Functional Inequalities in Stratified Lie groups with Sobolev, Besov, Lorentz and Morrey spaces

TL;DR

This paper investigates refined Sobolev, Gagliardo-Nirenberg, and Morrey-Sobolev inequalities within stratified Lie groups, extending classical results to more general function spaces like Lorentz and Morrey spaces.

Contribution

It introduces generalized inequalities in stratified Lie groups using Lorentz and Morrey spaces, covering both p=1 and p>1 cases with new relaxed and refined results.

Findings

Refined Sobolev inequalities for p=1 in stratified Lie groups.

Gagliardo-Nirenberg inequalities for p>1 with Lorentz spaces.

Extension to Morrey-Sobolev inequalities in the same setting.

Abstract

The study of Sobolev inequalities can be divided in two cases: p = 1 and 1 < p < +$\infty$. In the case p = 1 we study here a relaxed version of refined Sobolev inequalities. When p > 1, using as base space classical Lorentz spaces associated to a weight from the Arino-Muckenhoupt class Bp, we will study Gagliardo-Nirenberg inequalities. As a by-product we will also consider Morrey-Sobolev inequalities. These arguments can be generalized to many different frameworks, in particular the proofs are given in the setting of stratified Lie groups.