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arXiv · 1402.3104

Riesz transforms of non-integer homogeneity on uniformly disconnected sets

Maria Carmen Reguera·Xavier Tolsa
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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.

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Maria Carmen Reguera, Xavier Tolsa. 2014-02-13. Riesz transforms of non-integer homogeneity on uniformly disconnected sets. https://arxiv.org/abs/1402.3104

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