Hardy and Rellich inequalities, identities, and sharp remainders on homogeneous groups

Durvudkhan Suragan,Michael Ruzhansky

doi:10.1016/j.aim.2017.07.020

Abstract

We give sharp remainder terms of Lp and weighted Hardy and Rellich inequalities on one of most general subclasses of nilpotent Lie groups, namely the class of homogeneous groups. As consequences, we obtain analogues of the generalised classical Hardy and Rellich inequalities and the uncertainty principle on homogeneous groups. We also prove higher order inequalities of Hardy–Rellich type, all with sharp constants. A number of identities are derived including weighted and higher order types.

Highlights

In this paper we are interested in Hardy, Rellich, and higher order inequalities of Hardy–Rellich type in the setting of general homogeneous groups
For p = 2, we can iterate the exact representation formulae for the remainder that we obtained for the Rellich inequality and for the weighted Hardy inequalities
We present the Lp-Hardy inequality and the remainder formula on the homogeneous group G

Summary

Introduction

In this paper we are interested in Hardy, Rellich, and higher order inequalities of Hardy–Rellich type in the setting of general homogeneous groups. We are interested in questions of best constants, their attainability, and sharp expressions for the remainders

Hardy inequalities

Rellich inequalities

Higher order Hardy–Rellich inequalities

Review of results

Preliminaries

Lp-Hardy inequality and uncertainty principle

Weighted Hardy inequalities

Rellich inequality