Littlewood-Paley Characterizations of Fractional Sobolev Spaces via Averages on Balls

TL;DR

This paper provides new characterizations of fractional Sobolev spaces using Littlewood-Paley functions and ball averages, establishing near-sharp conditions on the integrability parameter p.

Contribution

It introduces novel Littlewood-Paley characterizations of fractional Sobolev spaces via ball averages, valid for specific p ranges, with implications for metric measure spaces.

Findings

Characterizations hold for p in ( ext{max}igrace{1, rac{2n}{2 ext{α}+n}igrace}), nearly sharp.

Characterizations fail outside the specified p range.

Provides a new approach to defining fractional Sobolev spaces on metric measure spaces.

Abstract

In this paper, the authors characterize Sobolev spaces $W^{\alpha,p}({\mathbb R}^n)$ with the smoothness order $\alpha\in(0,2]$ and $p\in(\max\{1, \frac{2n}{2\alpha+n}\},\infty)$, via the Lusin area function and the Littlewood-Paley $g_\lambda^\ast$-function in terms of centered ball averages. The authors also show that the condition $p\in(\max\{1, \frac{2n}{2\alpha+n}\},\infty)$ is nearly sharp in the sense that these characterizations are no longer true when $p\in (1,\max\{1, \frac{2n}{2\alpha+n}\})$. These characterizations provide a new possible way to introduce fractional Sobolev spaces with smoothness order in $(1,2]$ on metric measure spaces.