A class of bounded operators on Sobolev spaces
type: Item
instance of: scholarly article
title: A class of bounded operators on Sobolev spaces
zbMATH Open document ID: 1061.46034
author: Soulaymane Korry
published in: Archiv der Mathematik
publication date: 2004-09-22T00:00:00Z
review text: A result of J. Kinnunen states that the Hardy-Littlewood maximal operator is bounded on \(H^1_p({\mathbb R}^n)\) whenever \(1< p < \infty\). The main purpose of this paper is to extend this result by showing that the Hardy-Littlewood maximal operator is actually bounded on \(H^s_p({\mathbb R}^n)\) whenever \(1< p < \infty\) and \(0<s\leq 1\). For this purpose, the author introduces a class \(C\) of nonlinear bounded operators on \(L^r({\mathbb R}^n)\) for some \(1< r< \infty\) satisfying some additional conditions, and shows that each \(T\in C\) defines a bounded operator on the Sobolev spaces \(H^s_p({\mathbb R}^n)\) whenever \(1< p < \infty\) and \(0<s\leq 1\). The author obtains the main conclusion by showing that the Hardy-Littlewood maximal operator is in fact in \(C\).
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 42B25
zbMATH DE Number: 2101539
zbMATH Keywords: Hardy-Littlewood maximal operator
reviewed by: Shangquan Bu
MaRDI profile type: Publication
full work available at URL: s00013-003-0416-x
OpenAlex ID: W1989489546
DOI: 10.1007/S00013-003-0416-X
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description: scientific article; zbMATH DE number 2101539