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

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.