Characterizing spaces satisfying Poincaré Inequalities and applications to differentiability
We characterize complete RNP-differentiability spaces as those spaces which are rectifiable in terms of doubling metric measure spaces satisfying some local -Poincaré inequalities. This gives a full characterization of spaces admitting a strong form of a differentiability structure in the sense of Cheeger, and provides a partial converse to his theorem. The proof is based on a new “thickening” construction, which can be used to enlarge subsets into spaces admitting Poincaré inequalities. We also introduce a new notion of quantitative connectivity which characterizes spaces satisfying local Poincaré inequalities. This characterization is of independent interest, and has several applications separate from differentiability spaces. We resolve a question of Tapio Rajala on the existence of Poincaré inequalities for the class of -spaces which satisfy a weak Ricci-bound. We show that deforming a geodesic metric measure space by Muckenhoupt weights preserves the property of possessing a Poincaré inequality. Finally, the new condition allows us to show that many classes of weak, Orlicz and non-homogeneous Poincaré inequalities “self-improve” to classical -Poincaré inequalities for some , which is related to Keith’s and Zhong’s theorem on self-improvement of Poincaré inequalities.
This paper focuses on the problem of characterizing geometrically two analytic conditions on a metric space: the existence of a certain measurable differentiable structure and the property of possessing a Poincaré inequality. Since they were introduced, many fundamental questions about their relationships and geometric nature have remained open.
The concepts of differentiability and Poincaré inequalities involve somewhat different terminology, history and techniques. Our theorems relate to both of these and some independently interesting applications. Thus, in order to facilitate readability, we first give a general overview of our main results, followed by some more detailed discussion on the individual topics.
Poincaré inequalities for metric spaces were introduced by Heinonen and Koskela in , and have been central tools in the study of such concepts as Sobolev spaces and quasiconformal maps in metric spaces. Spaces satisfying such inequalities, and which are measure doubling, are called PI-spaces. For precise definitions see Definitions 2.13 and 2.11. While characterizations of a subclass of PI-spaces appeared in  and , and a general characterization in , it remained a question to find constructions of these spaces and flexible ways of proving these inequalities in particular contexts. For example, see the detailed discussion in .
Measurable differentiable structures, on the other hand, were introduced by Cheeger in . His main theorem showed that a PI-space possesses a measurable differentiable structure, which permits differentiation of Lipschitz functions almost everywhere. The spaces satisfying such a theorem, without the PI-space assumption, are called differentiability spaces (or Lipschitz differentiability spaces ). See below Definitions 5.2 and 5.3.
An early question was if the assumptions of Cheeger were in some sense necessary . Because positive measure subsets of PI-spaces may be totally disconnected while they remain differentiability spaces , strictly speaking, a Poincaré inequality cannot be necessary. However, it remained a question if a PI-space structure could be recovered in a weaker form such as by taking tangents or by covering the space in some form. This question was related to better understanding the local geometry of PI-spaces. Various authors, such as Cheeger, Heinonen, Kleiner and Schioppa, posed similar questions. A strong form of this question appeared in , where it was asked if every differentiability space is PI-rectifiable. Call a metric space PI-rectifiable if it can be covered up to measure zero by a countable number of subsets of PI-spaces. See Definition 5.1.
Our main result fully resolves the PI-rectifiability question for a subclass of RNP-differentiability spaces. Conversely, together with a result of Cheeger and Kleiner , it fully characterizes PI-rectifiable spaces.
A complete metric measure space is a RNP-Lipschitz differentiability space if and only if it is PI-rectifiable and every porous set has measure zero. 111The assumption of porous sets is somewhat technical and is similar to the discussion in .
RNP-differentiability is a priori a stronger assumption than Lipschitz differentiability. In a RNP-differentiability space one can differentiate all Lipschitz functions with values in RNP-Banach spaces, instead of just ones with finite dimensional targets. These spaces were introduced in the pioneering work of Bate and Li , where they showed that such differentiability spaces satisfied certain asymptotic and non-homogeneous versions of Poincaré inequalities. Also, an earlier paper by Cheeger and Kleiner  showed that every PI-space is a RNP-differentiability space.
Recent examples by Andrea Schioppa in  show that, indeed, there exist differentiability spaces which are not RNP-differentiability spaces. Or equivalently (by our result), they are not PI-rectifiable. Schioppa’s and our work together demonstrate that differentiability of Lipschitz functions depends on the target, and that a sufficiently strong assumption of differentiability is equivalent to possessing a Poincaré inequality in some sense. This work exposes an interesting problem of understanding how this dependence on the target is related to the local geometry of the space.
The proof of Theorem 1.1 rests on two contributions that are of independent interest. Our starting point is the work of Bate and Li , where they identified a decomposition of the space into pieces with asymptotic non-homogeneous Poincaré inequalities. For us, it is more important that these subsets satisfy a quantitative connectivity condition. In order to prove PI-rectifiability, we need to be able to enlarge, or “thicken” these possibly totally disconnected subsets into spaces with better connectivity properties. See Theorem 1.15 below for a more detailed discussion.
Once this enlarged space is constructed, one needs to verify that it satisfies a Poincaré inequality. This required a way of identifying a quantitative connectivity condition that is easier to establish for the space, and showing that this connectivity condition is equivalent to a Poincaré inequality. In other words, we needed a new and weaker characterization of PoincarÂ´e inequalities. We do this by introducing (in Definition 3.1) a novel condition on a metric measure space that we call –connectivity. An interesting feature of this condition is that it is formally very similar to the definition of Muckenhoupt weights, and thus our methods draw a close formal similarity between the theories of Poincaré inequalities and the theory of Muckenhoupt weights (see discussion following Definition 3.1). Some analogies between Poincaré inequalities and Muckenhoupt weights have already been observed in relation to self-improvement phenomenons by Keith and Zhong .
In terms of this condition, we show the following.
A -doubling complete metric measure space admits a local -Poincaré inequality for some if and only if it is locally –connected for some . Both directions of the theorem are quantitative in the respective parameters.
The variable in the above theorem is the exponent in the Poincaré inequality, which measures the quality of the inequality. A larger means worse connectivity. Notable from our perspective is that the characterization is for a -Poincaré inequality for some , and that our characterization applies for any doubling metric measure space. Previous characterizations either assumed Ahlfors regularity , or presumed knowledge of the exponent , such as in  and . As demonstrated by examples of Schioppa , it is possible for this exponent to be arbitrarily large. Thus, applying characterizations from  seem difficult in some cases where the exponent is a priori unknown. Further, our formalism avoids direct usage of modulus estimates, and seems easier to apply in our context.
The new characterization of Poincaré inequalities has several applications of independent interests. The first answers affirmatively a question of Tapio Rajala on the existence of Poincaré inequalities on certain metric measure spaces with weak synthetic Ricci curvature bounds. These spaces, called -spaces, were introduced by Ohta in . We show that, at least for a large enough exponent, these spaces satisfy a Poincaré inequality. One expects that the exponent could be chosen to be smaller.
If is a space, then it satisfies a local -Poincaré-inequality for . Further, if , then it satisfies a global -Poincaré inequality.
We next consider more general self-improvement phenomena for Poincaré inequalities. In a celebrated paper, Keith and Zhong proved that in a doubling complete metric measure space a -Poincaré-inequality, with , immediately improves to a -Poincaré inequality  for some depending on the constants in the doubling and Poincaré inequality. We ask if somewhat similarly more general Poincaré-type inequalities, say Orlicz-Poincaré inequalities, also imply some -Poincaré inequalities.
By a Poincaré type-inequality we will refer to inequalities that control the oscillation of a function by some, possibly non-linear, functional of its gradient. Such inequalities have appeared in the work of Semmes , Bate and Li , Feng-Yu Wang , Tuominen , Heikkinen [39, 40] and Jana Björn in . We will here show that on quasiconvex doubling metric measure spaces many types of “weak” Poincaré inequalities imply a Poincaré inequality. In particular, most of the definitions of Poincaré inequalities produce the same category of PI-spaces.
Suppose that is a -doubling metric measure space and satisfies a local non-homogeneous -Poincaré-inequality. Then the space is locally –connected and moreover admits a -Poincaré inequality for some and some . All the variables are quantitative in the parameters.
In particular, the non-homogeneous Poincaré inequalities considered by Bate and Li in  improve to -Poincaré inequalities for some . The terminology used in this theorem is defined in section 1.3. We also obtain the following result strengthening the conclusion of Dejarnette  and Tuominen [72, Theorem 5.7].
Suppose is a doubling metric measure space satisfying a strong -Orlicz-Poincaré inequality in the sense of , then it satisfies also a -Poincaré inequality for some .
Remark: We remark, that in some senses this result is weaker then  and , because we do not control effectively the range of exponents . Further, we remark that in  two different types of Poincaré inequalities are considered, a strong and a weak one. Since we do not want to distract the reader with a discussion of Luxembourg norms, we remark that the proof and statement also holds with minor modifications for the weak -Orlicz-Poincaré inequalities defined in . The strong Orlicz-Poincaré inequality coincides with the ones considered by Heikkinen and Tuominen in, for example, [39, 40, 72, 73]. The inequalities of Feng-Yu Wang  are of a different nature, since they are in fact stronger than regular -Poincaré inequalities. His inequalities are more related to Orlicz versions of Sobolev-Poincaré inequalities.
Most notably, the property of being a PI-space can be recovered even if the function decays arbitrarily fast at the origin. The exponent of the obtained Poincaré inequality in Theorem 1.5 will grow in such cases, but one cannot fully lose a Poincaré inequality by such examples.
Finally, we will show a theorem concerning -weights on metric measure spaces. These weights can vanish and blow-up on “small” subsets of the space, and thus allow flexibility in obtaining weighted Poincaré inequalities. This generalizes some aspects from  concerning sub-Riemannian metrics and vector fields satisfying the Hörmander condition. For the definition, see Section 4 or Definition 1.13.
Let be a geodesic -measure doubling metric measure space. If satisfies a -Poincaré inequality and , then there is some such that -satisfies a -Poincaré inequality.
In the following subsections, we present the above theorems in more detail and explain some historical connections and proof techniques. A reader who is only interested in the characterization of Poincaré inequalities and its applications could only read subsections 1.2,1.3. On the other hand, a reader mostly interested in the PI-rectifiability of RNP-differentiability spaces could simply read subsection 1.4.
All the results in this paper will be stated for complete separable metric measure spaces equipped with a Radon measure with for all balls . Most of the results could also be modified to apply to non-complete spaces.
1.2. Characterizing spaces with Poincaré inequalities
Poincaré inequalities and doubling measures are useful tools in analysis of differentiable manifolds and metric spaces alike (see Section 2 for the definitions). Following the convention established by Cheeger and Kleiner, a space with both a doubling measure and a Poincaré inequality is referred to as a PI-space or -Poincaré space (see Definition 2.13). For a general overview of these spaces and some later developments, see the beautiful expository article by Heinonen  and the book .
Due to the many desirable structural properties that PI-spaces have, much effort has been expended to understand what geometric properties guarantee a Poincaré inequality in some form. By now, several classes of spaces with Poincaré inequalities are known. See for example [53, 55, 20, 45, 60, 17, 24, 63, 64, 48]. The proofs that these examples satisfy Poincaré inequalities are often challenging, and make extensive use of the geometry of the underlying space. Our first theorem is thus motivated by the following question, which has also been posed in .
Which geometric properties characterize PI-spaces? Find new and weak conditions on a space that guarantee a Poincaré inequality.
As an answer to the above question, we introduce a new connectivity, or avoidance, property for a general metric measure space. This condition involves three parameters and is referred to as –connectivity. The condition is somewhat technical to state, and we defer to Section 3 and Definition 3.1 to state it rigorously. In the vaguest sense, it means that for each pair of points and every -low density “obstacle” set , there exists a “curve” of length making “jumps” of cumulative size at most while avoiding the obstacle set .
It turns out that Definition 3.1 fully characterizes doubling spaces admitting Poincaré inequalities. The rigorous statement is contained above in Theorem 1.2. Note, the doubling is not needed to show that –connectivity is sufficient for a Poincaré inequality, because by Lemma 3.4 doubling is implied by this condition. Similarly, the theorems below could dispense with the assumption of -doubling. However, we add this to explicate the dependence of the parameters on the implied doubling constant. Note that Lemma 3.4 gives a fairly bad bound for the doubling, and often the doubling constant may be much smaller.
The rigorous definition of connectivity uses curve fragments instead of curves. The usage of curve fragments may seem counter-intuitive and technical at first. However, as a motivation we note that it becomes easier for a curve to avoid an obstacle if we permit some jumps, or discontinuities. However, a conceptual difficulty arises as we have to allow infinitely many jumps, which leads to the notion of so called curve fragments. These curve fragments are defined on compact subsets of instead of intervals. To simplify, the reader might initially imagine a curve which is continuous except for finitely many jumps. Such curves with jumps in relation to Poincaré inequalities have appeared implicitly in prior work, such as in [43, 14, 19, 30].
The reason we also need curve fragments stems from the fact that differentiability spaces may be totally disconnected, and thus may only possess such curve fragments with infinitely many jumps. Further, Bate has shown in  that differentiability spaces possess a rich supply of curve fragments. Moreover, Bate and Li also used curve fragments to express connectivity in . We also remark that, if is quasiconvex, one can avoid the use of curve fragments in Theorem 1.2. This leads to a definition in terms of curves, which is discussed in .
Our connectivity condition is motivated by a similar condition appearing in the work of Bate and Li [8, Lemma 3.5]. The main difference is that Bate and Li treat this property only in connection to certain classes of differentiability spaces, while here we isolate it as a property of a general metric measure space (see below for more discussion). Another related condition appears in connection to the -Poincaré inequality discussed in [30, Theorem 3.1(f)], but there one must assume .
Theorem 1.2 fully characterizes PI-spaces. Other similar characterizations of spaces with Poincaré inequalities appear in the work of Keith  and (for some ranges of exponents and with assumption of homogeneity or Ahlfors regularity) in [43, 30], but there the characterization is for a fixed , and in terms of modulus estimates. In some cases, one might be interested in Poincaré inequalities without knowing a priori the value of sought, or without efficient control on the homogeneity of the space. Also, in general might be arbitrarily large . In such contexts, our characterization seems easier to apply. In particular, we give applications of our characterization below, which prove Poincaré inequalities in new contexts.
Next we outline how the connectivity condition implies a Poincaré inequality, the other direction being of a different nature and following from Theorem 1.4. The proof is crucially based on a general idea of iteration, where connectivity estimates are iterated via maximal-function type estimates to give stronger connectivity properties and ultimately the Poincaré inequality. Below, we use this iteration to first prove quasiconvexity in Proposition 3.6.
Following quasiconvexity, we use the same iteration scheme to obtain a finer notion of connectivity, which we call fine -connectivity. This is done in order to simplify the proof, and in order to quantify more effectively the exponent appearing in the Poincaré inequality. It should be compared to the relationship between -weights and -weights in classical analysis. The main content is that it allows for controlling the size of the jumps in curve fragments polynomially in terms of the density parameter for the obstacle . More precisely, if an obstacle is of relative size , then the total size of the gaps can be improved to . In particular, instead of the jumps being bounded by , we force them to decay with the size of the obstacle in a quantitative way. The precise definition is presented in Definition 3.5. In terms of this notion, the theorem reads as follows.
Assume and . If is a –connected -doubling metric measure space, then there exist and such that the space is also finely - connected with parameters . We can choose , and , where .
Remark: For example, note that is finely -connected. This means that any set of relative measure in the unit ball can be avoided as long as we permit jumps of total size of the order . The tightness of this can be seen by choosing an obstacle set E which is a ball centered at a point. Similarly, the space arising from gluing two copies of along the origin is finely -connected. If the Lebesgue measure on is deformed by an -Muckenhoupt weight , then the space is -connected. This follows from the results in [69, Chapter V].
We further remark, that while the connectivity conditions are very similar to Muckenhoupt conditions, a major difference holds. Both conditions admit a “selfimprovement” property of the form in Theorem 1.8. This can be thought of as the statement that -weights are -weights for some . However, Muckenhoupt weights and Poincaré inequalities also possess a different type of self-improvement. An -weight automatically belongs to for some , and a -Poincaré inequality on a doubling metric measure space improves to a -Poincaré inequality, when (see ). However, a finely -connected space may fail to be finely -connected for any . For example, is finely -connected but not finely -connected for any . Thus, fine connectivity is not an open condition.
Finally, the last step in showing that Definition 3.1 implies a Poincaré inequality is to iterate the estimate given by Theorem 1.8 to construct curves along which a function has small integral. This involves a summability condition for a geometric series, which leads to the restriction . The quantitative version of the aforementioned is the following.
Let be a locally -measure doubling locally finely -connected metric measure space with parameters . Then, for any the space satisfies a local -Poincaré inequality at scale with constants . In short, the space is a PI-space.
We can set , , and .
The range of ’s in this theorem is tight in general. Take the space arising as the gluing of two copies of through their origins. The resulting space , where is the glued metric and the sum of the measures on each component, is finely -connected and satisfies a -Poincaré inequality only for . On the other hand, for some particular examples, such as or Ricci-bounded manifolds , we know that the space satisfies a -Poincaré inequality, but are only finely -connected. Also, if this result is applied to Muckenhoupt weights , Theorem 1.9 would give a Poincaré inequality for , while a Poincaré inequality actually holds for the larger range (see e.g. [69, Chapter V] or ). Finally, we remark that this theorem is not an equivalence. A -PI-space might not be -connected for any .
The iteration scheme used to prove quasiconvexity (Proposition 3.6), improvement of connectivity (Theorem 1.8) and Poincaré inequalities (Theorem 1.9) all are based on an iterated gap-filling and limiting argument. Recall, that the curve fragments guaranteed by Definitions 3.1 and 3.5 can contain gaps, or “jumps”. However, the core idea is to re-use the connectivity condition at the scale of these gaps and to replace them with finer curve fragments. A similar argument appears in [8, Lemma 3.6], and earlier in [43, Proof of Theorem 5.17]. In order to prove quasiconvexity, we can always set the relevant obstacle set to be . However, for the application to Theorems 1.8 and 1.9, we will need to define obstacles at different scales. To do this successfully, we use a maximal function estimate. Such iterative arguments employing maximal functions resemble the proofs of Theorems V.3.1.4 and V.5.1.3 in .
Finally, while our connectivity condition and conclusion do not use the modulus estimates of Keith  and Heinonen-Koskela , it is not surprising that at the end we are able to connect our condition to a certain type of modulus estimate. We refer to Theorem 3.20 below and the discussion preceding it for the precise statement, as it is not relevant for most of our discussion. For similar modulus bounds in other contexts see . In fact, the techniques of this paper are generally useful for obtaining modulus estimates for certain curve families.
1.3. Applications of the Characterization
Our Therem 1.2 can be used in a variety of contexts to establish Poincaré inequalities under a priori weaker connectivity properties.
The first result concerns metric spaces with weak Ricci bounds, where Theorem 1.3 states that any MCP-space admits some Poincaré inequality. These spaces originally arose following the work of Cheeger and Colding on Ricci limits [20, 21]. Many different definitions appeared such as different definitions for [70, 71, 54],  and a strengthening [3, 1]. Ohta also defined a very weak form of a Ricci bound by the measure contraction property and introduced -spaces . Spaces satisfying one of the stronger conditions (, or ) were all known to admit -Poincaré inequalities . It was also known that a non-branching -space would admit a -Poincaré inequality. This was observed by Renesse , whose proof was essentially a repetition of classical arguments in . However, Rajala had conjectured that this assumption of non-branching was inessential. spaces are interesting partly because they are known to include some very non-Euclidean geometries such as the Heisenberg group and certain Carnot-groups [46, 61].
We are left with the following open problem.
Open Question: Does every -space admit a local -Poincaré inequality?
We next consider self-improvement phenomena for Poincaré inequalities, where we have Theorem 1.4 stating that a large family of weak Poincaré inequalities “self-improve”.
By a Poincaré-type inequality we will refer to inequalities that control the oscillation of a function by some, possibly non-linear, functional of its gradient. First, recall the definition of an upper gradient for metric measure spaces by Heinonen and Koskela .
Let be a metric measure space and a Lipschitz function. We call a non-negative Borel-measurable an upper gradient for if for every rectifiable curve we have
With this definition, we can define a non-homogeneous Poincaré inequality.
Let be a metric measure space. Let be increasing functions with the following properties.
We say that satisfies a non-homogeneous -Poincaré inequality if for every -Lipschitz222The constant is only used to simplify arguments below. Any fixed bound could be used. Also, by replacing the right hand side with the local Luxembourg norm we could simply assume that is Lipschitz. This would lead to the notion of “weak” Orlicz-Poincaré inequality considered in . The proof for Theorem 1.4 works just as well for these inequalities, and thus the Poincaré inequalities with Luxembourg norms on the right hand side also imply -Poincaré inequalities for . For simplicity, we omit this detail. Using Luxembourg norms may be somewhat more natural due to scaling invariance, which does not hold for Definition 1.11. However, these inequalities have not been studied or used in other contexts than . function , every upper gradient and every ball with we have
This class of Poincaré inequalities subsumes the ones of Heikkinen and Tuominen in [72, 73, 40], the ones considered by Björn in  (which include the ones of Heikkinen and Tuominen), and the Non-homogeneous Poincaré Inequalities (NPI) considered by Bate and Li in . In both of these classes, without any substantial additional assumptions we obtain -Poincaré inequalities for some finite through Theorem 1.4. As already discussed in sub-section 1.1, a corollary gives direct results for Orlicz-Poincaré inequalities.
The proof of Theorem 1.4 is based on defining a function , which is, roughly speaking, the smallest size of gaps along a curve fragment connecting a pair of points and avoiding a certain set. If the space doesn’t have many curve fragments, then this functional oscillates a lot. However, its gradient is concentrated on the a small subset of the space, which makes the right hand side of inequality (1.12) small, and forces the connectivity property to hold with some parameters.
The final application concerns weighted metric measure spaces. We observed above a formal similarity between Definition 3.1 and the definition of Muckenhoupt weights. For our purposes we define Muckenhoupt weights as follows.
Let be a -measure doubling metric measure space. We say that a Radon measure is a generalized -measure, or , if , and there exist such that for any and any Borel-set
If for some locally integrable , then we call a Muckenhoupt-weight.
There are different definitions in the literature. These variants and their equivalence is discussed in . For geodesic metric measure spaces many of them are equivalent. Also, there is the class of strong -weights introduced by David and Semmes in  (see also [67, 65]). These weights are somewhat different from Muckenhoupt weights, and usually form a sub-class of them .
For an unfamiliar reader, we remind that Muckenhoupt weights may vanish and blow-up on subsets of the space. See [69, Section V] for examples. Such weights are also somewhat flexible to construct, as alluded to in [34, Section 3.18]. Interestingly enough, Theorem 1.6 shows that deformations by such weights preserve on geodesic spaces the property of possessing a Poincaré inequality.
1.4. Relationship between differentiability spaces and PI-spaces
Cheeger  defined a metric measure analog of a differentiable structure and proved a powerful generalization of Rademacher’s theorem for PI-spaces. The spaces admitting differentiation are here called differentiability spaces. See below Definition 5.2. Our main result, Theorem 1.1, is that a subclass of these spaces, called RNP-differentiability spaces, are PI-rectifiable (see 5.1 for a definition). Thus, within this subclass the conditions of Cheeger are both necessary and sufficient.
This result builds on earlier work by Bate and Li , where the authors noticed that RNP-differentiability spaces satisfy certain asymptotic and non-homogeneous Poincaré inequalities. Even earlier, a number of similarities between PI-spaces and Poincaré inequalities were discovered: asymptotic doubling , an asymptotic lip-Lip equality almost everywhere [6, 62], large family of curve fragments representing the measure and “lines” in the tangents . These works developed ideas and gave strong support for some result like Theorem 1.1, which however requires a number of new techniques, such as Theorem 1.2 and Theorem 1.15.
The work of Bate and Li initiated the detailed study of RNP-differentiability spaces, which permit differentiation of Lipschitz functions with values in RNP-Banach spaces (Radon Nikodym Property). The definition is contained below in Section 5.3. For more information on RNP-Banach spaces see . The question of differentiability of RNP-Banach space valued Lipschitz functions had already arisen earlier in the work of Cheeger and Kleiner where it was shown that a PI-space admits a Rademacher theorem for such functions .
Let be a RNP-differentiability space. Then can be covered, up to measure zero, by countably many positive measure subsets , such that each is metric doubling, when equipped with its restricted distance, and for -a.e. each space admits a -Poincaré-inequality for some . 333The subsets are equipped with the restricted measure and metric.
Here, denotes the set of measured Gromov-Hausdorff tangents at for the space . See [37, 48] for the definition of pointed measured Gromov-Hausdorff convergence, and [19, 25] for the definition of tangent spaces.
We give a few remarks on the proof techniques. Our proof of Theorem 1.1 starts off with citing the result of Bate and Li that decomposes a RNP-differentiability space into parts with asymptotic and non-homogeneous forms of Poincaré inequalities, as well as a uniform and asymptotic form of Definition 3.1. We introduce two new ideas to use these pieces. On the one hand, we observe that our connectivity condition implies a Poincaré inequality using Theorem 1.2, and this already gives a Poincaré inequality for tangents of RNP-differentiability spaces (see the appendix A). To obtain PI-rectifiability we also need the ability to enlarge the pieces used by Bate and Li to satisfy, intrinsically, the connectivity property in 3.1.
The following theorem is used to construct the enlarged space. The terminology used is defined later in Sections 5 and 2. A crucial observation is that the assumptions involve a relative form of doubling and connectivity, and in the conclusion we construct a space with an intrinsic Poincaré inequality and an intrinsic doubling property. Thus, it can be used even for general subsets of PI-spaces to enlarge them to PI-spaces (the enlarged space being different from the original space).
Let be arbitrary. Assume is a metric measure space and subsets are given, where is measurable and is compact. Assume further that is -doubling along , is uniformly -dense in along , and with the restricted measure and distance is locally –connected along . There exist constants , and a complete metric space which is locally -doubling and –connected, and an isometry which preserves the measure. In particular, the resulting metric measure space is a PI-space.
For an intuitive, and slightly imprecise, overview of the proof of this construction one can consider the case of a compact subset . Here is well-connected when thought of as a subset of , but may not be intrinsically connected. To satisfy a local Poincaré-inequality the space must be locally quasiconvex. In order to make locally quasiconvex we will glue a metric space to it. The space is tree-like, and it’s vertices correspond to a discrete approximation of along with its neighborhood. The vertices exist at different scales, and near-by vertices are attached by edges to each other at comparable scales. By using net-points of , and Whitney centers for a neighborhood of , we prevent adding too many points or edges at any given scale or location. This construction is analogous to that of a hyperbolic filling [18, Section 2] (see also a more recent presentation in English ). Very similar ideas also appear in the work of Bonk, Bourdon and Kleiner related to problems of quasiconformal maps [14, 13, 16].
1.5. Structure of paper
We first cover some general terminology and frequently used lemmas in section 2. In section 3, we introduce our notion of connectivity and prove basic properties and finally derive Poincaré inequalities. In section 4, we apply the results in both new and classical settings. Finally, in section 5 we apply the results to the study of RNP-differentiability spaces and introduce the relevant concepts. In the appendix, we include a different proof that tangents of RNP-differentiability spaces are PI-spaces and that our connectivity condition is preserved under measured Gromov-Hausdorff-convergence.
As the paper consists of two related, but somewhat independent ideas, one concerning characterizing differentiability spaces and the other concerning characterization of Poincaré inequalities, we have tried to separate these in the structure of the paper. A reader interested only in the classification of spaces with Poincaré inequalities could read section 2 followed by the main proofs in section 3.
A reader interested merely in the application could read section 2 and Definition 3.1 after which one can read section 4 without needing that much from other parts of the paper. However, the proof of Theorem 1.4 is closely related to Theorem 1.2, and thus is included in section 3.
Finally, a reader who is simply interested in the PI-rectifiability of RNP-differentiability spaces and the involved “thickening construction” can move after section 2 directly to section 5, which is mostly self-contained except for references to Theorem 1.2.
Acknowledgments: The author is thankful to professor Bruce Kleiner for suggesting the problem on the local geometry of Lipschitz differentiability spaces, for numerous helpful discussions on the topic and several comments that improved the exposition of this paper. Kleiner was instrumental in restructuring the proofs in the third and fifth sections and thus helped greatly simplify the presentation.
The author also thanks a number of people who have given useful comments in the process of writing this paper, such as Sirkka-Liisa Eriksson, Jana Björn, Nagesvari Shanmugalingam, Pekka Koskela, Guy C. David and Ranaan Schul. Some results of the paper were heavily influenced by conversations with Tatiana Toro and Jeff Cheeger. An earlier draft of this paper had a more complicated construction used to resolve Theorem 1.15. This construction is here rephrased in terms of a modified hyperbolic filling which is much clearer than the earlier version. This modification was encouraged by Bruce Kleiner, and suggested to the author by Daniel Meyer. We also thank the anonymous referees for numerous comments and corrections. The research was supported by a NSF graduate student fellowship DGE-1342536 and NSF grant DMS-1405899.
2. Notational conventions and preliminary results
We will be studying the geometry of complete and separable metric measure spaces . Where not explicitly stated, all the measures considered in this paper will be Radon measures. An open ball in a metric space with center and radius will be denoted by , and for we will denote by . Throughout we will assume that for every ball .
A metric measure space , such that for all balls , is said to be (locally) -doubling if for all and any we have
We say that the space is -doubling if this property holds for every . Further, we simply call a space doubling if there is a constant such that it is -doubling, and locally doubling if there are constants such that it is locally -doubling.
There is also a metric notion of doubling that does not refer to the measure. Our definition is often referred to as measure doubling. However, throughout this paper, except briefly in relation to Theorem 1.14, we will only use this stronger version, and thus simply say doubling.
A metric measure space is said to be asymptotically doubling if for almost every we have
We also define a relative version of doubling for subsets.
A metric measure space is said to be -doubling along if for all and any we have
A set is called porous if there exist constants such that for every and every there exists a such that . A set is called -porous if there exist countably many porous sets (with possibly different constants ), such that
Let be a metric measure space, and a positive measure subset. A point is called an -density point of if for any we have
We say that is uniformly -dense along if every point is an -density point of .
A map between two metric spaces and is Lipschitz if there is a constant such that . We will denote by the optimal constant in this inequality and call it the global Lipschitz constant of a Lipschitz function. Further, for real-valued we define the two local Lipschitz constants
A map is called bi-Lipschitz if there is a constant such that . The smallest such constant is called the distortion of .
A continuous embedding is said to preserve measure, if is measurable and . Further, the push-forward of a measure is defined by .
Let , be constants. We say that a complete metric measure space equipped with a Radon measure satisfies a -Poincaré inequality with constants if for every , every and every Lipschitz function .
Additionally, we say that the metric measure space satisfies a local (–)Poincaré inequality at scale if the aforementioned property holds but only for all . A space satisfies a local -Poincaré inequality if the aforementioned holds for some .
Remark: There are different versions of Poincaré inequalities and their equivalence in various contexts has been studied in  and [38, 41]. The quantity on the right-hand side could also be replaces by (see equations (2.10) and (2.9) above), and on complete spaces by an upper gradient (in any of the senses discussed in [19, 43, 29]). For non-complete spaces, which we do not focus on, the issue is slightly more delicate (see counterexamples in ), but can often be avoided by taking completions. Further, as long as the space is complete, we do not need to constrain the inequality for Lipschitz functions but could also use appropriately defined Sobolev spaces.
Note that on the right-hand side the ball is enlarged by a factor . Some authors call inequalities with “weak” Poincaré inequalities, but we do not distinguish between these different terms. On geodesic metric spaces the inequality can be improved to have .
We use the following notion of PI-space.
A complete metric measure space equipped with a Radon measure is called a -PI space at scale with doubling constant and Poincaré constants if it is -doubling and satisfies a local -Poincaré inequality at scale with constants . Further, a space is called a -PI space or simply a PI-space if there exist remaining constants so that the space satisfies the aforementioned property.
A curve fragment in a metric space is a Lipschitz map , where is compact. We say the curve fragment connects points and if . Further, define . If is an interval we simply call a curve.
As default, and to simplify notation below, we will assume the curve fragment has been translated so that unless otherwise stated.
Frequently, we will observe that the open set can be expressed as a countable union of maximal disjoint open intervals as
We will employ this notation, up to some necessary subscripts, with only brief comments below. These intervals are also called “gaps” or “jumps”. We also define a measure of the size of these jumps
The length of a curve fragment is defined as
Since is assumed to be Lipschitz we have
Analogous to curves we can define an integral over a curve fragment . Denote . The function is Lipschitz on . Thus, it is differentiable for almost every density point and for such we set and call it the metric derivative (see  and ). We define an integral of a Borel function as follows
when the right-hand side makes sense. This is true for example if is bounded from below or above. Naturally, if has Lebesgue measure zero, then the integral vanishes and is useless. Thus, usually we are primarily interested in curve fragments with domains of positive measure.
We will need to take limits of curve fragments, and in order to do so we present two auxiliary lemmas on reparametrization and compactness.
Let be a complete metric space, points and a curve fragment connecting to . There exists a compact , an increasing Lipschitz function and a -Lipschitz curve fragment defined by for . Moreover, this curve fragment satisfies the following properties.
Either or , for almost every .
For every non-negative Borel function
Proof: First, by replacing the image space by we can reduce to the case where is compact. Next, consider the isometric distance embedding , where is the Banach space of continuous functions equipped with the supremum norm on , and . Then, define the gaps of as , and define the piecewise linearly extended Lipchitz curve by for and
for . This curve can be shown to be Lipschitz, and .
Then, let be the length-reparametrization, and the length-reparametrization of . Here, we use [2, Lemma 1.1.4]. Define , and .
Clearly , and the properties of the metric derivative and invariance of curve integrals follow from [2, Lemma 1.1.4]. Finally,
Also, we have , since is linear on the interval . Then, as well from the definition of . This gives
Finally, since , we also have .
If is a curve fragment, denote by its graph. Then, we say that a sequence of curve fragments converges to a curve fragment if . Here is the Hausdorff metric for compact sets defined by
Fix . Let X be a complete metric space, a compact subset and let be a sequence of -Lipschitz curve fragments with , , and . There exists a subsequence converging to a Lipschitz curve fragment with
Proof: Assume by passing to a subsequence that
Since the collection of all compact subsets of forms a complete metric space under the Hausdorff metric , we can choose a subsequence of that converges in the Hausdorff metric to a set . Since are uniformly -Lipschitz, the set is a graph of a -Lipschitz function444If it wasn’t a graph of a -Lipschitz function, then there would exist a such that for distinct and with . But by the definition of Hausdorff convergence of sets, there would exist such that and . However, then , and , which would lead to a contradiction to the -Lipschitz property of for large enough . and can be expressed as , where is a -Lipschitz curve fragment defined on . Now, if , we can show that along the subsequence, and any finite collection of gaps, there is a sequence of maximal open intervals which converge to as . Thus, the estimate
is easy to obtain.
For a locally integrable function and a ball with we define the average of a function as
We will frequently use the Hardy-Littlewood maximal function at scale :
and the unrestricted maximal function
Refer to  for a standard proof of the following result. The norm of a function is denoted by . The possible ambiguity of the space or the measure is not relevant for us, as the measure will be evident from the context.
Let be a -measure doubling metric measure space and and arbitrary, then for any non-negative integrable function and we have
and if the space is -doubling, then (2.21) holds for all , and moreover we have
A metric space is called -quasiconvex if for every there exists a Lipschitz curve such that and . A metric space is locally -quasiconvex if the same holds for all with . A metric space is called geodesic if it is -quasiconvex.
Any -quasiconvex space is -bi-Lipschitz to a geodesic space by defining a new distance
Finally, we will need an elementary fact concerning averages.
If is a metric measure space and is a locally integrable function, and , then for any
Proof: The result follows by twice applying the triangle inequality.
3. Proving Poincaré inequalities
We define a notion of connectivity in terms of an avoidance property. For the definition of curve fragments see Definition 2.14.
Let555The definition is only interesting for . However, we allow it to be larger to simplify some arguments below. If then the curve fragment could consist of two points, that is and . Indeed, any pair of points in any metric space is –connected, when and is arbitrary. Since the length of a curve fragment connecting to is at least , the definition also must assume to be meaningful. , and be given. If is a pair of points with , then we say that the pair is –connected if for every Borel set such that there exists a Lipschitz curve fragment connecting and , such that the following hold.
We call a –connected space, if every pair of points with is -connected.
We say that is (uniformly) –connected along , if every with and with is -connected in .
If is –connected for all , we simply say that is –connected.
General remarks: The set above will often be referred to as an “obstacle”. Since we are working with Radon measures on proper metric spaces, to verify the condition we would only need to consider “test sets” which are either all compact, or just open. For open sets this is trivial, since the measure is outer regular and any Borel obstacle can be approximated on the outside by an open set . For compact sets, the argument goes via exhausting an open set by compact sets and obtaining a sequence of curve fragments.
For certain purposes we could also restrict to using curves , and replace the second and third condition by
However, while less intuitive to state with curve fragments, that language is necessary for the application to differentiability spaces in section 5, since the spaces we construct may be a priori disconnected. Also, it is often easier to construct curve fragments than curves, since they permits certain jumps. The version of this definition with curves is presented in .
Remark on similarity with Muckenhoupt-weights: Our motivation for using Definition 3.1 and for some details of the proofs below stem from the theory of Muckenhoupt weights. One way of seeing this formal similarity is the following analogy. Consider for simplicity the Lebesgue measure on . Then, as in Definition 1.13 we say that if there exist, such that for any and any Borel-set
Notice, that we have switched the roles of and here compared to the definition before. The definition above is somewhat similar, if we set for the moment . As before, we have that is –connected if for every Borel and any with ,
and connecting to .
In this case the scale-invariant condition is replaced by , and two other conditions as well as an existential quantifier is added. However, in the simple case of the one dimension space one directly sees that a -doubling is an -measure with constants