Riesz transforms of non-integer homogeneity on uniformly disconnected sets

Abstract

In this paper we obtain precise estimates for the $L^2$ norm of the $s$-dimensional Riesz transforms on very general measures supported on Cantor sets in ${\mathbb{R}}^d$, with $d-1<s<d$. From these estimates we infer that, for the so called uniformly disconnected compact sets, the capacity ${\gamma }_s$ associated with the Riesz kernel $x/|x{|}^{s+1}$ is comparable to the capacity ${\dot{C}}_{\frac{2}{3}(d-s),\frac{3}{2}}$ from non-linear potential theory.

Minor corrections