# Analytic capacity and projections

### Alan Chang

Princeton University, USA### Xavier Tolsa

ICREA and Universitat Autónoma de Barcelona, Bellaterra (Barcelona), Spain

## 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 $\mu$ is a Borel measure supported on $E$, then the analytic capacity of $E$ satisfies

where $c$ is some positive constant, $I\subset [0,\pi)$ is an arbitrary interval, and $P_\theta\mu$ is the image measure of $\mu$ by $P_\theta$, the orthogonal projection onto the line $\{re^{i\theta}:r\in\mathbb R\}$. 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.

## Cite this article

Alan Chang, Xavier Tolsa, Analytic capacity and projections. J. Eur. Math. Soc. 22 (2020), no. 12, pp. 4121–4159

DOI 10.4171/JEMS/1004