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A NOTE ON THE ENDPOINT REGULARITY OF THE HARDY–LITTLEWOOD MAXIMAL FUNCTIONS

Part of: Linear function spaces and their duals Harmonic analysis in several variables

Published online by Cambridge University Press:  11 December 2015

FENG LIU ,
TING CHEN  and
HUOXIONG WU
FENG LIU*
Affiliation:
College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, Shandong 266590, PR China email liufeng860314@163.com
TING CHEN
Affiliation:
School of Mathematics, University of Edinburgh, EH9 3JZ, UK email t.chen-16@sms.ed.ac.uk
HUOXIONG WU
Affiliation:
School of Mathematical Sciences, Xiamen University, Xiamen, Fujian 361005, PR China email huoxwu@xmu.edu.cn
*
✉liufeng860314@163.com
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Abstract

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In this note we give a simple proof of the endpoint regularity for the uncentred Hardy–Littlewood maximal function on $\mathbb{R}$. Our proof is based on identities for the local maximum points of the corresponding maximal functions, which are of interest in their own right.

Keywords

maximal functionSobolev spaceslocal maximumvariation

MSC classification

Primary: 42B25: Maximal functions, Littlewood-Paley theory
Secondary: 46E35: Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 94 , Issue 1 , August 2016 , pp. 121 - 130
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

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