Any uniformly rectifiable set is contained in a set with Poincaré inequality

Introduction

In the late 1990s problems connected to quasiconformal and quasisymmetric maps led to the development of a Sobolev spaces theory on metric measure spaces (mms). In their foundational work [26], Heinonen and Koskela introduced the notion of upper gradient, meant to replace the norm of gradients of smooth functions. To define it, let $(X,\mu)$ be a mms. A nonnegative Borel function $g : X \to[0,\infty]$ is said to be an upper gradient of a real valued function $u$ if

M.H. is supported by the Research Council of Finland via the project Quantitative differentiability and rectifiability in metric spaces, grant no. 363800.

M.V. was supported by the European Union MSCA IF QuReViMal, grant no. 101108515, and by an IKERBASQUE starting grant. M.V. was also supported by the Academy of Finland via the project “Higher dimensional Analyst’s Traveling Salesman theorems and Dorronsoro estimates on non-smooth sets”, grant No. 347828.

I.Y.V. was supported by the European Union (ERC, Con-Fine, 101078057) and the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Pisa, CUP I57G22000700001. the inequality

is satisfied for all rectifiable curves $\gamma$ joining $x$ to $y$ in $X$. The path integral is computed with respect to arc length along $\gamma$. Coupling this notion with that of Poincaré inequality, Shanmugalingam developed a theory of what she termed Newton-Sobolev spaces [Sha00], denoted by $N^{1,p}(X)$ (see also [11]).

The importance of (1.1) for a first-order Sobolev theory lies in the fact that it ensures the presence of ‘thick’ families of rectifiable curves in $X$. When coupled with the doubling property of $\mu$, this ensures that upper gradients reflect appropriately local changes of the associated functions. Indeed, Cheeger proved that if $(X,\mu)$ supports a Poincaré inequality and is doubling, then it possesses an infinitesimal linear structure sufficiently rich to permit upper gradients of Lipschitz functions to be controlled by their point-wise Lipschitz constant. In other words, Cheeger proved in this setting a version of Rademacher’s differentiation theorem about Euclidean Lipschitz functions. Spaces $(X,\mu)$ which support this infinitesimal linear structure needed for Cheeger’s Rademacher’s result to apply were termed Lipschitz differentiability spaces. A converse of Cheeger’s theorem was proved in the remarkable works by Bate and Li, and Eriksson-Bique [10, 21], where they showed that any Lipschitz differentiability space (with the Radon-Nikodym Property) is PI-rectifiable – it can be ‘covered’ by a countable number of PI-spaces.

The property of supporting a Poincaré inequality is quantitative and multiscale. As such, one would expect it to have quantitative and multiscale consequences. Cheeger’s theorem, however, is qualitative: it says that Lipschitz functions on $X$ are infinitesimally linear in an appropriate generalised sense. In Euclidean setting, there is a quantitative version of Rademacher’s theorem stating that, in fact, Lipschitz functions are close to being affine at the typical location and scale – this is Dorronsoro’s theorem [18]. This begs the question: does an appropriate version of this theorem hold on PI spaces? And, conversely, do spaces with a Dorronsoro theorem also have PI?

To proceed, recall that a metric measure space $(X,\mu)$ is Ahlfors $d$-regular if $X=\operatorname{supp}(\mu)$ and $\mu$ is Ahlfors regular, meaning that there is $A\geq1$ such that

When $(X,\mu)$ is Ahlfors $d$-regular, and $X\subset\mathbb{R}^n$, we now know precisely the condition for which $(X,\mu)$ supports a Dorronsoro type theorem: uniform $d$-rectifiability is that condition. This is due to David and Semmes [20] and the proof of the WALA conjecture by Merlo [Mer26]. We recall the definition of uniform rectifiability (introduced in [19]) below.

Returning to the first question above, Azzam [5] proved that if $X \subset\mathbb{R}^n$ (equipped with the Euclidean metric) is Ahlfors $d$-regular and supports a $(1,d)$-Poincaré, then it is uniformly $d$-rectifiable. In this article, we prove an appropriate converse. Namely:

For Azzam, it was enough to assume a $(1,d)$-PI to guarantee uniform rectifiability. Since we construct a space which supports a $(1,1)$-PI, and since a $(1,p)$-PI is stronger than a $(1,q)$-PI whenever $p<q$, it follows that the space we construct is also $d$-UR. One observes also that Theorem 1.3 and Azzam’s result imply that any $d$-ADR $(1,d)$-PI space in $\mathbb{R}^n$ is a subset of a $d$-ADR $(1,1)$-PI space in $\mathbb{R}^{n+1}$.

Remarks on the literature

The literature on Lipschitz differentiability, PI spaces and Sobolev spaces on mms is very broad and we only scratch the surface here. First, we should remark that, aside from those already mentioned, two other definition of Sobolev spaces in metric measure spaces where proposed: that of Hajłasz [Haj96] and of Ambrosio-Gigli-Savaré [1]. All the definitions coincide for a general mms, except for the one of Hajłasz which coincides with the others in the presence of a PI. Cheeger, after his seminal result [11], proved quantitative analogues [12, 13]. While a notion of uniform rectifiability for general metric measure spaces has been introduced in [8] by Bate, Schul and the first author, it is unclear under which conditions Cheeger’s quantitative result may imply this notion. For geometric quantified flatness estimates across scales related to uniform rectifiability in metric spaces see also [23]. As we mentioned above, if $X \subset\mathbb{R}^n$, quantitative affine approximation of Lipschitz function does imply uniform rectifiability. This is the content of [Mer26]. In fact, if $X \subset\mathbb{R}^n$ is $d$-ADR, one may recover the full Donoroso characterisation of Sobolev spaces [3]. See also [29] for an analogous result on RCD spaces.

In general, the qualitative theory of Lipschitz differentiability is much better understood for general metric measure spaces, thanks to the collective efforts of several mathematicians, see e.g. the work of Bate [7, 6], of Bate-Li [9], of Cheeger-Kleiner-Schioppa [15], of Schioppa [34], of Keith [30, 31], of Alberti-Marchese [2], and of David, Eriksson-Bique and Schul [17] (in no particular order).

The theory of Sobolev spaces has much progressed, too: we refer the reader to the monograph [27]. Remarkable, and connected to the progress on Lipschitz differentiability and directions present in $X$ mentioned above, is the construction of Sobolev differential by Eriksson-Bique and Soultanis [22].

Outline of the proof. Consider $E$ as in the statement of Theorem 1.3. Since $E$ is uniformly rectifiable, it admits a corona decomposition, i.e., it is well-approximated in most locations and scales by a controlled ‘number’ of Lipschitz graphs (and contained in their union up to a set of measure zero). It is well-known that Lipschitz graphs admit $(1,1)$-Poincaré inequalities. Thus, the basic idea is build bridges between the graphs from the corona decomposition in such a way that generates a global Poincaré inequality.

There are several problems we need to overcome. First, we only have access to graphs in the good scales and locations for $E$ but we need to describe the coarse behaviour of the enveloping set in all cubes. This begs the question, what does the set look like in the bad locations and scales and how do you glue this to the good graphs in a way that is consistent with a Poincaré inequality? We solve this by considering approximations by $d$-dimensional dyadic skelata, i.e., $d$-dimensional boundary of dyadic cubes. Instead of using the graphs directly, our approximations in the good regions will be unions of dyadic skelata which are ‘aligned’ along the original graphs and appear at the scale of the minimal cubes. In the bad locations we have no geometry to deal with, so we just look at a fine-mesh approximation of ambient $n$-dimensional ball. See § 6. The benefit of considering such a approximations is the following: if two dyadic skelata with comparable size have large intersection, then their union supports a Poincaré inequality (see § 5). Since the good and bad regions are built using dyadic skelata, this makes the gluing procedure far easier. One caveat of using skeletal approximations is that it is not immediate that they support a Poincaré inequality. Thus, we must check that the skeletal approximations admit a Poincaré inequality. We check this condition in Proposition 5.7. Note that, since we want to make use of the above good intersection properties of nearby skelata, we will want to consider smooth trees where nearby minimal cubes have comparable sizes. These are defined in § 4.

The next problem is to avoid bad intersections of the skeletal approximations and the bridges that we build between each of the skeletal approximations. For this we add an additional dimension and ‘lift’ the approximations in a controlled way which depends on how well it approximates $E$, see (69). To connect one skeletal approximation to the next layer of skeletal approximations, we pick a single $d$-dimensional skeleton for each minimal cube and ‘drag’ one of its $(d-1)$-dimensional faces down to the approximation below it (here, down and below refer the height in this additional dimension). One should have in mind connecting the two sets with a bridge which is bi-Lipschitz to a $d$-dimensional cube, although the reality is a little more complicated that. These bridges are built in § 8.

Finally, we have to verify a Poincaré inequality for the set we construct as above. We start in § 9 by showing a local Poincaré inequality, i.e., we check that a skeletal approximation, together with all its connecting bridges, supports a Poincaré inequality. To check a global Poincaré inequality we find, for each pair of points, a connecting ‘Harnack chain’ of balls such that consecutive balls lie in one of the spaces that already have Poincaré estimates. This gives us a way to estimate the oscillation of functions in consecutive balls and, thus, this type of connectivity with local Poincaré estimates is enough to conclude a global Poincaré inequality. This is carried out in Section 10 and relies on the results in § 3.

Statement on AI. No AI tools were used for this article.

Preliminaries

Notation

Throughout the paper $C$ will denote some absolute constant (depending perhaps on $n$) which may change from line to line. If there exists $C \ge1$ such that $a \le Cb$, then we will write $a \lesssim b$. If $C$ depends on parameters $(t_1,\ldots,t_m)$, then we will write $a \lesssim_{t_1,\ldots,t_m} b$. We will write $a \sim b$ if $a \lesssim b$ and $b \lesssim a$. The notation $a \sim_{t_1,\ldots,t_m} b$ is defined similarly.

For a metric space $(X,d)$, we let $B(x,r)$ denote the open in $X$ centred at $x$ with radius $r$, that is,

If $B$ is a ball in $X$, we will often write $x_B$ and $r_B$ to denote its centre and radius, respectively (although such objects are not necessarily uniquely defined). For sets $E,F \subseteq X$, let

The $k$-dimensional Hausdorff measure of $E$ is defined as

For a set $A \subseteq\mathbb{R}^n \times\{0\}$ and a function $f:A \to\mathbb{R}$, denote the graph of $f$ over $A$ by $\operatorname{Graph}_A(f)$, that is,

Dyadic cubes and their skeleta

Let $\Delta$ denote the collection of dyadic cubes on $\mathbb{R}^n$. For $I \in\Delta$, let $x_I$ denote its centre and $\ell(I)$ denote its side-length. For $c > 0$, let $cI$ denote the cube concentric to $I$ with side-length equal $\ell(cI) = c\ell(I)$. Let

We let $\partial_d I$ denote its $d$-dimensional skeleton, i.e., if $I = \prod_{j=1}^{n} I_j$ for some (half open, half closed) intervals $I_j \subseteq\mathbb{R}$, then

These dyadic skeleta will be the basic building block from which we construct the set $Z$ from Theorem 1.3. We work with these sets because it is easy to describe how nearby skeleta interact with each other. The properties we need are proven below. First, we show that the gluing metric between nearby cubes respect the ambient Euclidean distance (a sort of orthogonality). Second, we show that skeleta corresponding to nested cubes have large intersections. Both of these properties will be useful in showing that unions of such skeleta support a Poincaré inequality.