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

Liu, Feng ; Mao, Suzhen

Czechoslovak Mathematical Journal, Tome 67 (2017) no. 1, pp. 219-234

Résumé

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 $1$, 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 $.

Détail

DOI : 10.21136/CMJ.2017.0475-15

Classification : 42B25, 46E35
Keywords: one-sided maximal operator; Sobolev space; bounded variation; continuity