z-logo
Premium
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.

This content is not available in your region!

Continue researching here.

Having issues? You can contact us here