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 $\hat{I}¼$ is a Borel measure supported on $E$, then the analytic capacity of $E$ satisfies $${\hat{I}}^3(E)\ge c\frac{\hat{I}¼(E{)}^2}{{\int }_I\Vert P_{\hat{I}}¸\hat{I}¼{\Vert }_2^{2}d\hat{I}¸},$$ where $c$ is some positive constant, $I\subset [0,\ddot{I}€)$ is an arbitrary interval, and $P_{\hat{I}}¸\hat{I}¼$ is the image measure of $\hat{I}¼$ by $P_{\hat{I}}¸$, the orthogonal projection onto the line $\{r{e}^{i\hat{I}¸}:r\in \mathbb{R}\}$. 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.

Minor corrections and adjustments. An additional appendix where we estimate $L^2$ densities of projections in terms of a conical Riesz energy plus an 'error term'