On the endpoint regularity of discrete maximal operators

TL;DR

This paper proves that the discrete maximal operator associated with convex sets is bounded and continuous from l^1 to l^1, including its non-centered version, establishing endpoint regularity results.

Contribution

It establishes the boundedness and continuity of the gradient of the discrete maximal operator on l^1 spaces, including non-centered variants, which was previously unknown.

Findings

Boundedness of the gradient of the maximal operator on l^1

Continuity of the operator from l^1 to l^1

Results hold for both centered and non-centered operators

Abstract

Given a discrete function $f:\Z^d \to \R$ we consider the maximal operator $$Mf(\vec{n}) = \sup_{r\geq0} \frac{1}{N(r)} \sum_{\vec{m} \in \bar{\Omega}_r} \big|f(\vec{n} + \vec{m})\big|,$$ where $\big\{\bar{\Omega}_r\big\}_{r \geq 0}$ 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 $f \mapsto \nabla M f$ is bounded and continuous from $l^1(\Z^d)$ to $l^1(\Z^d)$. We also prove the same result for the non-centered version of this discrete maximal operator.