Principal values for the signed Riesz kernels of non-integer dimension
type: Item
instance of: scholarly article
title: Principal values for the signed Riesz kernels of non-integer dimension
zbMATH Open document ID: 1225.42007
DOI: 10.1216/RMJ-2011-41-3-869
author: L. Prat
published in: Rocky Mountain Journal of Mathematics
publication date: 2011-08-02T00:00:00Z
review text: This paper consists of a study of the \(\alpha\)-Riesz transform \(R^\alpha\) of a positive Radon measure \(\mu\) on \(\mathbb R^n\) for \(0<\alpha<1\). If \(\mu\) has compact support, the author first proves that the \(L^2(\mu)\)-boundedness of \(R^\alpha\) implies the \(\mu\)-almost everywhere existence of the principal values of the \(\alpha\)-Riesz transform of \(\mu\). The proof follows some ideas given by \textit{X. Tolsa} in [J. Reine Angew. Math. 502, 199--235 (1998; Zbl 0912.42009)]. Then, the author shows that the \(\mu\)-measure of the set where the principal value of \(R^\alpha\) does not exist is positive when \(\mu\) has a positive and finite upper \(\alpha\)-density \(\mu\)-almost everywhere. This is done by adapting a powerful result of \textit{F. Nazarov, S. Treil} and \textit{A. Volberg} [Acta Math. 190, No. 2, 151--239 (2003; Zbl 1065.42014)] on the \(L^2(\mu)\)-boundedness of singular integrals on non-homogeneous spaces.
reviewed by: Luc Deleaval
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 28A75
zbMATH DE Number: 5934874
zbMATH Keywords: Riesz transform
zbMATH Keywords: principal values
zbMATH Keywords: \(L^2(\mu)\)-boundedness
zbMATH Keywords: symmetrization method
zbMATH Keywords: upper density
MaRDI profile type: Publication
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full work available at URL: rmj-2011-41-3-869
OpenAlex ID: W2069854355
description: scientific article; zbMATH DE number 5934874