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Vector‐valued Riesz potentials: Cartan‐type estimates and related capacities
Author(s) -
Eiderman V.,
Nazarov F.,
Volberg A.
Publication year - 2010
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/pdq003
Subject(s) - mathematics , gauge (firearms) , type (biology) , hausdorff measure , infinity , pure mathematics , measure (data warehouse) , riesz potential , borel set , argument (complex analysis) , hausdorff space , borel measure , countable set , point (geometry) , mathematical analysis , hausdorff dimension , probability measure , geometry , ecology , history , biochemistry , chemistry , archaeology , database , computer science , biology
We give sharp upper bounds for the size of the set of points in ℝ d , with d ⩾ 1, where the Riesz transform of a Borel measure ν is large. This size is measured by the Hausdorff content with various gauge functions. We begin with the case when ν is a linear combination of N point masses. Among other things, we characterize all gauge functions for which the estimates do not blow up as N tends to infinity. In this case a routine limiting argument allows us to extend our bounds to all finite Borel measures. We also show how our techniques can be applied to estimates for certain capacities.