Riesz transforms of non-integer homogeneity on uniformly disconnected sets

TL;DR

This paper provides precise $L^2$ estimates for Riesz transforms on measures supported on Cantor sets and relates Riesz capacity to nonlinear potential theory capacities for uniformly disconnected sets.

Contribution

It establishes sharp $L^2$ bounds for Riesz transforms on general measures and links Riesz capacity with nonlinear potential theory capacities for uniformly disconnected sets.

Findings

$L^2$ norm estimates for Riesz transforms on Cantor set measures

Comparison between Riesz capacity and nonlinear potential theory capacity

Results applicable to uniformly disconnected compact 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.