On the endpoint regularity of discrete maximal operators

Given a discrete function $f:\mathbb{Z}^d \to \mathbb{R}$, we consider the maximal operator\[Mf(\vec{n}) = \sup_{r\geq0} \frac{1}{N(r)} \sum_{\vec{m} \in \overline{\Omega}_r} \big|f(\vec{n} + \vec{m})\big|,\]where $\big\{\overline{\Omega}_r\big\}_{r \geq 0}$ are dilations of a convex set $\Omega$ (open, bounded and with Lipschitz boundary) containing the origin and $N(r)$ is the number of lattice points inside $\overline{\Omega}_r$. We prove here that the operator $f \mapsto \nabla M f$ 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.

Keywords

discrete maximal operators, Hardy–Littlewood maximal operator, Sobolev spaces, bounded variation

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Published 18 July 2013