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Sharp Hardy and spectral gap inequalities on special irreversible Finsler manifolds

Part of: Spectral theory and eigenvalue problems Local differential geometry Calculus on manifolds; nonlinear operators Partial differential equations on manifolds; differential operators Inequalities

Published online by Cambridge University Press:  09 September 2025

Sándor Kajántó
Sándor Kajántó*
Affiliation:
Department of Mathematics and Computer Science, Babeş-Bolyai University, Cluj-Napoca, Romania Institute of Applied Mathematics, Óbuda University, Budapest, Hungary (sandor.kajanto@ubbcluj.ro)
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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. Kristály et al. [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.

Keywords

Finlser manifoldsHardy inequalitiesRiccati pairssharpnessspectral gap estimates

MSC classification

Primary: 26D10: Inequalities involving derivatives and differential and integral operators
Secondary: 35P15: Estimation of eigenvalues, upper and lower bounds 53B40: Finsler spaces and generalizations (areal metrics) 58C40: Spectral theory; eigenvalue problems 58J60: Relations with special manifold structures (Riemannian, Finsler, etc.)

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 36
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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