Vector-valued Riesz potentials: Cartan-type estimates and related capacities

被引：16

作者：

Eiderman, V. ^{[1
]}

Nazarov, F. ^{[2
]}

Volberg, A. ^{[3
]}

机构：

[1] Univ Kentucky, Dept Math, Lexington, KY 40506 USA

[2] Univ Wisconsin, Dept Math, Madison, WI 53706 USA

[3] Michigan State Univ, Dept Math, E Lansing, MI 48823 USA

来源：

PROCEEDINGS OF THE LONDON MATHEMATICAL SOCIETY | 2010年 / 101卷

基金：

美国国家科学基金会;

关键词：

CALDERON-ZYGMUND OPERATORS; ANALYTIC CAPACITY; KERNELS; SETS;

D O I：

10.1112/plms/pdq003

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

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 nu is large. This size is measured by the Hausdorff content with various gauge functions. We begin with the case when nu 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.

引用

页码：727 / 758

页数：32

共 29 条

[1]

Adams D.R., 1996, GRUNDLEHREN MATH WIS, V314

[2] Cauchy transforms of point masses: The logarithmic derivative of polynomials [J].