Rough maximal functions and rough singular integral operators applied to integrable radial functions
type: Item
instance of: scholarly article
title: Rough maximal functions and rough singular integral operators applied to integrable radial functions
zbMATH Open document ID: 0880.42010
DOI: 10.4171/RMI/216
author: Peter Sjögren
author: F. Soria
published in: Revista Matemática Iberoamericana
publication date: 1998-02-08T00:00:00Z
full work available at URL: https://eudml.org/doc/39531
review text: Consider the maximal operator \[ M_\Omega f(x)= \sup_{R>0} R^{-n}\int|\Omega(y') f(x-y)|dy \] and the singular integral operator \[ T_\Omega f(x)= \int\Omega(y')|y|^{-n} f(x-y)dy, \] where \(\Omega\in L^1(S^{n-1})\) is extended to a function homogeneous of degree 0; in the definition of \(T_\Omega\) one assumes cancellation: \(\int\Omega(y') d\sigma(y')=0\). It is an open question whether \(M_\Omega\) (or \(T_\Omega\) for odd \(\Omega\)) is of weak type (1,1) although this has been proved under the stronger assumption \(\Omega\in L\log L\). In this interesting paper, the authors prove that \(M_\Omega\) and \(T_\Omega\) are of weak (1,1) on \textit{radial} functions in \(\mathbb{R}^n\), where \(\Omega\) is merely assumed to be in \(L^1(S^{n-1})\). The assumption that \(\Omega\) be odd is not needed in the result for radial functions. The authors also prove the almost everywhere existence of the principal value integral, for radial \(f\in L^1\) by establishing the weak-type (1,1) estimate for the corresponding maximal singular integral (the corresponding question for general \(f\) seems to be open even under the assumption \(\Omega\in L^\infty\)). The proof is quite delicate; it partially relies on methods developed by the authors in a previous paper [Isr. J. Math. 95, 211-229 (1996)].
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 42B25
zbMATH DE Number: 1052954
zbMATH Keywords: maximal operator
zbMATH Keywords: singular integral operator
zbMATH Keywords: weak type (1,1)
reviewed by: Andreas Seeger
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1052954