Hardy-Littlewood-Sobolev and Stein-Weiss inequalities on homogeneous Lie groups

TL;DR

This paper extends the Stein-Weiss and Hardy-Littlewood-Sobolev inequalities to general homogeneous Lie groups, utilizing properties of homogeneous norms, and provides simplified proofs for these classical inequalities in this broader context.

Contribution

It generalizes key integral inequalities to homogeneous Lie groups and offers simplified proofs, expanding their applicability beyond Euclidean spaces.

Findings

Stein-Weiss inequality proven on homogeneous Lie groups

Hardy-Littlewood-Sobolev inequality simplified for these groups

Extensions of classical inequalities to non-commutative settings

Abstract

In this note we prove the Stein-Weiss inequality on general homogeneous Lie groups. The obtained results extend previously known inequalities. Special properties of homogeneous norms play a key role in our proofs. Also, we give a simple proof of the Hardy-Littlewood-Sobolev inequality on general homogeneous Lie groups.