Logarithmic Hardy-Rellich inequalities on Lie groups

TL;DR

This paper establishes new logarithmic Hardy-Rellich inequalities on Lie groups, including refinements for graded groups and applications to stratified groups, extending known results to broader geometric contexts.

Contribution

It introduces a family of weighted logarithmic Hardy-Rellich inequalities on Lie groups, with refinements for graded groups and applications to fractional operators, advancing the theoretical framework.

Findings

Derived logarithmic Hardy-Rellich inequalities on Lie groups.

Refined inequalities for graded and stratified groups using Sobolev norms.

Extended inequalities to fractional p-sub-Laplacian on homogeneous groups.

Abstract

In this paper we obtain logarithmic Hardy and Rellich inequalities on general Lie groups. In the case of graded groups, we also show their refinements using the homogeneous Sobolev norms. In fact, we derive a family of weighted logarithmic Hardy-Rellich inequalities, for which logarithmic Hardy and Rellich inequalities are special cases. As a consequence of these inequalities, we also derive a Gross type logarithmic Hardy inequality on general stratified groups. An interesting feature of such estimate is that we consider the measure which is Gaussian only on the first stratum of the group. Such choice of the measure is natural in view of the known Gross type logarithmic Sobolev inequalities on stratified groups. The obtained results are new already in the setting of the Euclidean space $\mathbb R^n.$ Finally, we also present a simple argument for getting a logarithmic Poincar\'e inequality, as well as the logarithmic Hardy inequality for the fractional $p$-sub-Laplacian on homogeneous groups.