Improved Sobolev Inequalities and Muckenhoupt weights on stratified Lie groups

TL;DR

This paper extends Improved Sobolev inequalities with Muckenhoupt weights to stratified Lie groups, introducing new techniques for the critical case p=1 and generalizations involving weak-type Sobolev spaces.

Contribution

It develops novel methods for proving inequalities on stratified Lie groups, especially for the critical case p=1, and introduces a new generalization using weak-type Sobolev spaces.

Findings

Established Improved Sobolev inequalities on stratified Lie groups.

Developed alternative techniques for the critical case p=1.

Generalized inequalities with weak-type Sobolev spaces.

Abstract

We study in this article the Improved Sobolev inequalities with Muckenhoupt weights within the framework of stratified Lie groups. This family of inequalities estimate the Lq norm of a function by the geometric mean of two norms corresponding to Sobolev spaces W(s;p) and Besov spaces B(-b, infty, infty). When the value p which characterizes Sobolev space is strictly larger than 1, the required result is well known in R^n and is classically obtained by a Littlewood-Paley dyadic blocks manipulation. For these inequalities we will develop here another totally different technique. When p = 1, these two techniques are not available anymore and following M. Ledoux in R^n, we will treat here the critical case p = 1 for general stratified Lie groups in a weighted functional space setting. Finally, we will go a step further with a new generalization of Improved Sobolev inequalities using weak-type Sobolev spaces.