Endpoint Sobolev and BV Continuity for maximal operators, II

TL;DR

This paper investigates the continuity properties of classical and fractional maximal operators in Sobolev and BV spaces, providing positive answers to recent open questions in both continuous and discrete settings.

Contribution

It establishes the continuity of the fractional maximal operator's derivative in Sobolev spaces and the discrete centered maximal operator in BV spaces, advancing understanding of these operators.

Findings

Proves continuity of the fractional maximal operator's derivative from W^{1,1} to L^q.

Shows continuity of the discrete centered maximal operator in BV(Z).

Complements recent boundedness results with new continuity insights.

Abstract

In this paper we study some questions about the continuity of classical and fractional maximal operators in the Sobolev space $W^{1,1}$, in both continuous and discrete setting, giving a positive answer to two questions posed recently, one of them regarding the continuity of the map $f \mapsto \big(\widetilde M_{\beta}f\big)'$ from $W^{1,1}(\mathbb{R})$ to $L^q(\mathbb{R})$, for $q=\frac{1}{1-\beta}$. Here $\widetilde M_{\beta}$ denotes the non-centered fractional maximal operator on $\mathbb{R}$ with $\beta\in(0,1)$. The second one regarding the continuity of the discrete centered maximal operator in the space of functions of bounded variation BV$(\mathbb{Z})$, complementing some recent boundedness results.