Improved Hardy and Rellich inequalities on nonreversible Finsler manifolds

TL;DR

This paper investigates sharp Hardy and Rellich inequalities on nonreversible Finsler manifolds, providing refined inequalities with remainder terms under specific curvature conditions, applicable also to reversible metrics.

Contribution

It establishes globally refined Hardy and Rellich inequalities with remainder terms on nonreversible Finsler manifolds, extending previous results to broader curvature conditions and reversible cases.

Findings

Sharp constants for inequalities derived

Refined inequalities with remainder terms established

Results valid under specific curvature and reversibility conditions

Abstract

In this paper, we study the sharp constants of quantitative Hardy and Rellich inequalities on nonreversible Finsler manifolds equipped with arbitrary measures. In particular, these inequalities can be globally refined by adding remainder terms like the Brezis-V\'azquez improvement, if Finsler manifolds are of strictly negative flag curvature, vanishing S-curvature and finite uniformity constant. Furthermore, these results remain valid when Finsler metrics are reversible.