Czechoslovak Mathematical Journal, Vol. 67, No. 1, pp. 219-234, 2017

On the regularity of the one-sided Hardy-Littlewood maximal functions

Feng Liu, Suzhen Mao

Received September 2, 2015.   First published February 24, 2017.

Feng Liu (corresponding author), College of Mathematics and Systems Science, Shandong University of Science and Technology, No. 579 Qianwangang Road Economic and Technical Development Zone, Qingdao 266590, Shandong, P. R. China, e-mail: liufeng860314@163.com; Suzhen Mao, School of Sciences, Nanchang Institute of Technology, No. 289 Tianxiang Road, Nanchang 330099, Jiangxi, P. R. China, e-mail: suzhen.860606@163.com

Abstract: In this paper we study the regularity properties of the one-dimensional one-sided Hardy-Littlewood maximal operators $\mathcal{M}^+$ and $\mathcal{M}^-$. More precisely, we prove that $\mathcal{M}^+$ and $\mathcal{M}^-$ map $W^{1,p}(\mathbb{R})\rightarrow W^{1,p}(\mathbb{R})$ with $1<p<\infty$, boundedly and continuously. In addition, we show that the discrete versions $M^+$ and $M^-$ map ${\rm BV}(\mathbb{Z})\rightarrow{\rm BV}(\mathbb{Z})$ boundedly and map $l^1(\mathbb{Z})\rightarrow{\rm BV}(\mathbb{Z})$ continuously. Specially, we obtain the sharp variation inequalities of $M^+$ and $M^-$, that is, {\rm Var}(M^+(f))\leq{\rm Var}(f)\quad\text{and}\quad{\rm Var}(M^-(f))\leq{\rm Var}(f) if $f\in{\rm BV}(\mathbb{Z})$, where ${\rm Var}(f)$ is the total variation of $f$ on $\mathbb{Z}$ and ${\rm BV}(\mathbb{Z})$ is the set of all functions $f \mathbb{Z}\rightarrow\mathbb{R}$ satisfying ${\rm Var}(f)<\infty$.

Keywords: one-sided maximal operator; Sobolev space; bounded variation; continuity

Classification (MSC 2010): 42B25, 46E35

DOI: 10.21136/CMJ.2017.0475-15

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