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arXiv · 2102.10206

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

David Beltran·Cristian González-Riquelme·José Madrid·Julian Weigt
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Abstract

We establish that the map $f\mapsto |\nabla \mathcal{M}_{\alpha}f|$ 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 $\mathcal{M}_{\alpha}$ denotes either the centered or non-centered fractional Hardy--Littlewood maximal operator. In particular, we cover the cases $d >1$ and $\alpha \in (0,1)$ in full generality, for which results were only known for radial functions.

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David Beltran, Cristian González-Riquelme, José Madrid, Julian Weigt. 2021-02-19. Continuity of the gradient of the fractional maximal operator on $W^{1,1}(\mathbb{R}^d)$. https://arxiv.org/abs/2102.10206

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