Ahlfors-David regular sets and bilipschitz maps

Given two Ahlfors-David regular sets in metric spaces, we study the question whether one of them has a subset bilipschitz equivalent with the other

Strong Marstrand theorems and dimensions of sets formed by subsets of hyperplanes

We present strong versions of Marstrand's projection theorems and other related theorems. For example, if E is a plane set of positive and finite s-dimensional Hausdorff measure, there is a set X of directions of Lebesgue measure 0, such that the projection onto any line with direction outside X, of any subset F of E of positive s-dimensional measure, has Hausdorff dimension min(1,s), i.e. the set of exceptional directions is independent of F. Using duality this leads to results on the dimension of sets that intersect families of lines or hyperplanes in positive Lebesgue measure.Comment: 8 page

Boundedness and convergence for singular integrals of measures separated by Lipschitz graphs

We shall consider the truncated singular integral operators T_{\mu, K}^{\epsilon}f(x)=\int_{\mathbb{R}^{n}\setminus B(x,\epsilon)}K(x-y)f(y)d\mu y and related maximal operators $T_{\mu,K}^{\ast}f(x)=\underset{\epsilon >0}{\sup}| T_{\mu,K}^{\epsilon}f(x)|$. We shall prove for a large class of kernels $K$ and measures $\mu$ and $\nu$ that if $\mu$ and $\nu$ are separated by a Lipschitz graph, then $T_{\nu,K}^{\ast}:L^p(\nu)\to L^p(\mu)$ is bounded for $1<p<\infty$. We shall also show that the truncated operators $T_{\mu, K}^{\epsilon}$ converge weakly in some dense subspaces of $L^2(\mu)$ under mild assumptions for the measures and the kernels.Comment: To appear in the Bulletin of the LM

On the analytic capacity and curvature of some cantor sets with non-σ-finite length

We show that if a Cantor set $E$ as considered by Garnett in \cite{G2} has positive Hausdorff $h$-measure for a non-decreasing function $h$ satisfying \int^1_0r^{-3}\,h(r)^2\,dr < \infty, then the analytic capacity of $E$ is positive. Our tool will be the Menger three-point curvature and Melnikov's identity relating it to the Cauchy kernel. We shall also prove some related more general results