Riesz transforms of non-integer homogeneity on uniformly disconnected sets

In this paper we obtain precise estimates for the L2 norm of the s-dimensional Riesz transforms on very general measures supported on Cantor sets in Rd, with d − 1 <s<d. From these estimates we infer that, for the so-called uniformly disconnected compact sets, the capacity γs associated with the Riesz kernel x/|x| s+1 is comparable to the capacity C˙ 2 3 (d−s), 3 2 from non-linear potential theory.