On the endpoint regularity of discrete maximal operators

Abstract

Given a discrete function $f:{\mathbb{Z}}^d\to \mathbb{R}$ we consider the maximal operator $$Mf(\vec{n})=\underset{r\ge 0}{\sup }\frac{1}{N(r)}\sum _{\vec{m}\in {\bar{\Omega }}_r}|f(\vec{n}+\vec{m})|,$$ where $\{{\bar{\Omega }}_r{\}}_{r\ge 0}$ are dilations of a convex set $\Omega$ (open, bounded and with Lipschitz boudary) containing the origin and $N(r)$ is the number of lattice points inside ${\bar{\Omega }}_r$. We prove here that the operator $f\mapsto \nabla Mf$ is bounded and continuous from $l^1({\mathbb{Z}}^d)$ to $l^1({\mathbb{Z}}^d)$. We also prove the same result for the non-centered version of this discrete maximal operator.

References in corpus (3)

• On the regularity of maximal operators
• On the variation of maximal operators of convolution type
• On regularity of the discrete Hardy-Littlewood maximal function

Cited by in corpus (10)

• Derivative bounds for fractional maximal functions
• On the variation of maximal operators of convolution type
• Endpoint Sobolev and BV continuity for maximal operators
• Regularity of maximal functions on Hardy-Sobolev spaces
• Gradient bounds for radial maximal functions
• Regularity of maximal operators: recent progress and some open problems
• Pseudo-differential operators on ${\mathbb{Z}}^n$ with applications to discrete fractional integral operators
• Regularity and continuity of the multilinear strong maximal operators
• Fefferman-Stein inequalities for the Hardy-Littlewood maximal function on the infinite rooted $k$-ary tree
• The Variation of the Uncentered Maximal Operator with respect to Cubes