Generalized Littlewood-Paley characterizations of fractional Sobolev spaces

In this paper, the authors characterize the Sobolev spaces W-alpha,W-p(R-n) with alpha is an element of(0, 2] and p is an element of(max{1, 2n/2 alpha+n},infinity) via a generalized Lusin area function and its corresponding Littlewood-Paley g(lambda)*-function. The range p is an element of(max{1, 2n/2 alpha+n},infinity) is also proved to be nearly sharp in the sense that these new characterizations are not true when 2n/2 alpha+n > 1 and p is an element of(1, 2n/2 alpha+n). Moreover, in the endpoint case p = 2n/2 alpha+n, the authors also obtain some weak type estimates. Since these generalized Littlewood-Paley functions are of wide generality, these results provide some new choices for introducing the notions of fractional Sobolev spaces on metric measure spaces.