Mean Value Inequalities and Characterizations of Sobolev spaces on graded groups

TL;DR

This paper introduces new characterizations of Sobolev spaces on graded Lie groups using Littlewood-Paley and Strichartz functionals, emphasizing mean value inequalities that extend to $L^p$ spaces and second-order differences.

Contribution

It develops novel point-wise mean value inequalities and characterizations of Sobolev spaces on graded Lie groups, expanding existing methods with second-order differences.

Findings

Mean value inequalities generalized to $L^p$-spaces.

Characterizations of Sobolev spaces via Littlewood-Paley and Strichartz functionals.

New inequalities of independent interest for analysis 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.