# Endpoint Sobolev continuity of the fractional maximal function in higher dimensions

## TL;DR

This paper proves endpoint Sobolev continuity properties of the fractional maximal function in higher dimensions, extending known boundedness results and establishing new continuity mappings for radial and general functions.

## Contribution

It establishes the Sobolev continuity of the fractional maximal operator at the endpoint space in higher dimensions, including radial and general cases, and relates boundedness conjectures to continuity in one dimension.

## Key findings

- Continuity of the fractional maximal operator in Sobolev spaces for radial functions in higher dimensions.
- Extension of results to the centered fractional maximal operator.
- Implication of boundedness conjecture for continuity in one dimension.

## Abstract

We establish continuity mapping properties of the non-centered fractional maximal operator $M_{\beta}$ in the endpoint input space $W^{1,1}(\mathbb{R}^d)$ for $d \geq 2$ in the cases for which its boundedness is known. More precisely, we prove that for $q=d/(d-\beta)$ the map $f \mapsto |\nabla M_\beta f|$ is continuous from $W^{1,1}(\mathbb{R}^d)$ to $L^{q}(\mathbb{R}^d)$ for $ 0 < \beta < 1$ if $f$ is radial and for $1 \leq \beta < d$ for general $f$. The results for $1\leq \beta < d$ extend to the centered counterpart $M_\beta^c$. Moreover, if $d=1$, we show that the conjectured boundedness of that map for $M_\beta^c$ implies its continuity.