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New sharp Hardy and Rellich type inequalities on Cartan–Hadamard manifolds and their improvements

Part of: Inequalities Global differential geometry Other generalizations

Published online by Cambridge University Press:  23 August 2019

Van Hoang Nguyen
Van Hoang Nguyen*
Affiliation:
Institut de Mathématiques de Toulouse, Université Paul Sabatier, 118 Route de Narbonne, 31062 Toulouse cédex 09, France and Institute of Mathematics, Vietnam Academy of Science and Technology, Hanoi, Vietnam (vanhoang0610@yahoo.com; nvhoang@math.ac.vn)
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Abstract

In this paper, we prove several new Hardy type inequalities (such as the weighted Hardy inequality, weighted Rellich inequality, critical Hardy inequality and critical Rellich inequality) related to the radial derivation (i.e., the derivation along the geodesic curves) on the Cartan–Hadamard manifolds. By Gauss lemma, our new Hardy inequalities are stronger than the classical ones. We also establish the improvements of these inequalities in terms of sectional curvature of the underlying manifolds which illustrate the effect of curvature to these inequalities. Furthermore, we obtain some improvements of Hardy and Rellich inequalities on the hyperbolic space ℍn. Especially, we show that our new Rellich inequalities are indeed stronger than the classical ones on the hyperbolic space ℍn.

Keywords

Hardy inequalityRellich inequalitycritical Hardy inequalitycritical Rellich inequalityCartan–Hadamard manifoldssharp constant

MSC classification

Primary: 26D10: Inequalities involving derivatives and differential and integral operators
Secondary: 31C12: Potential theory on Riemannian manifolds 53C20: Global Riemannian geometry, including pinching 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 6 , December 2020 , pp. 2952 - 2981
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Copyright
Copyright © 2019 The Royal Society of Edinburgh

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