IUMJ

Title: Capacities associated with scalar signed Riesz kernels, and analytic capacity

Authors: Joan Mateu, Laura Prat and Joan Verdera

Issue: Volume 60 (2011), Issue 4, 1319-1362

Abstract:

Analytic capacity is associated with the Cauchy kernel $1/z$ and the space $L^{\infty}$. One has likewise capacities associated with the real and imaginary parts of the Cauchy kernel and $L^{\infty}$. Striking results of Tolsa and a simple remark show that these three capacities are comparable. We present an extension of this fact to $\mathbb{R}^n$, $n \geq 3$, involving the vector-valued Riesz kernel of homogeneity $-1$ and $n-1$ of its components.