Operators interpolating between Riesz potentials and maximal operators
type: Item
instance of: scholarly article
title: Operators interpolating between Riesz potentials and maximal operators
zbMATH Open document ID: 0688.42018
author: Daniel M. Oberlin
published in: Illinois Journal of Mathematics
publication date: 1989-01-01T00:00:00Z
review text: Let \(\lambda\) be normalized Lebesgue measure on either the unit ball or the unit sphere in \({\mathbb{R}}^ n\) and \(\lambda_ r\) denote the dilate of \(\lambda\). For a continuous function f with compact support, define in the unit ball case \[ S_{p,q,s}f(x)=[\int^{\infty}_{0}| r^{n/p-n/q}\lambda_ r*f(x)|^ s dr/r]^{1/s}, \] \[ S_{p,q,\infty}f(x)=\sup_{r>0}r^{n/p-n/q}| \lambda_ r*f(x)|. \] In the unit sphere case, define operators \(T_{p,q,s}\) and \(T_{p,q,\infty}\) analogously. For nonnegative f, both \(S_{p,q,1}\) and \(T_{p,q,1}\) are multiples of the Riesz potential \(I_{\alpha}(f)\) when \(\alpha =n/p-n/q\). On the other hand, \(S_{p,q,\infty}\) and \(T_{p,q,\infty}\) are maximal operators, especially \(S_{p,p,\infty}\) is the Hardy-Littlewood maximal operator and \(T_{p,p,\infty}\) is the spherical maximal operator. So, \(S_{p,q,s}'s\) \((T_{p,q,s}'s)\) are operators interpolating between Riesz potentials and maximal operators. The author gives a necessary and sufficient condition for \(S_{p,q,s}\) to be bounded from \(L^ p\) to \(L^ q\). As for the boundedness of \(T_{p,q,s}\), he gives several necessary conditions and sufficient ones.
Mathematics Subject Classification ID: 42B25
Mathematics Subject Classification ID: 42B15
Mathematics Subject Classification ID: 41A05
zbMATH DE Number: 4127852
zbMATH Keywords: Riesz potential
zbMATH Keywords: Hardy-Littlewood maximal operator
zbMATH Keywords: spherical maximal operator
reviewed by: Kôzô Yabuta
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4127852