Let $μ$ be a Radon measure on $\mathbb{R}^d$. We define and study conical energies $\mathcal{E}_{μ,p}(x,V,α)$, which quantify the portion of $μ$ lying in the cone with vertex $x\in\mathbb{R}^d$, direction $V\in G(d,d-n)$, and aperture $α\in (0,1)$. We use these energies to characterize rectifiability and the big pieces of Lipschitz graphs property. Furthermore, if we assume that $μ$ has polynomial growth, we give a sufficient condition for $L^2(μ)$-boundedness of singular integral operators with smooth odd kernels of convolution type.
We give a geometric criterion for a topological surface in the first Heisenberg group to be an intrinsic Lipschitz graph, using planar cones instead of the usual open cones.
We introduce a notion of rectifiability modeled on Carnot groups. Precisely, we say that a subset E of a Carnot group M and N is a subgroup of M, we say E is N-rectifiable if it is the Lipschitz image of a positive measure subset of N. First, we discuss the implications of N-rectifiability, where N is a Carnot group (not merely a subgroup of a Carnot group), which include N-approximability and the existence of approximate tangent cones isometric to N almost everywhere in E. Second, we prove that, under a stronger condition concerning the existence of approximate tangent cones isomorphic to N a…
The well-known curvature method initiated in works of Melnikov and Verdera is now commonly used to relate the $L^2(μ)$-boundedness of certain singular integral operators to the geometric properties of the support of measure $μ$, e.g. rectifiability. It can be applied however only if Menger curvature-like permutations, directly associated with the kernel of the operator, are non-negative. We give an example of an operator in the plane whose corresponding permutations change sign but the $L^2(μ)$-boundedness of the operator still implies that the support of $μ$ is rectifiable. To the best of our…
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…
Let $E \subset \mathbb{C}$ be a Borel set such that $0<\mathcal{H}^1(E)<\infty$. David and Léger proved that the Cauchy kernel $1/z$ (and even its coordinate parts $\textrm{Re}\, z/|z|^2$ and $\textrm{Im}\, z/|z|^2$, $z\in \mathbb{C}\setminus\{0\}$) has the following property $(*)$: the $L^2(\mathcal{H}^1\lfloor E)$-boundedness of the corresponding singular integral operator implies the rectifiability of $E$. Recently Chousionis, Mateu, Prat and Tolsa extended this result to any kernel of the form $(\textrm{Re}\, z)^{2n-1}/|z|^{2n}$, $n\in \mathbb{N}$. In this paper, we prove that the property…
We continue our study of the reflectionless measures associated to an $s$-dimensional Calderón-Zygmund operator (CZO) acting in $\mathbb{R}^d$ with $s\in (0,d)$. Here, our focus will be the study of CZOs that are rigid, in the sense that they have few reflectionless measures associated to them. Our goal is to prove that the rigidity properties of a CZO $T$ impose strong geometric conditions upon the support of any measure $μ$ for which $T$ is a bounded operator in $L^2(μ)$. In this way, we shall reduce certain well-known problems at the interface of harmonic analysis and geometric measure theo…
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…
Let $E \subset \C$ be a Borel set with finite length, that is, $0<\mathcal{H}^1 (E)<\infty$. By a theorem of David and Léger, the $L^2 (\mathcal{H}^1 \lfloor E)$-boundedness of the singular integral associated to the Cauchy kernel (or even to one of its coordinate parts $x / |z|^2,y / |z|^2,z=(x,y) \in \C$) implies that $E$ is rectifiable. We extend this result to any kernel of the form $x^{2n-1} /|z|^{2n}, z=(x,y) \in \C,n \in \mathbb{N}$. We thus provide the first non-trivial examples of operators not directly related with the Cauchy transform whose $L^2$-boundedness implies rectifiability.
Let $E$ be an $1$-Ahlfors regular subset of the Heisenberg group $\mathbb{H}$. We prove that there exists a $-1$-homogeneous kernel $K_1$ such that if $E$ is contained in a $1$-regular curve the corresponding singular integral is bounded in $L^2(E)$. Conversely, we prove that there exists another $-1$-homogeneous kernel $K_2$, such that the $L^2(E)$-boundedness of its corresponding singular integral implies that $E$ is contained in an $1$-regular curve. These are the first non-Euclidean examples of kernels with such properties. Both $K_1$ and $K_2$ are weighted versions of the Riesz kernel cor…
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 consider the Calderón-Zygmund kernels $K_ {α,n}(x)=(x_i^{2n-1}/|x|^{2n-1+α})_{i=1}^d$ in $\mathbb{R}^n$ for $0<α\leq 1$ and $n\in\mathbb{N}$. We show that, on the plane, for $0<α<1$, the capacity associated to the kernels $K_{α,n}$ is comparable to the Riesz capacity $C_{\frac23(2-α),\frac 3 2}$ of non-linear potential theory. As consequences we deduce the semiadditivity and bi-Lipschitz invariance of this capacity. Furthermore we show that for any Borel set $E\subset\mathbb{R}^n$ with finite length the $L^2(\mathcal{H}^1 \lfloor E)$-boundedness of the singular integral associated to $K_{1,…
In the late `90s there was a flurry of activity relating $1$-rectifiable sets, boundedness of singular integral operators, the analytic capacity of a set, and the integral Menger curvature in the plane. In `99 Leger extended the results for Menger curvature to $1$-rectifiable sets in higher dimension, as well as to the codimension one case. A decade later, Lerman and Whitehouse, and later Meurer, found higher-dimensional geometrically motivated generalizations of Menger curvature that yield results about the uniform rectifiability of measures and the rectifiability of sets respectively. In thi…
Let $μ$ be a finite Radon measure in $\mathbb{R}^d$ with polynomial growth of degree $n$, although not necessarily $n$-AD-regular. We prove that under some geometric conditions on $μ$ that are closely related to rectifiability and involve the so-called $β$-numbers of Jones, David and Semmes, all singular integral operators with an odd and sufficiently smooth Calderón-Zygmund kernel are bounded in $L^2(μ)$. As a corollary, we obtain a lower bound for the Lipschitz harmonic capacity of a compact set in $\mathbb{R}^d$ only in terms of its metric and geometric properties.
We repurpose tools from the theory of quantitative rectifiability to study the qualitative rectifiability of measures in $\Bbb{R}^n$, $n\geq 2$. To each locally finite Borel measure $μ$, we associate a function $\widetilde J_2(μ, x)$ which uses a weighted sum to record how closely the mass of $μ$ is concentrated on a line in the triples of dyadic cubes containing $x$. We show that $\widetilde J_2(μ, x) < \infty$ $μ$-a.e. is a necessary condition for $μ$ to give full mass to a countable family of rectifiable curves. This confirms a conjecture of Peter Jones from 2000. A novelty of this result i…
