On the endpoint regularity of discrete maximal 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

On the variation of maximal operators of convolution type II

“…42B25, 46E35, 35B50, 31B05, 35J05, 35K08.holds for the one-dimensional uncentered version of this operator, as proved by Aldaz and Pérez Lázaro [1].Higher dimensional analogues of (1.2) for the Hardy-Littlewood maximal operator, centered or uncentered, are open problems (see, for instance, the work of Haj lasz and Onninen [9]). Other interesting works related to the regularity of the Hardy-Littlewood maximal operator and its variants, when applied to Sobolev and BV functions,are [2,3,4,5,8,10,11,12,14,15,20,22].In the precursor of this work [6, Theorems 1 and 2], Carneiro and Svaiter proved the variation-diminishing property, i.e. inequality (1.2) with C = 1, for the maximal operators associated to the Poisson kernel(1.3) and the Gauss kernelTheir proof is based on an interplay between the analysis of the maximal functions and the structure of the underlying partial differential equations (Laplace's equation and heat equation).…”

Regularity of Maximal Operators: Recent Progress and Some Open Problems

“…Then, in the instances where we assumed a W 1,1 (R)-convergence in the continuous setting, we will be assuming an ℓ 1 (Z)-convergence in the discrete setting. As a particular case of the general framework of Theorem 6.4 above (which is [9, Theorem 3]) we see that the maps f → (M β f ) ′ and f → ( M β f ) ′ are continuous from ℓ 1 (Z) to ℓ q (Z) for 0 ≤ β < 1 and q = 1/(1 − β) (the case β = 0 of these results had previously been obtained in [8]). Therefore, we have an affirmative answer for the discrete analogues of Theorems 7.2 and 7.3 and Questions 12 and 15.…”

Geometric properties of infinite graphs and the Hardy–Littlewood maximal operator

“…Note that for j ≥ n + 1, we have 2 j−1 ≤ 2 j − 2 n , so we get that (6) |B((2 n , 0), r j )| ≤ 4r j .…”