Non existence of principal values of signed Riesz transforms of non integer dimension

2008/12/12 by Aleix Ruiz de Villa, Xavier Tolsa, de Villa, Aleix Ruiz +1
Mathematics · #28A75 #31B10 #42B25 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #math.CA #math.FA #msc:28A75 #msc:31B10 #msc:42B25

paper · pdf · doi:10.48550/arxiv.0812.2421

arxiv created 2008/12/12 · arxiv updated 2009/12/01

Abstract

In this paper we prove that, given s> 0, if E is a subset of R^m with positive and bounded s-dimensional Hausdorff measure H^s and the principal values of the s-dimensional signed Riesz transform of H^s|E exist H^s-almost everywhere in E, then s is integer. Other more general variants of this result are also proven.

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