Remarks on the Hardy type inequalities with remainder terms in the framework of equalities

TL;DR

This paper explores Hardy type inequalities through equalities in Hilbert spaces, providing new characterizations and insights into the structure of these inequalities and the conditions under which extremizers do not exist.

Contribution

It introduces a framework of equalities that derive Hardy inequalities and characterizes functions for which the remainder term vanishes, using orthogonality in Hilbert spaces.

Findings

Equalities imply Hardy inequalities by removing the remainder term.

Characterization of functions with vanishing remainder term.

Nonexistence of nontrivial extremizers for these inequalities.

Abstract

We study the Hardy type inequalities in the framework of equalities. We present equalities which immediately imply Hardy type inequalities by dropping the remainder term. Simultaneously we give a characterization of the class of functions which makes the remainder term vanish. A point of our observation is to apply an orthogonality properties in general Hilbert space, and which gives a simple and direct understanding of the Hardy type inequalities as well as the nonexistence of nontrivial extremizers.