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 $P_{lambda}=-Delta-lambda$ where $0 leq lambda leq lambda_1$ and $lambda_1$ is the bottom of the L^2 spectrum of the laplacian, a problem that had been studied in Berchio, Ganguly, and Grillo (2017) only for $lambda=lambda_1$. A different, critical and new inequality on the hyperbolic space, 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 $P_{lambda}$.

An optimal improvement for the Hardy inequality on the hyperbolic space and related manifolds / Berchio, Elvise; Ganguly, Debdip; Grillo, Gabriele; Pinchover, Yehuda. - In: PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH. SECTION A. MATHEMATICS. - ISSN 0308-2105. - STAMPA. - 150:(2020), pp. 1699-1736. [10.1017/prm.2018.139]

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

Berchio, Elvise;
2020

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 $P_{lambda}=-Delta-lambda$ where $0 leq lambda leq lambda_1$ and $lambda_1$ is the bottom of the L^2 spectrum of the laplacian, a problem that had been studied in Berchio, Ganguly, and Grillo (2017) only for $lambda=lambda_1$. A different, critical and new inequality on the hyperbolic space, 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 $P_{lambda}$.
File in questo prodotto:
File Dimensione Formato  
BGGP_preprint.pdf

accesso aperto

Descrizione: Articolo principale
Tipologia: 1. Preprint / submitted version [pre- review]
Licenza: PUBBLICO - Tutti i diritti riservati
Dimensione 402.15 kB
Formato Adobe PDF
402.15 kB Adobe PDF Visualizza/Apri
2560155.pdf

non disponibili

Tipologia: 2a Post-print versione editoriale / Version of Record
Licenza: Non Pubblico - Accesso privato/ristretto
Dimensione 9.72 MB
Formato Adobe PDF
9.72 MB Adobe PDF   Visualizza/Apri   Richiedi una copia
Pubblicazioni consigliate

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11583/2731784