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A REMARK ON THE REGULARITY OF THE DISCRETE MAXIMAL OPERATOR

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

Published online by Cambridge University Press:  01 December 2016

FENG LIU
FENG LIU*
Affiliation:
College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, Shandong 266590, PR China email liufeng860314@163.com
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Abstract

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We study the regularity properties of several classes of discrete maximal operators acting on $\text{BV}(\mathbb{Z})$ functions or $\ell ^{1}(\mathbb{Z})$ functions. We establish sharp bounds and continuity for the derivative of these discrete maximal functions, in both the centred and uncentred versions. As an immediate consequence, we obtain sharp bounds and continuity for the discrete fractional maximal operators from $\ell ^{1}(\mathbb{Z})$ to $\text{BV}(\mathbb{Z})$.

Keywords

discrete maximal operatorfractional maximal operatorbounded variationcontinuity

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 95 , Issue 1 , February 2017 , pp. 108 - 120
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

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