Riesz transforms of non-integer homogeneity on uniformly disconnected sets

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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.

Bibliographic note

Minor corrections


Original languageEnglish
Pages (from-to)7045–7095
JournalTransactions of the American Mathematical Society
Issue number10
Early online date10 Feb 2016
Publication statusPublished - Oct 2016


  • math.CA, math.AP