Sharp Hardy and spectral gap inequalities on special irreversible Finsler manifolds

TL;DR

This paper constructs examples of irreversible Finsler manifolds where classical Hardy and spectral gap inequalities are sharp, providing new insights into the geometry of such spaces and extending analytical methods.

Contribution

It introduces specific irreversible Finsler manifolds where Hardy and spectral gap inequalities are sharp, and develops a Finslerian Riccati pairs method for proving these inequalities.

Findings

Examples of irreversible Finsler manifolds with sharp Hardy inequalities.

Examples of irreversible Finsler manifolds with sharp spectral gap estimates.

Development of a Finslerian Riccati pairs method for Hardy inequalities.

Abstract

The sharpness of various Hardy-type inequalities is well-understood in the reversible Finsler setting; while infinite reversibility implies the failure of these functional inequalities, cf. Krist\'aly, Huang, and Zhao [Trans. Am. Math. Soc., 2020]. However, in the remaining case of irreversible manifolds with finite reversibility, there is no evidence on the sharpness of Hardy-type inequalities. In fact, we are not aware of any particular examples where the sharpness persists. In this paper we present two such examples involving two celebrated inequalities: the classical/weighted Hardy inequality (assuming non-positive flag curvature) and the McKean-type spectral gap estimate (assuming strong negative flag curvature). In both cases, we provide a family of Finsler metric measure manifolds on which these inequalities are sharp. We also establish some sufficient conditions, which guarantee the sharpness of more involved Hardy-type inequalities on these spaces. Our relevant technical tool is a Finslerian extension of the method of Riccati pairs (for proving Hardy inequalities), which also inspires the main ideas of our constructions.