Related papers: Reflectionless measures and the Mattila-Melnikov-V…

In this article, we close a gap in the literature by proving existence of invariant measures for reflected SPDEs with only one reflecting barrier. This is done by arguing that the sequence (u(t, .)) is tight in the space of probability…

The well-known curvature method initiated in works of Melnikov and Verdera is now commonly used to relate the $L^2(\mu)$-boundedness of certain singular integral operators to the geometric properties of the support of measure $\mu$, e.g.…

In this paper we establish a Besicovitch-Federer type projection theorem for general measures. Specifically, let $\mu$ be a finite Borel measure on $\mathbb{R}^n$ and let $0 < m < n$ be an integer. We show that, under the sole assumption…

Let $A(\cdot)$ be an $(n+1)\times (n+1)$ uniformly elliptic matrix with H\"older 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…

We prove an equidistribution theorem a la Bader-Muchnik for operator-valued measures associated with boundary representations in the context of discrete groups of isometries of CAT(-1) spaces thanks to an equidistribution theorem of T.…

Uniform bounds are developed for derivatives of solutions of the $2$-dimensional constant negative curvature equation and the Weil-Petersson metric for the Teichm\"{u}ller and moduli spaces. The dependence of the bounds on the geometry of…

We consider a uniformly rectifiable set $\Gamma \subset \mathbb R^n$ of dimension $d<n-1$. By using degenerate elliptic operators on the complement $\Omega = \mathbb R^n \setminus \Gamma$, Guy David, Svitlana Mayboroda, and the author…

We introduce new flatness coefficients, which we call $\iota$-numbers, for Ahlfors $k$-regular sets in metric spaces ($k\in \mathbb{N}$). Using these coefficients for $k=1$, we characterize uniform $1$-rectifiability in rather general…

In much discussed work Artemov has recently shown that, for $\mathrm{PA}$, the consistency schema admits a form of uniform verification via selector proofs, despite the unprovability of the corresponding uniform consistency sentence…

We resolve a long-standing open problem posed by Federer concerning the rectifiability of the integral geometric measure with exponent p >1, thereby settling a question that has persisted since its formulation. While the main theorem is…

Suppose that $\Omega \subset \mathbb{R}^{n+1}$, $n \ge 2$, is an open set satisfying the corkscrew condition with an $n$-dimensional ADR boundary, $\partial \Omega$. In this note, we show that if harmonic functions are…

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 $L^p(\Omega)$ for every $p \in (1,\infty)$, where $\Omega := \mathbb{R}^{n+1}…

We present a natural and simple proof of the Radon - Nikodym theorem for measures with values in the space of bounded linear operators on a separable Hilbert space. This space is not separable, that is why it is essential to assume in the…

In this paper we prove a correspondence principle between multivariate functions of bounded variation in the sense of Hardy and Krause and signed measures of finite total variation, which allows us to obtain a simple proof of a generalized…

This paper is devoted to the proof of two related results. The first one asserts that if $\mu$ is a Radon measure in $\mathbb R^d$ satisfying $$\limsup_{r\to 0} \frac{\mu(B(x,r))}{r}>0\quad \text{ and }\quad…

We give an elementary proof of the Gel'fand-Kapranov-Zelevinsky theorem that non-resonant A-hypergeometric systems are irreducible. We also provide a proof of a converse statement In this second version we have removed the condition of…

It is proved that for any $0<\beta<\alpha$, any bounded Ahlfors $\alpha$-regular space contains a $\beta$-regular compact subset that embeds biLipschitzly in an ultrametric with distortion at most $O(\alpha/(\alpha-\beta))$. The bound on…

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…

To prove that a measure, linearly representable by means of a finite set of nonnegative matrices $\mathcal M$, has the weak-Gibbs property, one check the uniform convergence (on $\mathcal M^\mathbb N$) of the sequence of vectors…

We establish a simple and powerful lemma that provides a criterion for sequences in metric spaces to be Cauchy. Using the lemma, it is then easily verified that the Picard iterates $\{T^nx\}$, where $T$ is a contraction or asymptotic…