Remarks on the Rellich inequality

TL;DR

This paper offers a new perspective on Rellich inequalities by deriving them from equalities, clarifying their structure and explaining why nontrivial extremisers do not exist.

Contribution

It introduces equalities that imply Rellich inequalities when remainders are omitted, providing a clearer understanding of their mathematical foundation.

Findings

Rellich inequalities can be derived from specific equalities.

Nonexistence of nontrivial extremisers is explained.

Provides a straightforward approach to understanding Rellich inequalities.

Abstract

We study the Rellich inequalities in the framework of equalities. We present equalities which imply the Rellich inequalities by dropping remainders. This provides a simple and direct understanding of the Rellich inequalities as well as the nonexistence of nontrivial extremisers.