A note on stability of Hardy inequalities

TL;DR

This paper extends stability results for Hardy inequalities to the setting of homogeneous groups, providing remainder estimates for Rellich inequalities and emphasizing the applicability to any homogeneous quasi-norm.

Contribution

It reformulates Hardy inequality stability results within the framework of homogeneous groups, broadening their scope beyond Euclidean spaces.

Findings

Stability estimates hold for any homogeneous quasi-norm.

Remainder estimates are obtained for Rellich inequalities.

Results generalize Euclidean stability results to homogeneous groups.

Abstract

In this note we formulate recent stability results for Hardy inequalities in the language of Folland and Stein's homogeneous groups. Consequently, we obtain remainder estimates for Rellich type inequalities on homogeneous groups. Main differences from the Euclidean results are that the obtained stability estimates hold for any homogeneous quasi-norm.