Related papers: Rectifiability via a square function and Preiss' t…

Let $\Omega$ be a domain in $\mathbb{R}^{d+1}$, $d \geq 1$. In the paper's references [HMM2] and [GMT] it was proved that if $\Omega$ satisfies a corkscrew condition and if $\partial \Omega$ is $d$-Ahlfors regular, i.e. Hausdorff measure…

We prove that, given an $RCD^{*}(K,N)$-space $(X,d,m)$, then it is possible to $m$-essentially cover $X$ by measurable subsets $(R_{i})_{i\in \mathbb{N}}$ with the following property: for each $i$ there exists $k_{i} \in \mathbb{N}\cap…

This paper is devoted to show that the flatness of tangents of $1$-codimensional measures in Carnot Groups implies $C^1_\mathbb{G}$-rectifiability. As applications we prove that measures with $(2n+1)$-density in the Heisenberg groups…

We prove that if the Hausdorff dimension of a compact set $E \subset {\Bbb R}^2$ is greater than 7/4, then the set of {\ag three-point configurations determined by $E$ has positive three-dimensional measure}. We establish this by showing…

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 L\'eger, the $L^2 (\mathcal{H}^1 \lfloor E)$-boundedness of the singular integral associated to the Cauchy kernel (or…

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…

We study multilinear generalized Radon transforms using a graph-theoretic paradigm that includes the widely studied linear case. These provide a general mechanism to study Falconer-type problems involving $(k+1)$-point configurations in…

Let $n, m$ be positive integers, $n\geq m$. We make several remarks on the relationship between approximate differentiability of higher order and Morse-Sard properties. For instance, among other things we show that if a function…

Robin's criterion states that the Riemann Hypothesis (RH) is true if and only if Robin's inequality sum_{d|n}d<e^{gamma}n loglog n is satisfied for n>=5041, where gamma denotes the Euler(-Mascheroni) constant. We show by elementary methods…

Let $t \in (1,2)$, and let $B \subset \mathbb{R}^{2}$ be a Borel set with $\dim_{\mathrm{H}} B > t$. I show that $$\mathcal{H}^{1}(\{e \in S^{1} : \dim_{\mathrm{H}} (B \cap \ell_{x,e}) \geq t - 1\}) > 0$$ for all $x \in \mathbb{R}^{2} \,…

Let $\mathbb{S} \subset \mathbb{C}$ be the circle in the plane, and let $\Omega: \mathbb{S} \to \mathbb{S}$ be an odd bi-Lipschitz map with constant $1+\delta_\Omega$, where $\delta_\Omega>0$ is small. Assume also that $\Omega$ is twice…

In some former works of Azzam and Tolsa it was shown that $n$-rectifiability can be characterized in terms of a square function involving the David-Semmes $\beta_2$ coefficients. In the present paper we construct some counterexamples which…

In this note it is shown that if $\mu$ is an $n$-Ahlfors regular measure in $\mathbb R^{n+1}$ such that the $n$-dimensional Riesz transform is bounded in $L^2(\mu)$ and the so-called BAUPP (bilateral approximation by unions of parallel…

For a given decreasing positive real function $\psi$, let $\mathcal{A}_n(\psi)$ be the set of real numbers for which there are infinitely many integer polynomials $P$ of degree up to $n$ such that $\left\lvert P(x) \right\rvert \leq…

In this paper, we provide a proof that functions belonging to Besov spaces $B^{r}_{q,\infty}(\mathbb{R}^N,\mathbb{R}^d)$, $q\in [1,\infty)$, $r\in(0,1)$, satisfy the following formula under a certain condition: \begin{equation}…

Given a compact set $E \subset \mathbb{R}^{d - 1}$, $d \geq 1$, write $K_{E} := [0,1] \times E \subset \mathbb{R}^{d}$. A theorem of C. Bishop and J. Tyson states that any set of the form $K_{E}$ is minimal for conformal dimension: if…

In the setting of a metric space equipped with a doubling measure that supports a Poincar\'e inequality, we show that a set $E$ is of finite perimeter if and only if $\mathcal H(\partial^1 I_E)<\infty$, that is, if and only if the…

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}…

For every e>0, any subset of R^n with Hausdorff dimension larger than (1-e)n must have ultrametric distortion larger than 1/(4e).

The inversion theorem for the k-plane Radon transform in R^n is often stated for Schwartz functions, and lately for smooth functions on R^n fulfilling that f(x)=O(|x|^{-N}) for some N>n. In this paper it will be shown, that it suffices to…