Related papers: Rectifiability of harmonic measure

We consider a generalized Cantor set $E(\omega)$ for an infinite sequence $\omega=(q_n)_{n=1}^{\infty}$ of positive numbers with $0<q_n<1$, and examine the quasiconformal equivalence to the standard middle one-third Cantor set…

The classical Besicovitch-Federer projection theorem implies that the d-dimensional Hausdorff measure of a set in Euclidean space with non-negligible d-unrectifiable part will strictly decrease under orthogonal projection onto almost every…

We study some variants of the Erd\H{o}s similarity problem. We pose the question if every measurable subset of the real line with positive measure contains a similar copy of an infinite geometric progression. We construct a compact subset…

In 1977 the celebrated theorem of B. Dahlberg established that the harmonic measure is absolutely continuous with respect to the Hausdorff measure on a Lipschitz graph of dimension $n-1$ in $\mathbb R^n$, and later this result has been…

We provide a sufficient geometric condition for $\mathbb{R}^n$ to be countably $(\mu,m)$ rectifiable of class $\mathscr{C}^{1,\alpha}$ (using the terminology of Federer), where $\mu$ is a Radon measure having positive lower density and…

It is well-known that quantitative, scale invariant absolute continuity (more precisely, the weak-$A_\infty$ property) of harmonic measure with respect to surface measure, on the boundary of an open set $ \Omega\subset \mathbb{R}^{n+1}$…

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…

We prove the continuity of logarithmic capacity under Hausdorff convergence of uniformly perfect planar sets. The continuity holds when the Hausdorff distance to the limit set tends to zero at sufficiently rapid rate, compared to the decay…

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 $\Omega \subset \mathbb{R}^{n+1}$ be an open set satisfying the corkscrew…

We show that if $\mathfrak{c} = \aleph_2$ then all covering numbers of Hausdorff measures $\operatorname{cov}(\mathcal{N}^s(\mathbb{R}^d))$ ($0 < s < d, d \in \omega$) are equal and all uniformity numbers…

For any bounded convex domain \Omega in R^N, we assign a positive finite Borel measure associated with the solution to a su-blinear elliptic equation in \Omega. We prove that this measure is weakly continuous in the sense of measure with…

David and Semmes proved that if all CZOs (of suitable dimension) are bounded with respect to an Ahlfors regular measure, then the measure is uniformly rectifiable. We extend this theorem to the parabolic space and the first Heisenberg…

Let $\Omega \subset \mathbb{R}^d$ be a set with finite Lebesgue measure such that, for a fixed radius $r>0$, the Lebesgue measure of $\Omega \cap B_r (x)$ is equal to a positive constant when $x$ varies in the essential boundary of…

We prove that for all $s\in(0,d)$ and $c\in (0,1)$ there exists a self-similar set $E\subset \mathbb{R}^d$ with Hausdorff dimension $s$ such that $\mathcal{H}^s(E)=c|E|^s$. This answers a question raised by Zhiying Wen[16].

Let $f \colon \Omega \to \Omega' $ be a Sobolev mapping of finite distortion between planar domains $\Omega $ and $\Omega'$, satisfying the $(INV)$ condition and coinciding with a homeomorphism near $\partial\Omega $. We show that $f$…

In this work we are going to establish H\"older continuity of harmonic maps from an open set $\Omega$ in an ${\rm RCD}(K,N)$ space valued into a ${\rm CAT}(\kappa)$ space, with the constraint that the image of $\Omega$ via the map is…

In relation to the Erd\H os similarity problem (show that for any infinite set $A$ of real numbers there exists a set of positive Lebesgue measure which contains no affine copy of $A$) we give some new examples of infinite sets which are…

A topological space $G$ is said to be a {\it rectifiable space} provided that there are a surjective homeomorphism $\phi :G\times G\rightarrow G\times G$ and an element $e\in G$ such that $\pi_{1}\circ \phi =\pi_{1}$ and for every $x\in G$…

In this paper we start a detailed study of a new notion of rectifiability in Carnot groups: we say that a Radon measure is $\mathscr{P}_h$-rectifiable, for $h\in\mathbb N$, if it has positive $h$-lower density and finite $h$-upper density…

We consider second order divergence form elliptic operators with $W^{1,1}$ coefficients, in a uniform domain $\Omega$ with Ahlfors regular boundary. We show that the $A_\infty$ property of the elliptic measure associated to any such…