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

TL;DR

This paper proves the continuity of the gradient of the fractional maximal operator on Sobolev space $W^{1,1}$ in $R^d$, extending known results beyond radial functions for certain parameters.

Contribution

It establishes the continuity of the map $f o | abla M_a f|$ from $W^{1,1}(R^d)$ to $L^q(R^d)$ for fractional maximal operators, generalizing previous radial-only results.

Findings

Proves continuity of the gradient map in Sobolev spaces.

Extends results to non-radial functions for $d>1$ and $a o 1$.

Applicable to both centered and non-centered fractional maximal operators.

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.