Analytic capacity and projections

Abstract

In this paper we study the connection between the analytic capacity of a set and the size of its orthogonal projections. More precisely, we prove that if $E\subset \mathbb C$ is compact and $μ$ is a Borel measure supported on $E$, then the analytic capacity of $E$ satisfies $$ γ(E) \geq c\,\frac{μ(E)^2}{\int_I \|P_θμ\|_2^2\,dθ}, $$ where $c$ is some positive constant, $I\subset [0,π)$ is an arbitrary interval, and $P_θμ$ is the image measure of $μ$ by $P_θ$, the orthogonal projection onto the line $\{re^{iθ}:r\in\mathbb R\}$. This result is related to an old conjecture of Vitushkin about the relationship between the Favard length and analytic capacity. We also prove a generalization of the above inequality to higher dimensions which involves related capacities associated with signed Riesz kernels.