Weak differentiability for fractional maximal functions of general $L^{p}$ functions on domains

TL;DR

This paper establishes new regularity results for fractional maximal functions on bounded domains, showing they map $L^p$ functions to Sobolev spaces for a range of $p$, including previously unknown cases.

Contribution

It proves that fractional maximal operators on bounded domains are weakly differentiable for all $p > 1$ when the smoothness index is at least 1, extending known results.

Findings

Fractional maximal operators map $L^p$ to $W^{1,p}$ for $p > 1$ on bounded domains.

Results include the previously unknown range $p o (1, n/(n-1)]$.

Endpoint regularity in domain setting is established.

Abstract

Let $\Omega \subset \mathbb{R}^{n}$ be bounded a domain. We prove under certain structural assumptions that the fractional maximal operator relative to $\Omega$ maps $L^{p}(\Omega) \to W^{1,p}(\Omega)$ for all $p > 1$, when the smoothness index $\alpha \geq 1$. In particular, the results are valid in the range $p \in (1, n/(n-1)]$ that was previously unknown. As an application, we prove an endpoint regularity result in the domain setting.