Analytic capacity and projections

20 Jan 2019  ·  Chang Alan, Tolsa Xavier ·

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. read more

PDF Abstract
No code implementations yet. Submit your code now


Classical Analysis and ODEs Analysis of PDEs