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

Liu, Feng, and Mao, Suzhen. "On the regularity of the one-sided Hardy-Littlewood maximal functions." Czechoslovak Mathematical Journal 67.1 (2017): 219-234. <http://eudml.org/doc/287907>.

@article{Liu2017,
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, \[\{\rm Var\}(M^\{+\}(f))\le \{\rm Var\}(f)\quad \text\{and\}\quad \{\rm Var\}(M^\{-\}(f))\le \{\rm Var\}(f)\] 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 $f\colon \mathbb \{Z\}\rightarrow \mathbb \{R\}$ satisfying $\{\rm Var\}(f)<\infty $.},
author = {Liu, Feng, Mao, Suzhen},
journal = {Czechoslovak Mathematical Journal},
keywords = {one-sided maximal operator; Sobolev space; bounded variation; continuity},
language = {eng},
number = {1},
pages = {219-234},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the regularity of the one-sided Hardy-Littlewood maximal functions},
url = {http://eudml.org/doc/287907},
volume = {67},
year = {2017},
}

TY - JOUR
AU - Liu, Feng
AU - Mao, Suzhen
TI - On the regularity of the one-sided Hardy-Littlewood maximal functions
JO - Czechoslovak Mathematical Journal
PY - 2017
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 67
IS - 1
SP - 219
EP - 234
AB - 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, \[{\rm Var}(M^{+}(f))\le {\rm Var}(f)\quad \text{and}\quad {\rm Var}(M^{-}(f))\le {\rm Var}(f)\] 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 $f\colon \mathbb {Z}\rightarrow \mathbb {R}$ satisfying ${\rm Var}(f)<\infty $.
LA - eng
KW - one-sided maximal operator; Sobolev space; bounded variation; continuity
UR - http://eudml.org/doc/287907
ER -