Riesz transforms of noninteger 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 $d1<s<d$. 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 $\dot{C}_{\frac{2}{3}(ds),\frac{3}{2}}$ from nonlinear potential theory.
 Publication:

arXiv eprints
 Pub Date:
 February 2014
 arXiv:
 arXiv:1402.3104
 Bibcode:
 2014arXiv1402.3104R
 Keywords:

 Mathematics  Classical Analysis and ODEs;
 Mathematics  Analysis of PDEs;
 31C45;
 42B20
 EPrint:
 Minor corrections