On the endpoint regularity of discrete maximal operators

By Emanuel Carneiro, Kevin A. Hughes. 2013

56 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/057bc1d6925dec585667d0cda45226bd0c8fcc68

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. Jos'e Madrid
2. Emanuel Carneiro
3. Shuanglin Shao
4. Aynur Bulut
5. William Beckner

In Computational Theory and Mathematics

1. Jos'e Madrid
2. Emanuel Carneiro

Citations in the corpus

• Derivative bounds for fractional maximal functions (2015, background): “8The local extrema are defined slightly differently in [4, 6], but used with the meaning stated here.”