Littlewood–Paley functions and Sobolev spaces

Abstract

We establish Littlewood–Paley characterizations of the Sobolev spaces Wα,p in Euclidean spaces using several square functions defined via the spherical average, the ball average, the Bochner–Riesz means and some other well known operators. We provide a simple proof so that we are able to extend and improve many results published in recent papers.