Mean Value Inequalities and Characterizations of Sobolev spaces on graded groups

Abstract

We develop characterizations for Sobolev spaces of potential type on graded Lie groups, by means of Littlewood-Paley square functions, and Strichartz functionals involving second-order differences. A key role is played by some mean value inequalities that may be of independent interest. Indeed, the point-wise first-order inequality is generalized in two directions: on $L^p$-spaces and by considering second-order differences.

We extended the abstract. We corrected a definition of Sobolev spaces defined by derivatives ${\mathbb{W}}_{p;k}(\mathbb{G})$ on page 1 (suitable for Theorem 3.6). We added the corresponding reference [R Burns. "Sobolev spaces on Lie groups". PhD thesis. The Australian National University, 1991. The definition of inhomogeneous Sobolev spaces of potential type remains the same