Analytic capacity and projections

TL;DR

This paper establishes a lower bound for the analytic capacity of a compact set in the complex plane using the size of its orthogonal projections and extends the results to higher dimensions with related capacities.

Contribution

It introduces a new inequality linking analytic capacity with projection measures and generalizes it to higher-dimensional capacities involving Riesz kernels.

Findings

Proved a lower bound for analytic capacity based on projection measures.

Extended the inequality to higher dimensions with Riesz kernel capacities.

Connected the results to Vitushkin's conjecture on Favard length and analytic capacity.

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 $$ \gamma(E) \geq c\,\frac{\mu(E)^2}{\int_I \|P_\theta\mu\|_2^2\,d\theta}, $$ 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.