Local maximal operators on fractional Sobolev spaces

TL;DR

This paper investigates the boundedness of local maximal operators on fractional Sobolev spaces and characterizes fractional Hardy inequalities using localized testing conditions, advancing understanding of fractional analysis on open sets.

Contribution

It establishes boundedness criteria for local maximal operators on fractional Sobolev spaces and links fractional Hardy inequalities to localized testing conditions.

Findings

Boundedness of local maximal operators on $W^{s,p}(G)$ established.

Fractional $(s,p)$-Hardy inequality characterized via Whitney cube testing.

Provides new tools for fractional Sobolev space analysis on open sets.

Abstract

In this note we establish the boundedness properties of local maximal operators $M_G$ on the fractional Sobolev spaces $W^{s,p}(G)$ whenever $G$ is an open set in $\mathbb{R}^n$, $0<s<1$ and $1<p<\infty$. As an application, we characterize the fractional $(s,p)$-Hardy inequality on a bounded open set $G$ by a Maz'ya-type testing condition localized to Whitney cubes.