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arXiv · 2508.13793

Sharp Hardy and spectral gap inequalities on special irreversible Finsler manifolds

Sándor Kajántó
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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.

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Sándor Kajántó. 2025-08-19. Sharp Hardy and spectral gap inequalities on special irreversible Finsler manifolds. https://doi.org/10.1017/prm.2025.10072

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