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

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.

arXiv comment29 pages, Submitted