Mathematical Research Letters

Volume 30 (2023)

Number 3

Continuity of the gradient of the fractional maximal operator on $W^{1,1} (\mathbb{R}^d)$

Pages: 689 – 707

DOI: https://dx.doi.org/10.4310/MRL.2023.v30.n3.a3

Authors

David Beltran (Departament d’Anàlisi Matemàtica, Universitat de València, Spain)

Cristian González-Riquelme (Departamento de matemática, Instituto Superior Técnico, Lisboa, Portugal)

José Madrid (Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, Va., U.S.A.)

Julian Weigt (Mathematics Institute, University of Warwick, Coventry, United Kingdom)

Abstract

We establish that the map $f \mapsto {\lvert \nabla \mathcal{M}_\alpha f \rvert}$ is continuous from $W^{1,1} (\mathbb{R}^d)$ to $L^q (\mathbb{R}^d)$, where $\alpha \in (0, d), q = \frac{d}{d-\alpha}$ and $M_\alpha$ denotes either the centered or non-centered fractional Hardy–Littlewood maximal operator. In particular, we cover the cases $d \gt 1$ and $\alpha \in (0, 1)$ in full generality, for which results were only known for radial functions.

Received 3 March 2021

Accepted 1 November 2021

Published 15 December 2023