Capacities associated with Calder\'on-Zygmund kernels

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