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An optimal improvement for the Hardy inequality on the hyperbolic space and related manifolds

Part of: Inequalities Linear function spaces and their duals Other generalizations

Published online by Cambridge University Press:  01 February 2019

Elvise Berchio ,
Debdip Ganguly ,
Gabriele Grillo  and
Yehuda Pinchover
Elvise Berchio
Affiliation:
Dipartimento di Scienze Matematiche, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129Torino, Italy (elvise.berchio@polito.it)
Debdip Ganguly
Affiliation:
Department of Mathematics, Indian Institute of Science Education and Research, Dr. Homi Bhabha Road, Pashan Pune411008, India (debdipmath@gmail.com)
Gabriele Grillo
Affiliation:
Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy (gabriele.grillo@polimi.it)
Yehuda Pinchover
Affiliation:
Department of Mathematics, Technion – Israel Institute of Technology, Haifa3200003, Israel (pincho@technion.ac.il)
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Abstract

We prove optimal improvements of the Hardy inequality on the hyperbolic space. Here, optimal means that the resulting operator is critical in the sense of Devyver, Fraas, and Pinchover (2014), namely the associated inequality cannot be further improved. Such inequalities arise from more general, optimal ones valid for the operator $ P_{\lambda }:= -\Delta _{{\open H}^{N}} - \lambda $ where 0 ⩽ λ ⩽ λ1(ℍN) and λ1(ℍN) is the bottom of the L2 spectrum of $-\Delta _{{\open H}^{N}} $, a problem that had been studied in Berchio, Ganguly, and Grillo (2017) only for the operator $P_{\lambda _{1}({\open H}^{N})}$. A different, critical and new inequality on ℍN, locally of Hardy type is also shown. Such results have in fact greater generality since they are proved on general Cartan-Hadamard manifolds under curvature assumptions, possibly depending on the point. Existence/nonexistence of extremals for the related Hardy-Poincaré inequalities are also proved using concentration-compactness technique and a Liouville comparison theorem. As applications of our inequalities, we obtain an improved Rellich inequality and we derive a quantitative version of Heisenberg-Pauli-Weyl uncertainty principle for the operator $P_\lambda.$

Keywords

Hyperbolic spaceoptimal Hardy inequalityextremals

MSC classification

Primary: 46E35: Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
Secondary: 26D10: Inequalities involving derivatives and differential and integral operators 31C12: Potential theory on Riemannian manifolds
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 4 , August 2020 , pp. 1699 - 1736
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Copyright
Copyright © Royal Society of Edinburgh 2019

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