Capacities associated with Calder\'on-Zygmund kernels

TL;DR

This paper establishes that capacities linked to certain Calderón-Zygmund kernels in the plane are comparable to classical analytic capacity, extending understanding of capacity measures associated with singular integral kernels.

Contribution

It proves the comparability of capacities related to specific vectorial Calderón-Zygmund kernels with analytic capacity, revealing their equivalence in measuring geometric properties.

Findings

Capacities associated with the kernels are comparable to analytic capacity.

The results extend the understanding of capacities linked to Calderón-Zygmund kernels.

The paper provides a new connection between vectorial kernels and classical capacity measures.

Abstract

Analytic capacity is associated with the Cauchy kernel $1/z$ and the $L^\infty$-norm. For $n\in\mathbb{N}$, one has likewise capacities related to the kernels $K_i(x)=x_i^{2n-1}/|x|^{2n}$, $1\le i\le 2$, $x=(x_1,x_2)\in\mathbb{R}^2$. The main result of this paper states that the capacities associated with the vectorial kernel $(K_1, K_2)$ are comparable to analytic capacity.