Hardy inequalities on constant-order noncommutative Vilenkin groups

TL;DR

This paper extends classical integral inequalities, including Hardy and Hardy-Littlewood-Pólya inequalities, to the setting of noncommutative Vilenkin groups, establishing sharp estimates and functional inequalities for related operators.

Contribution

It introduces new Hardy-type inequalities and functional inequalities for noncommutative Vilenkin groups and graded -Lie groups, linking pseudo-differential operators with these inequalities.

Findings

Established sharp weak and strong type estimates for Hardy operators.

Extended Hardy-Littlewood-Pólya and Sobolev inequalities to noncommutative Vilenkin groups.

Linked pseudo-differential operators with Hardy inequalities.

Abstract

In this note we extend several integral inequalities to the context of noncommutative Vilenkin groups. We prove some sharp weak and strong type estimates for the Hardy operator and the Hardy-Littlewood-P{\'o}lya operator on constant-order noncommutative Vilenkin groups. In particular for graded $\K$-Lie groups, where $\K$ is a non-archimedean local field, we additionally provide some functional inequalities, like the Hardy-Littlewood-Sobolev unequality and the Stein-Weiss inequality, linking some classes of homogeneous pseudo-differential operators, like the Vladimirov-Taibleson operator and the Vladimirov Laplacian, with Hardy inequalities.