L²-norm and estimates from below for Riesz transforms on Cantor sets

2010/12/04 by Vladimir Eiderman, Alexander Volberg, Eiderman, Vladimir +1
Mathematics · #42A61 #42B20 #Classical Analysis and ODEs (math.CA) #F.2.2 #FOS: Mathematics #acm:42A61 #acm:42B20 #math.CA #msc:42A61 #msc:42B20

paper · pdf · doi:10.48550/arxiv.1012.0941

28 pages, 1 figure

arxiv created 2010/12/04 · arxiv updated 2010/12/07

Abstract

The aim of this paper is to estimate the L²-norms of vector-valued Riesz transforms R_ν^s and the norms of Riesz operators on Cantor sets in \R^d, as well as to study the distribution of values of R_ν^s. Namely, we show that this distribution is "uniform" in the following sense. The values of |R_ν^s|² which are comparable with its average value are attended on a "big" portion of a Cantor set. We apply these results to give examples demonstrating the sharpness of our previous estimates for the set of points where Riesz transform is large, and for the corresponding Riesz capacities. The Cantor sets under consideration are different from the usual corner Cantor sets. They are constructed by means a certain process of regularization introduced in the paper.

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