Capacities associated with Calderón-Zygmund kernels

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