + 12 ( x y − y x )) . There are many standard, bi-Lipschitz equivalent ways to equip H with a metric. For simplicity, we fix the so-calledKorányi distance, though it will make little difference below. Definition 5.3.
The
Korányi norm of ( x, y, z ) ∈ H is (cid:107) ( x, y, z ) (cid:107) = (( x + y ) + 16 z ) / . The
Korányi distance between p, q ∈ H is d ( p, q ) = (cid:107) p − q (cid:107) . The Korányi distance on H induces the usual topology from R and has the following features (see, e.g. [40,Example 1.3]):(i) Left-invariance: d ( p, q ) = d ( p (cid:48) · p, p (cid:48) · q ) for all p, q, p (cid:48) ∈ H .(ii) Dilations: For each t > , the map δ t ( x, y, z ) = ( tx, tx, t z ) is a group homomorphism with the property that d ( δ t ( p ) , δ t ( q )) = td ( p, q ) for all p, q ∈ H (iii) Doubling: The Lebesgue measure L on R is doubling on ( H , d ) . In fact, it satisfies the Ahlfors -regularityproperty L ( B ( p, r )) ≈ r for all p ∈ H , r > and a fixed positive implied constant.In particular, item (iii) implies that every open set in H has Hausdorff dimension .The Kora´nyi distance is also bi-Lipschitz eqiuvalent to the Carnot-Carathéodory distance, which we do not definehere, as both satisfy properties (i)-(iii) above [40].From these properties, we first derive the following: Lemma 5.4.
There is an open set U in the Heisenberg group such that L | U supports two independent Alberti repre-sentations. NFINITESIMAL SPLITTING FOR SPACES WITH THICK CURVE FAMILIES AND EUCLIDEAN EMBEDDINGS 23
This fact is well-known, and of course much more is true. We include a brief proof only to show that no sophisti-cated tools are needed.
Proof.
Let φ : H → R be the canonical Lipschitz chart, given by ( x, y, z ) (cid:55)→ ( x, y ) . Fix any disjoint, independentcones, C x , C y in R containing the x and y -axis respectively.For each p = ( a, b ) ∈ R , the curves α p ( t ) = (cid:18) t, a, b − at (cid:19) and β p ( t ) = (cid:18) a, t, b + 12 at (cid:19) are bi-infinite geodesics in H . Indeed, α ( a,b ) ( t ) = (0 , a, b ) · ( t, , and β ( a,b ) ( t ) = ( a, , b ) · (0 , t, , and the curves ( t, , and (0 , t, are clearly geodesics.Define the maps Φ α and Φ β from (cid:2) − , (cid:3) to R by ( p, t ) (cid:55)→ α p ( t ) and ( p, t ) (cid:55)→ β p ( t ) , respectively. One directlyverifies that these maps are injective and are open mappings on the interiors of their domains. Since Φ α (0) = Φ β (0) =(0 , , , there is a non-empty open set U contained in Φ α (cid:32)(cid:20) − , (cid:21) (cid:33) ∩ Φ β (cid:32)(cid:20) − , (cid:21) (cid:33) . We will obtain two independent Alberti representations of the measure µ = L | U on H .Computing the Jacobians gives immediately that Φ α and Φ β are volume preserving. Writing λ for Lebesguemeasure on (cid:2) − , (cid:3) , we therefore obtain by change of variables that(5.2) µ ( A ) = ˆ R ˆ R A ( α p ( t )) dt dλ ( p ) and(5.3) µ ( A ) = ˆ R ˆ R A ( β p ( t )) dt dλ ( p ) for any Borel set A ⊆ U .Thus, define a probabilty measure P on Frag( H ) by the pushforward of λ | [ − , ] under the map from (cid:2) − , (cid:3) given by p (cid:55)→ α p | [ − , ] ∈ Frag( H ) . Note that this map is continuous on (cid:2) − , (cid:3) .For γ ∈ Frag( X ) , define ν γ = 0 if γ is not in the support of P . Otherwise, γ = α p | [ − , ] for some p , and we set ν γ = 1 U · H | im( γ ) . The pair ( P, ν ) defines an Alberti representation of µ by (5.2), supported on curves of the form α p | [ − , ] . Theexact same procedure applied to the curves β p yields an Alberti representation of µ supported on curves of the form β p | [ − , ] .Since ( φ ◦ α p ) (cid:48) ( t ) = (1 , ∈ C x and ( φ ◦ β p ) (cid:48) ( t ) = (0 , ∈ C y for each p ∈ R and t ∈ R , the two Albertirepresentations are independent. (cid:3) Proof of Corollary 1.11.
There are a number of proofs of Corollary 1.11 in the literature. We give a proof belowthat avoids Pansu’s differentiation theorem from [49]. It relies only on Theorem 1.2, invariance of domain, and thebasic properties of the Heisenberg group stated in the previous subsection. Of course, we still use a blowup argument,so the ideas are similar in spirit.
Proof of Corollary 1.11.
Suppose that the Heisenberg group (with the Korányi metric) admitted a bi-Lipschitz embed-ding into some Euclidean space. By Lemma 5.4 and Corollary 1.3, some tangent of the Heisenberg group would bebi-Lipschitz equivalent to Z × R , for some complete metric space Z .By the homogeneity and dilation structure of the metric, every tangent of the Heisenberg group is isometric to theHeisenberg group itself. Thus, in this case, H is bi-Lipschitz equivalent to Z × R . Since H is proper and quasiconvex,so is Z .It follows that Z is bi-Lipschitz equivalent to a geodesic metric space, simply by replacing the metric on Z by theassociated length metric. Therefore, in particular, Z contains a Lipschitz embedding γ : [0 , → Z .Let φ : Z × R → H be bi-Lipschitz. The map from (0 , × (0 , into H given by ( t, p ) (cid:55)→ φ ( γ ( t ) , p ) is therefore a Lipschitz homeomorphism from an open set of R into H . By invariance of domain, the image of thismap must be open in H . On the other hand, this set has Hausdorff dimension at most , as the Lipschitz image of asubset of R . This violates the Ahlfors -regularity of H . (cid:3)
6. A
PPENDIX : P
ROOF OF P ROPOSITION outer measure µ ∗ ( A ) = inf { µ ( B ) : B Borel , B ⊇ A } , and the associated notion of a point of outer density x of a set A , where lim r → µ ∗ ( B ( x, r ) ∩ A ) µ ( B ( x, r )) → . As explained briefly in [21], every set of positive outer measure has a point of outer density.The following lemma is a simple extension of (vi) of Lemma 2.4. See also [20, Lemma 9.6] or [39, Proposition3.1] for closely related statements whose proofs can easily be modified to yield this one.
Lemma 6.1.
Let A ⊆ R n support a doubling measure µ and a Lipschitz f : A → R m . Let E ⊆ A have a point ofouter µ -density at a ∈ A . Then Tan R n ( E, f, a ) = Tan R n ( A, f, a ) . We now define a notion of distance that yields the correct topology. As in [21], the distance we define will notprecisely be a metric, but it will suffice for our purposes.
Definition 6.2.
Let
A, B ⊆ R N be sets and f, g : R N → R M , Lipschitz functions. Define ˜ D (( A, f ) , ( B, g )) = inf { (cid:15) > d /(cid:15) ( A, B ) < (cid:15) and | f − g | < (cid:15) on ( A ∪ B ) ∩ B (0 , /(cid:15) ) } . Then define D = min( ˜ D, / . NFINITESIMAL SPLITTING FOR SPACES WITH THICK CURVE FAMILIES AND EUCLIDEAN EMBEDDINGS 25
To simplify notation, given A ⊆ R N , f : R N → R M Lipschitz, λ > , and p ∈ R N , we set A p,λ = λ − ( A − p ) and f p,λ ( x ) = λ − ( f ( λx + p ) − f ( p )) . Note that f p,λ is Lipschitz with the same constant as f .The following is analogous to [21, Lemma 2.3]. Lemma 6.3.
The function D has the following properties:(i) It is non-negative and symmetric.(ii) If D (( A, f ) , ( B, g )) = 0 then A = B and f = g on A = B .(iii) For all pairs ( A, f ) , ( B, g ) , ( C, h ) , we have the quasi-triangle inequality D (( A, f ) , ( C, h )) ≤ D (( A, f ) , ( B, g )) + D (( B, g ) , ( C, h ))) (iv) ( ˆ A, ˆ f ) ∈ Tan R N ( A, f, a ) if and only if some Lipschitz extensions of f and ˆ f to all of R N and some sequence λ i → satisfy (6.1) D (( A a,λ i , f a,λ i ) , ( ˆ A, ˆ f )) → Proof.
The first two items are simple, the third follows exactly as in [21, Lemma 2.3], and the fourth follows from[20, Lemma 8.7]. (cid:3)
The next lemma is an analog of [21, Lemma 2.6].
Lemma 6.4.
For each
N, M ∈ N and L, η > , the collection S = { ( B, g ) : B ⊆ R N , g : R N → R M L -Lipschitz } is contained in a countable collection of sets B (cid:96) with D -diameter at most η . The “ D -diameter” of a collection of pairs { ( B, g ) } is the supremum of the D -distance between pairs of elementsin the collection. Proof.
Consider all pairs ( K, h ) such that K ⊆ Q N ⊆ R N is finite, and h : K → Q M .By Lemma 6.3(iii), it suffices to show, given η ∈ (0 , and ( B, g ) ∈ S , that ( B, g ) is within D -distance η ofsome ( K, ˆ h ) , where ( K, h ) is as above and ˆ h is a Lipschitz extension of h to all R N . We may also assume L > .Fix K ⊆ B (0 , η − ) ∩ Q N to be η/L -separated and finite such that d η − ( K, B ) ≤ η/L. Then, for x ∈ K , set h ( x ) to be an element of Q M within distance η/L of g ( x ) . Note that h is L -Lipschitz. Extend h to a L -Lipschitz map ˆ h on R N by Kirszbraun’s theorem. For x ∈ K ∩ B (0 , η − ) , | g ( x ) − ˆ h ( x ) | ≤ η/L < η and if x ∈ B ∩ B (0 , η − ) , then | g ( x ) − ˆ h ( x ) | ≤ η + | g ( y ) − h ( y ) | ≤ η, where y is a closest element in K to x . This completes the proof. (cid:3) Proof of Proposition 2.5.
We closely follow the argument in [21]. Let A ⊆ R n support a doubling measure µ . Let f : A → R m be a Lipschitz mapping. Extend f to be an L -Lipschitz function defined on all of R N by the standardMcShane-Whitney extension theorem.Our goal is to show that the set { a ∈ A : there exists ( B, g ) ∈ Tan R N ( A, f ) and b ∈ B such that ( B − b, g ( · + b ) − g ( b )) / ∈ Tan R N ( A, f ) } has outer measure zero.Consider the collection S = { ( B, g ) : B ⊆ R N , g : R N → R M L -Lipschitz } . Note that every rescaling or tangent of ( A, f ) lies in S .Fix k ≥ . Apply Lemma 6.4 to obtain countably many collections B l such that, for all l ,diam D ( B l ) < / k and S ⊆ ∪ B l .It therefore suffices to show that, for all k, l, m ∈ N , the set of “bad” points with parameters k, l, m , namely (cid:26) a ∈ A : there exists ( B, g ) ∈ Tan R n ( A, f ) and b ∈ B such that(6.2) ( B − b, g ( · + b ) − g ( b )) ∈ B l and D (( B − b, g ( · + b ) − g ( b )) , ( A a,t , f a,t )) > k for all t ∈ (0 , /m ) (cid:27) has outer measure zero.Suppose that, for some k, l, m ∈ N , the set above has positive outer measure, and call it A (cid:48) ⊆ A . Let a be a pointof outer density of A (cid:48) . Then there exist ( B, g ) ∈ Tan R N ( A, f, a ) and b ∈ B such that ( B − b, g ( · + b ) − g ( b )) ∈ B l and D (cid:18) ( B − b, g ( · + b ) − g ( b )) , (cid:18) t ( A − a ) , t ( f − f ( a )) (cid:19)(cid:19) > k , for all t ∈ (0 , /m ) . Because ( B, g ) ∈ Tan R N ( A, f, a ) = Tan R N ( A (cid:48) , f, a ) by Lemma 6.1, there is a sequence λ i → such that(6.3) (cid:15) i := D (cid:0) ( B, g ) , ( A (cid:48) a,λ i , f a,λ i ) (cid:1) → . In particular, we may choose a i ∈ A (cid:48) such that(6.4) | b − λ − i ( a i − a ) | < (cid:15) i , when i is sufficiently large. Claim 6.5.
We have D (( B − b, g ( · + b ) − g ( b )) , ( A λ i ,a i , f λ i ,a i )) → . Proof of Claim.
Observe that if q − b ∈ ( B − b ) ∩ B (0 , (2 (cid:15) i ) − ) , then q ∈ B ∩ B (0 , (cid:15) − i ) and therefore there is p ∈ A such that | q − λ − i ( p − a ) | < (cid:15) i . It follows that | ( q − b ) − λ − i ( p − a i ) | ≤ | q − λ − i ( p − a ) | + | b − λ − i ( a i − a ) | < (cid:15) i . NFINITESIMAL SPLITTING FOR SPACES WITH THICK CURVE FAMILIES AND EUCLIDEAN EMBEDDINGS 27
A similar argument also shows, conversely, that if λ − i ( p − a i ) is an arbitrary point in λ − i ( A − a i ) ∩ B (0 , (2 (cid:15) i ) − ) ,then there is a point q − b ∈ B − b such that | ( q − b ) − λ − i ( p − a i ) | < (cid:15) i . Together, these show that d (2 (cid:15) i ) − ( B − b, A a i ,λ i ) < (cid:15) i . By (6.3), the functions g and f a,λ i agree up to error (cid:15) i on ( B ∪ A a,λ i ) ∩ B (0 , (cid:15) − i ) .Note that if q − b ∈ ( B − b ) ∩ B (0 , (2 (cid:15) i ) − ) , then q ∈ B ∩ B (0 , (cid:15) − i ) . Similarly, if λ − i ( p − a i ) ∈ A a i ,λ i ∩ B (0 , (2 (cid:15) i ) − ) , then λ − i ( p − a ) ∈ A a,λ i ∩ B (0 , (cid:15) − ) . Thus, a basic calculation using (6.4) and the triangle inequality, extremely similar to that on [21, p. 565], yieldsthat the functions g ( · ) − g ( b ) and f λ i ,a i agree up to error (cid:46) (cid:15) i on (( B − b ) ∪ A a i ,λ i ) ∩ B (0 , (2 (cid:15) i ) − ) , where the implied constant depends only on the Lipschitz constants of f and g . This proves the claim. (cid:3) With the claim proven, one reaches a contradiction exactly as on [21, p. 566]. Therefore, the set of “bad points”defined in (6.2) has outer measure zero, which completes the proof. (cid:3) R EFERENCES[1] G. Alberti. Rank one property for derivatives of functions with bounded variation.
Proc. Roy. Soc. Edinburgh Sect. A , 123(2):239–274, 1993.[2] G. Alberti, M. Csörnyei, and D. Preiss. Structure of null sets in the plane and applications. In
European Congress of Mathematics , pages 3–22.Eur. Math. Soc., Zürich, 2005.[3] Giovanni Alberti and Andrea Marchese. On the differentiability of Lipschitz functions with respect to measures in the Euclidean space.
Geom.Funct. Anal. , 26(1):1–66, 2016.[4] L. Ambrosio, S. Di Marino, and G. Savaré. On the duality between p -modulus and probability measures. J. Eur. Math. Soc. (JEMS) ,17(8):1817–1853, 2015.[5] L. Ambrosio, B. Kleiner, and E. Le Donne. Rectifiability of sets of finite perimeter in Carnot groups: existence of a tangent hyperplane.
J.Geom. Anal. , 19(3):509–540, 2009.[6] Luigi Ambrosio and Bernd Kirchheim. Rectifiable sets in metric and Banach spaces.
Math. Ann. , 318(3):527–555, 2000.[7] Luigi Ambrosio and Paolo Tilli.
Topics on analysis in metric spaces , volume 25 of
Oxford Lecture Series in Mathematics and its Applications .Oxford University Press, Oxford, 2004.[8] D. Bate. Structure of measures in Lipschitz differentiability spaces.
J. Amer. Math. Soc. , 28(2):421–482, 2015.[9] Christopher J. Bishop and Yuval Peres.
Fractals in probability and analysis , volume 162 of
Cambridge Studies in Advanced Mathematics .Cambridge University Press, Cambridge, 2017.[10] Mario Bonk. Quasiconformal geometry of fractals. In
International Congress of Mathematicians. Vol. II , pages 1349–1373. Eur. Math. Soc.,Zürich, 2006.[11] Mario Bonk and Bruce Kleiner. Conformal dimension and Gromov hyperbolic groups with 2-sphere boundary.
Geom. Topol. , 9:219–246,2005.[12] Mario Bonk and Sergei Merenkov. Quasisymmetric rigidity of square Sierpi´nski carpets.
Ann. of Math. (2) , 177(2):591–643, 2013.[13] Marc Bourdon and Hervé Pajot. Cohomologie l p et espaces de Besov. J. Reine Angew. Math. , 558:85–108, 2003.[14] D. Burago, Y. Burago, and S. Ivanov.
A course in metric geometry , volume 33 of
Graduate Studies in Mathematics . American MathematicalSociety, Providence, RI, 2001.[15] Luca Capogna, Donatella Danielli, Scott D. Pauls, and Jeremy T. Tyson.
An introduction to the Heisenberg group and the sub-Riemannianisoperimetric problem , volume 259 of
Progress in Mathematics . Birkhäuser Verlag, Basel, 2007.[16] J. Cheeger. Differentiability of Lipschitz functions on metric measure spaces.
Geom. Funct. Anal. , 9(3):428–517, 1999.[17] J. Cheeger and B. Kleiner. On the differentiability of Lipschitz maps from metric measure spaces to Banach spaces. In
Inspired by S. S. Chern ,volume 11 of
Nankai Tracts Math. , pages 129–152. World Sci. Publ., Hackensack, NJ, 2006.[18] J. Cheeger and B. Kleiner. Differentiability of Lipschitz maps from metric measure spaces to Banach spaces with the Radon-Nikodým property.
Geom. Funct. Anal. , 19(4):1017–1028, 2009.[19] J. Cheeger, B. Kleiner, and A. Schioppa. Infinitesimal structure of differentiability spaces, and metric differentiation.
Anal. Geom. Metr.Spaces , 4(1):104–159, 2016. [20] G. David and S. Semmes. “Fractured fractals and broken dreams”, volume 7 of
Oxford Lecture Series in Mathematics and its Applications .The Clarendon Press, Oxford University Press, New York, 1997.[21] G. C. David. Tangents and rectifiability of Ahlfors regular Lipschitz differentiability spaces.
Geom. Funct. Anal. , 25(2):553–579, 2015.[22] Guy C. David and Sylvester Eriksson-Bique. Regular mappings and non-existence of bi-Lipschitz embeddings for slit carpets.
Adv. Math. ,364:107047, 2020.[23] Guy C. David and Kyle Kinneberg. Rigidity for convex-cocompact actions on rank-one symmetric spaces.
Geom. Topol. , 22(5):2757–2790,2018.[24] E. Durand-Cartagena, S. Eriksson-Bique, R. Korte, and N. Shanmugalingam. Equivalence of two bv classes of functions in metric spaces, andexistence of a semmes family of curves under a 1-poincaré inequality.
Advances in Calculus of Variations , 2019.[25] Vendula Honzlová Exnerová, Ondˇrej FK Kalenda, Jan Mal`y, and Olli Martio. Plans on measures and am-modulus. arXiv preprintarXiv:1904.04527 , 2019.[26] J. Heinonen. “Lectures on analysis on metric spaces”. Universitext. Springer-Verlag, New York, 2001.[27] J. Heinonen. Nonsmooth calculus.
Bull. Amer. Math. Soc. (N.S.) , 44(2):163–232, 2007.[28] J. Heinonen and S. Semmes. Thirty-three yes or no questions about mappings, measures, and metrics.
Conform. Geom. Dyn. , 1:1–12 (elec-tronic), 1997.[29] Juha Heinonen and Pekka Koskela. Quasiconformal maps in metric spaces with controlled geometry.
Acta Math. , 181(1):1–61, 1998.[30] Juha Heinonen, Pekka Koskela, Nageswari Shanmugalingam, and Jeremy T. Tyson.
Sobolev spaces on metric measure spaces , volume 27 of
New Mathematical Monographs . Cambridge University Press, Cambridge, 2015. An approach based on upper gradients.[31] Piotr Hajlasz (https://mathoverflow.net/users/121665/piotr hajlasz). Non embedding of the Heisenberg group. MathOverflow.URL:https://mathoverflow.net/q/297806 (version: 2018-04-16).[32] Esa Järvenpää, Maarit Järvenpää, Kevin Rogovin, Sari Rogovin, and Nageswari Shanmugalingam. Measurability of equivalence classes and
MEC p -property in metric spaces. Rev. Mat. Iberoam. , 23(3):811–830, 2007.[33] S. Keith. Modulus and the Poincaré inequality on metric measure spaces.
Math. Z. , 245(2):255–292, 2003.[34] S. Keith. A differentiable structure for metric measure spaces.
Adv. Math. , 183(2):271–315, 2004.[35] S. Keith and T. Laakso. Conformal Assouad dimension and modulus.
Geom. Funct. Anal. , 14(6):1278–1321, 2004.[36] Kyle Kinneberg. Conformal dimension and boundaries of planar domains.
Trans. Amer. Math. Soc. , 369(9):6511–6536, 2017.[37] Bernd Kirchheim. Rectifiable metric spaces: local structure and regularity of the Hausdorff measure.
Proc. Amer. Math. Soc. , 121(1):113–123,1994.[38] T. Laakso. Ahlfors Q -regular spaces with arbitrary Q > admitting weak Poincaré inequality. Geom. Funct. Anal. , 10(1):111–123, 2000.[39] Enrico Le Donne. Metric spaces with unique tangents.
Ann. Acad. Sci. Fenn. Math. , 36(2):683–694, 2011.[40] Enrico Le Donne. A primer on Carnot groups: homogenous groups, Carnot-Carathéodory spaces, and regularity of their isometries.
Anal.Geom. Metr. Spaces , 5(1):116–137, 2017.[41] John M. Mackay and Jeremy T. Tyson.
Conformal dimension , volume 54 of
University Lecture Series . American Mathematical Society,Providence, RI, 2010. Theory and application.[42] P. Mattila. Measures with unique tangent measures in metric groups.
Math. Scand. , 97(2):298–308, 2005.[43] Pertti Mattila.
Geometry of sets and measures in Euclidean spaces , volume 44 of
Cambridge Studies in Advanced Mathematics . CambridgeUniversity Press, Cambridge, 1995. Fractals and rectifiability.[44] Sergei Merenkov. A Sierpi´nski carpet with the co-Hopfian property.
Invent. Math. , 180(2):361–388, 2010.[45] J Nagata.
Modern dimension theory , volume 2 of
Sigma Series in Pure Mathematics . Heldermann Verlag, Berlin, revised edition, 1983.[46] Mikhail Ostrovskii. Radon-Nikodým property and thick families of geodesics.
J. Math. Anal. Appl. , 409(2):906–910, 2014.[47] Mikhail I. Ostrovskii. On metric characterizations of the Radon-Nikodým and related properties of Banach spaces.
J. Topol. Anal. , 6(3):441–464, 2014.[48] Pierre Pansu. Dimension conforme et sphère à l’infini des variétés à courbure négative.
Ann. Acad. Sci. Fenn. Ser. A I Math. , 14(2):177–212,1989.[49] Pierre Pansu. Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un.
Ann. of Math. (2) , 129(1):1–60, 1989.[50] D. Preiss. Geometry of measures in R n : distribution, rectifiability, and densities. Ann. of Math. (2) , 125(3):537–643, 1987.[51] W. Rudin.
Function theory in the unit ball of C n . Classics in Mathematics. Springer-Verlag, Berlin, 2008. Reprint of the 1980 edition.[52] A. Schioppa. Derivations and Alberti representations. Adv. Math. , 293:436–528, 2016.[53] Stephen Semmes. On the nonexistence of bi-Lipschitz parameterizations and geometric problems about A ∞ -weights. Rev. Mat. Iberoameri-cana , 12(2):337–410, 1996.[54] M. Sion. On general minimax theorems.
Pacific Journal of Mathematics , 8(1):171–176, 1958.[55] Jeremy T. Tyson. Sets of minimal Hausdorff dimension for quasiconformal maps.
Proc. Amer. Math. Soc. , 128(11):3361–3367, 2000.[56] William C. Waterhouse. Uniform convergence and graph convergence.
The American Mathematical Monthly , 83(8):641–643, 1976.[57] N. Weaver. Lipschitz algebras and derivations. II. Exterior differentiation.
J. Funct. Anal. , 178(1):64–112, 2000.[58] W.P. Ziemer. Extremal length and p-capacity.
Michigan Math. J. , 16:43–51, 1969.
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