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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
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