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Keywords:
one-sided maximal operator; Sobolev space; bounded variation; continuity
one-sided maximal operator; Sobolev space; bounded variation; continuity
Summary:
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 $f\colon \mathbb {Z}\rightarrow \mathbb {R}$ satisfying ${\rm Var}(f)<\infty $.
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