We show that an Ahlfors $d$-regular set $E$ in $\mathbb{R}^{n}$ is uniformly rectifiable if the set of pairs $(x,r)\in E\times (0,\infty)$ for which there exists $y \in B(x,r)$ and $0<t<r$ satisfying $\mathscr{H}^{d}_{\infty}(E\cap B(y,t))<(2t)^{d}-\varepsilon(2r)^d$ is a Carleson set for every $\varepsilon>0$.
To prove this, we generalize a result of Schul by proving, if $X$ is a $C$-doubling metric space, $\varepsilon,ρ\in (0,1)$, $A>1$, and $X_{n}$ is a sequence of maximal $2^{-n}$-separated sets in $X$, and $\mathscr{B}=\{B(x,2^{-n}):x\in X_{n},n\in \mathbb{N}\}$, then \[ \sum \left\{r_{B}^{s}: B\in \mathscr{B}, \frac{\mathscr{H}^{s}_{ρr_{B}}(X\cap AB)}{(2r_{B})^{s}}>1+\varepsilon\right\} \lesssim_{C,A,\varepsilon,ρ,s} \mathscr{H}^{s}(X). \] This is a quantitative version of the classical result that for a metric space $X$ of finite $s$-dimensional Hausdorff measure, the upper $s$-dimensional densities are at most $1$ $\mathscr{H}^{s}$-almost everywhere.
We show that any $d$-Ahlfors regular subset of $\mathbb{R}^{n}$ supporting a weak $(1,d)$-Poincaré inequality with respect to surface measure is uniformly rectifiable.
We present the converse to a higher dimensional, scale-invariant version of a classical theorem of F. and M. Riesz. More precisely, for $n\geq 2$, for an ADR domain $Ω\subset \re^{n+1}$ which satisfies the Harnack Chain condition plus an interior (but not exterior) Corkscrew condition, we show that absolute continuity of harmonic measure with respect to surface measure on $\partialΩ$, with scale invariant higher integrability of the Poisson kernel, is sufficient to imply uniformly rectifiable of $\partialΩ$.
We present a higher dimensional, scale-invariant version of a classical theorem of F. and M. Riesz. More precisely, we establish scale invariant absolute continuity of harmonic measure with respect to surface measure, along with higher integrability of the Poisson kernel, for a domain $Ω\subset \mathbb{R}^{n+1},\, n\geq 2$, with a uniformly rectifiable boundary, which satisfies the Harnack Chain condition plus an interior (but not exterior) corkscrew condition. In a companion paper to this one [HMU], we also establish a converse, in which we deduce uniform rectifiability of the boundary, assum…
Let $E\subset \mathbb{R}^{n+1}$, $n\ge 2$, be an Ahlfors-David regular set of dimension $n$. We show that the weak-$A_\infty$ property of harmonic measure, for the open set $Ω:= \mathbb{R}^{n+1}\setminus E$, implies uniform rectifiability of $E$.
A new proof is given of the Mattila-Melnikov-Verdera theorem on the uniform rectifiability of an Ahlfors-David regular measure whose associated Cauchy transform operator is bounded.
Let $E\subset \mathbb{R}^{n+1}$, $n\ge 2$, be a closed, Ahlfors-David regular set of dimension $n$ satisfying the "Riesz Transform bound" $$\sup_{\varepsilon>0}\int_E\left|\int_{\{y\in E:|x-y|>\varepsilon\}}\frac{x-y}{|x-y|^{n+1}} f(y) dH^n(y)\right|^2 dH^n(x) \leq C \int_E|f|^2 dH^n .$$ Assume further that $E$ is the boundary of a domain $Ω\subset \mathbb{R}^{n+1}$ satisfying the Harnack Chain condition plus an interior (but not exterior) Corkscrew condition. Then $E$ is uniformly rectifiable.
In this paper we study some questions in connection with uniform rectifiability and the $L^2$ boundedness of Calderon-Zygmund operators. We show that uniform rectifiability can be characterized in terms of some new adimensional coefficients which are related to the Jones' $β$ numbers. We also use these new coefficients to prove that n-dimensional Calderon-Zygmund operators with odd kernel of type $C^2$ are bounded in $L^2(μ)$ if $μ$ is an n-dimensional uniformly rectifiable measure.
For 0<n<d integers and r>2, we prove that an n-dimensional Ahlfors-David regular measure M in R^d is uniformly n-rectifiable if and only if the r-variation for the Riesz transform with respect to M is a bounded operator in L^2(M). This result can be considered as a partial solution to a well known open problem posed by G. David and S. Semmes which relates the L^2(M) boundedness of the Riesz transform to the uniform rectifiability of M.
In this paper it is shown that an Ahlfors-David $n$-dimensional measure $μ$ on $\mathbb{R}^d$ is uniformly $n$-rectifiable if and only if for any ball $B(x_0,R)$ centered at $\operatorname{supp}(μ)$, $$ \int_0^R \int_{x\in B(x_0,R)} \left|\frac{μ(B(x,r))}{r^n} - \frac{μ(B(x,2r))}{(2r)^n} \right|^2\,dμ(x)\,\frac{dr}r \leq c\, R^n.$$ Other characterizations of uniform $n$-rectifiability in terms of smoother square functions are also obtained.
We prove that if $μ$ is a d-dimensional Ahlfors-David regular measure in $\R^{d+1}$, then the boundedness of the $d$-dimensional Riesz transform in $L^2(μ)$ implies that the non-BAUP David-Semmes cells form a Carleson family. Combined with earlier results of David and Semmes, this yields the uniform rectifiability of $μ$.
In this paper it is shown that if $μ$ is an n-dimensional Ahlfors-David regular measure in $R^d$ which satisfies the so-called weak constant density condition, then $μ$ is uniformly rectifiable. This had already been proved by David and Semmes in the cases n=1, 2 and d-1, and it was an open problem for other values of n. The proof of this result relies on the study of the n-uniform measures in $R^d$. In particular, it is shown here that they satisfy the "big pieces of Lipschitz graphs" property.
Let $E\subset \mathbb{R}^{n+1}$, $n\ge 2$, be an Ahlfors-David regular set of dimension $n$. We show that the weak-$A_\infty$ property of harmonic measure, for the open set $Ω:= \mathbb{R}^{n+1}\setminus E$, implies uniform rectifiability of $E$. More generally, we establish a similar result for the Riesz measure, $p$-harmonic measure, associated to the $p$-Laplace operator, $1<p<\infty$.
We study Lipschitz differentiability spaces, a class of metric measure spaces introduced by Cheeger. We show that if an Ahlfors regular Lipschitz differentiability space has charts of maximal dimension, then, at almost every point, all its tangents are uniformly rectifiable. In particular, at almost every point, such a space admits a tangent that is isometric to a finite-dimensional Banach space. In contrast, we also show that if an Ahlfors regular Lipschitz differentiability space has charts of non-maximal dimension, then these charts are strongly unrectifiable in the sense of Ambrosio-Kirchh…
Suppose that $E \subset \mathbb{R}^{n+1}$ is a uniformly rectifiable set of codimension $1$. We show that every harmonic function is $\varepsilon$-approximable in $L^p(Ω)$ for every $p \in (1,\infty)$, where $Ω:= \mathbb{R}^{n+1} \setminus E$. Together with results of many authors this shows that pointwise, $L^\infty$ and $L^p$ type $\varepsilon$-approximability properties of harmonic functions are all equivalent and they characterize uniform rectifiability for codimension $1$ Ahlfors-David regular sets. Our results and techniques are generalizations of recent works of T. Hytönen and A. Rosén …
In this paper we characterize the so called uniformly rectifiable sets of David and Semmes in terms of the Wasserstein distance $W_2$ from optimal mass transport. To obtain this result, we first prove a localization theorem for the distance $W_2$ which asserts that if $μ$ and $ν$ are probability measures in $R^n$, $ϕ$ is a radial bump function smooth enough so that $\intϕdμ\gtrsim1$, and $μ$ has a density bounded from above and from below on the support of ϕ, then $W_2(ϕμ,aϕν)\leq c W_2(μ,ν),$ where $a=\intϕdμ/ \intϕ\,dν$.
Let $Ω\subset\mathbb R^{n+1}$, $n\geq1$, be a corkscrew domain with Ahlfors-David regular boundary. In this paper we prove that $\partialΩ$ is uniformly $n$-rectifiable if every bounded harmonic function on $Ω$ is $\varepsilon$-approximable or if every bounded harmonic function on $Ω$ satisfies a suitable square-function Carleson measure estimate. In particular, this applies to the case when $Ω=\mathbb R^{n+1}\setminus E$ and $E$ is Ahlfors-David regular. Our results solve a conjecture posed by Hofmann, Martell, and Mayboroda in a recent work where they proved the converse statements. Here we …
We study metric measure spaces that have quantitative topological control, as well as a weak form of differentiable structure. In particular, let $X$ be a pointwise doubling metric measure space. Let $U$ be a Borel subset on which the blowups of $X$ are topological planes. We show that $U$ can admit at most $2$ independent Alberti representations. Furthermore, if $U$ admits $2$ Alberti representations, then the restriction of the measure to $U$ is $2$-rectifiable. This is a partial answer to the case $n=2$ of a question of the second author and Schioppa.
In this work we extend many classical results concerning the relationship between densities, tangents and rectifiability to the parabolic spaces, namely $\mathbb{R}^{n+1}$ equipped with parabolic dilations. In particular we prove a Marstrand-Mattila rectifiability criterion for measures of general dimension, we provide a characterisation through densities of intrinsic rectifiable measures, and we study the structure of $1$-codimensional uniform measures. Finally, we apply some of our results to the study of a quantitative version of parabolic rectifiability: we prove that the weak constant den…
Aim of this paper is a finer analysis of the group of flat chains with coefficients in $Z_p$ introduced in a recent paper by Ambrosio-Katz, by taking quotients of the group of integer rectifiable currents, along the lines of the the Ziemer and Federer approach. We investigate the typical questions of the theory of currents, namely rectifiability of the measure-theoretic support and boundary rectifiability. Our main result can also be interpreted as a closure theorem for the class of integer rectifiable currents with respect to a (much) weaker convergence, induced by flat distance mod $p$, and …
We are interested in quantitative rectifiability results for subsets of infinite dimensional Hilbert space $H$. We prove a version of Azzam and Schul's $d$-dimensional Analyst's Travelling Salesman Theorem in this setting by showing for any lower $d$-regular set $E \subseteq H$ that \[ \text{diam}(E)^d + β^d(E) \sim \mathscr{H}^d(E) + \text{Error}, \] where $β^d(E)$ give a measure of the curvature of $E$ and the error term is related to the theory of uniform rectifiability (a quantitative version of rectifiability introduced by David and Semmes). To do this, we show how to modify the Reifenber…
The present paper establishes the correspondence between the properties of the solutions of a class of PDEs and the geometry of sets in Euclidean space. We settle the question of whether (quantitative) absolute continuity of the elliptic measure with respect to the surface measure and uniform rectifiability of the boundary are equivalent, in an optimal class of divergence form elliptic operators satisfying a suitable Carleson measure condition. The result can be viewed as a quantitative analogue of the Wiener criterion adapted to the singular $L^p$ data case. We split our proof on two main ste…
The present paper, along with its sequel, establishes the correspondence between the properties of the solutions of a class of PDEs and the geometry of sets in Euclidean space. We settle the question of whether (quantitative) absolute continuity of the elliptic measure with respect to the surface measure and uniform rectifiability of the boundary are equivalent, in an optimal class of divergence form elliptic operators satisfying a suitable Carleson measure condition. The result can be viewed as a quantitative analogue of the Wiener criterion adapted to the singular $L^p$ data case. This paper…
For a Radon measure $μ$ on $\mathbb{R}^d$, define $C^n_μ(x, t)= \ (\frac{1}{t^n} \ |\int_{B(x,t)} \frac{x-y}{t} \, dμ(y)\ | \ )$. This coefficient quantifies how symmetric the measure $μ$ is by comparing the center of mass at a given scale and location to the actual center of the ball. We show that if $μ$ is $n$-rectifiable, then $ \int_0^\infty |C^n_μ(x,t)|^2 \frac{dt}{t} < \infty \, \, μ\mbox{-almost everywhere}. $ Together with a previous result of Mayboroda and Volberg, where they showed that the converse holds true, this gives a characterisation of $n$-rectifiability. To prove our main re…
Let $A(\cdot)$ be an $(n+1)\times (n+1)$ uniformly elliptic matrix with Hölder continuous real coefficients and let $\mathcal E_A(x,y)$ be the fundamental solution of the PDE $\mathrm{div} A(\cdot) \nabla u =0$ in $\mathbb R^{n+1}$. Let $μ$ be a compactly supported $n$-AD-regular measure in $\mathbb R^{n+1}$ and consider the associated operator $$T_μf(x) = \int \nabla_x\mathcal E_A(x,y)\,f(y)\,dμ(y).$$ We show that if $T_μ$ is bounded in $L^2(μ)$, then $μ$ is uniformly $n$-rectifiable. This extends the solution of the codimension $1$ David-Semmes problem for the Riesz transform to the gradient…
The present paper, along with its companion [Hofmann, Martell, Mayboroda, Toro, Zhao, arXiv:1710.06157], establishes the correspondence between the properties of the solutions of a class of PDEs and the geometry of sets in Euclidean space. We settle the question of whether (quantitative) absolute continuity of the elliptic measure with respect to the surface measure and uniform rectifiability of the boundary are equivalent, in an optimal class of divergence form elliptic operators satisfying a suitable Carleson measure condition. The result can be viewed as a quantitative analogue of the Wiene…
We assume that $Ω_1, Ω_2 \subset \mathbb{R}^{n+1}$, $n \geq 1$ are two disjoint domains whose complements satisfy the capacity density condition and the intersection of their boundaries $F$ has positive harmonic measure. Then we show that in a fixed ball $B$ centered on $F$, if the harmonic measure of $Ω_1$ satisfies a scale invariant $A_\infty$-type condition with respect to the harmonic measure of $Ω_2$ in $B$, then there exists a uniformly $n$-rectifiable set $Σ$ so that the harmonic measure of $Σ\cap F$ contained in $B$ is bounded below by a fixed constant independent of $B$. A remarkable …
We construct extensions of Varopolous type for functions $f \in \text{BMO}(E)$, for any uniformly rectifiable set $E$ of codimension one. More precisely, let $Ω\subset \mathbb{R}^{n+1}$ be an open set satisfying the corkscrew condition, with an $n$-dimensional uniformly rectifiable boundary $\partial Ω$, and let $σ:= \mathcal{H}^n\lfloor_{\partial Ω}$ denote the surface measure on $\partial Ω$. We show that if $f \in \text{BMO}(\partial Ω,dσ)$ with compact support on $\partial Ω$, then there exists a smooth function $V$ in $Ω$ such that $|\nabla V(Y)| \, dY$ is a Carleson measure with Carleson…
We obtain Dini conditions with "exponent 2" that guarantee that an asymptotically conformal quasisphere is rectifiable. In particular, we show that for any e>0 integrability of (esssup_{1-t < |x| < 1+t} K_f(x)-1)^{2-e} dt/t implies that the image of the unit sphere under a global quasiconformal homeomorphism f is rectifiable. We also establish estimates for the weak quasisymmetry constant of a global K-quasiconformal map in neighborhoods with maximal dilatation close to 1.
Let $Ω\subset\mathbb{R}^{n+1}$, $n\geq2$, be an open set with Ahlfors-David regular boundary that satisfies the corkscrew condition. We consider a uniformly elliptic operator $L$ in divergence form associated with a matrix $A$ with real, merely bounded and possibly non-symmetric coefficients, which are also locally Lipschitz and satisfy suitable Carleson type estimates. In this paper we show that if $L^*$ is the operator in divergence form associated with the transpose matrix of $A$, then $\partialΩ$ is uniformly $n$-rectifiable if and only if every bounded solution of $Lu=0$ and every bounded…
In a recent paper, Csörnyei and Wilson prove that curves in Euclidean space of $σ$-finite length have tangents on a set of positive $\mathscr{H}^{1}$-measure. They also show that a higher dimensional analogue of this result is not possible without some additional assumptions. In this note, we show that if $Σ\subseteq \mathbb{R}^{d+1}$ has the property that each ball centered on $Σ$ contains two large balls in different components of $Σ^{c}$ and $Σ$ has $σ$-finite $\mathscr{H}^{d}$-measure, then it has $d$-dimensional tangent points in a set of positive $\mathscr{H}^{d}$-measure. We also give s…
Our purpose is to state quantitative conditions ensuring the rectifiability of a $d$--varifold $V$ obtained as the limit of a sequence of $d$--varifolds $(V_i)_i$ which need not to be rectifiable. More specifically, we introduce a sequence $\left\lbrace \mathcal{E}_i \right\rbrace_i$ of functionals defined on $d$--varifolds, such that if $\displaystyle \sup_i \mathcal{E}_i (V_i) < +\infty$ and $V_i$ satisfies a uniform density estimate at some scale $β_i$, then $V = \lim_i V_i$ is $d$--rectifiable. \noindent The main motivation of this work is to set up a theoretical framework where curves, su…
We identify a set of sufficient local conditions under which a significant portion of a Radon measure $μ$ on $\mathbb{R}^{n+1}$ with compact support can be covered by an $n$-uniformly rectifiable set at the level of a ball $B\subset \mathbb{R}^{n+1}$ such that $μ(B)\approx r(B)^n$. This result involves a flatness condition, formulated in terms of the so-called $β_1$-number of $B$, and the $L^2(μ|_B)$-boundedness, as well as a control on the mean oscillation on the ball, of the operator \begin{equation} T_μf(x)=\int \nabla_x\mathcal{E}(x,y)f(y)\,dμ(y). \end{equation} Here $\mathcal{E}(\cdot,\cd…
Solvability of the $L^p$ Dirichlet problem for the heat equation is equivalent to parabolic uniform rectifiability in the case of a parabolic Lipschitz graph
Let $Ω\subset \mathbb{R}^{n+1}$, $n\geq 2$, be 1-sided NTA domain (aka uniform domain), i.e. a domain which satisfies interior Corkscrew and Harnack Chain conditions, and assume that $\partialΩ$ is $n$-dimensional Ahlfors-David regular. We characterize the rectifiability of $\partialΩ$ in terms of the absolute continuity of surface measure with respect to harmonic measure. We also show that these are equivalent to the fact that $\partialΩ$ can be covered $\mathcal{H}^n$-a.e. by a countable union of portions of boundaries of bounded chord-arc subdomains of $Ω$ and to the fact that $\partialΩ$ p…
In this paper it is shown that if $μ$ is a finite Radon measure in $\mathbb R^d$ which is $n$-rectifiable and $1\leq p\leq 2$, then $$\int_0^\infty β_{μ,p}^n(x,r)^2\,\frac{dr}r<\infty \quad {for $μ$-a.e. $x\in\mathbb R^d$,}$$ where $$β_{μ,p}^n(x,r) = \inf_L (\frac1{r^n} \int_{\bar B(x,r)} (\frac{\mathrm dist(y,L)}{r})^p\,dμ(y))^{1/p},$$ with the infimum taken over all the $n$-planes $L\subset \mathbb R^d$. The $β_{μ,p}^n$ coefficients are the same as the ones considered by David and Semmes in the setting of the so called uniform $n$-rectifiability. An analogous necessary condition for $n$-rect…
In this paper we show that if $μ$ is a Borel measure in $\mathbb R^{n+1}$ with growth of order $n$, so that the $n$-dimensional Riesz transform $R_μ$ is bounded in $L^2(μ)$, and $B\subset\mathbb R^{n+1}$ is a ball with $μ(B)\approx r(B)^n$ such that: (a) there is some $n$-plane $L$ passing through the center of $B$ such that for some $δ>0$ small enough, it holds $\int_B \frac{dist(x,L)}{r(B)}\,dμ(x)\leq δ\,μ(B),$ (b) for some constant $ε>0$ small enough, $\int_B |R_\mu1(x) - m_{μ,B}(R_\mu1)|^2\,dμ(x) \leq ε\,μ(B)$, where $m_{μ,B}(R_\mu1)$ stands for the mean of $R_\mu1$ on $B$ with respect to …
A well-known class of questions asks the following: If $X$ and $Y$ are metric measure spaces and $f:X\rightarrow Y$ is a Lipschitz mapping whose image has positive measure, then must $f$ have large pieces on which it is bi-Lipschitz? Building on methods of David (who is not the present author) and Semmes, we answer this question in the affirmative for Lipschitz mappings between certain types of Ahlfors $s$-regular, topological $d$-manifolds. In general, these manifolds need not be bi-Lipschitz embeddable in any Euclidean space. To prove the result, we use some facts on the Gromov-Hausdorff con…
In this paper, we characterize the rectifiability (both uniform and not) of an Ahlfors regular set, E, of arbitrary co-dimension by the behavior of a regularized distance function in the complement of that set. In particular, we establish a certain version of the Riesz transform characterization of rectifiability for lower-dimensional sets. We also uncover a special situation in which the regularized distance is itself a solution to a degenerate elliptic operator in the complement of E. This allows us to precisely compute the harmonic measure of those sets associated to this degenerate operato…
We prove a structure theorem for any $n$-rectifiable set $E\subset \mathbb{R}^{n+1}$, $n\ge 1$, satisfying a weak version of the lower ADR condition, and having locally finite $H^n$ ($n$-dimensional Hausdorff) measure. Namely, that $H^n$-almost all of $E$ can be covered by a countable union of boundaries of bounded Lipschitz domains contained in $\mathbb{R}^{n+1}\setminus E$. As a consequence, for harmonic measure in the complement of such a set $E$, we establish a non-degeneracy condition which amounts to saying that $H^n|_E$ is "absolutely continuous" with respect to harmonic measure in the …
We study the properties of harmonic measure in semi-uniform domains. Aikawa and Hirata showed in \cite{AH08} that, for John domains satisfying the capacity density condition (CDC), the doubling property for harmonic measure is equivalent to the domain being semi-uniform. Our first result removes the John condition by showing that any domain satisfying the CDC whose harmonic measure is doubling is semi-uniform. Next, we develop a substitute for some classical estimates on harmonic measure in nontangentially accessible domains that works in semi-uniform domains. We also show that semi-uniform do…
We prove that coronizations with respect to arbitrary d-regular sets (not necessarily graphs) imply big pieces squared of these (approximating) sets. This is known (and due to David and Semmes in the case of sufficiently large co-dimension, and to Azzam and Schul in general) in the (classical) setting of Euclidean spaces with Hausdorff measure of integer dimension, where the approximating sets are Lipschitz graphs. Our result is a far reaching generalization of these results and we prove that coronizations imply big pieces squared is a generic property. In particular, our result applies, when …
We study the minimizers of a functional on the set of partitions of a domain $Ω\subset R^n$ into $N$ subsets $W_j$ of locally finite perimeter in $Ω$, whose main term is $\sum_{j=1^N} \int_{Ω\cap \partial W_j} a(x) dH^{n--1}(x)$. Here the positive bounded function $a$ may for instance be related to the Landscape function of some Schr{ö}dinger operator. We prove the existence of minimizers through the equivalence with a weak formulation, and the local Ahlfors regularity and uniform rectifiability of the boundaries $Ω\cap \partial W_j$.
