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 Laplacian shifted by a constant $lambda$, a problem that had been studied in Berchio, Ganguly, and Grillo (2017) only when $lambda$ is the bottom of the spactrum of the Laplacian. 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 shifted Laplacian.

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

Grillo G.;
2020-01-01

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 Laplacian shifted by a constant $lambda$, a problem that had been studied in Berchio, Ganguly, and Grillo (2017) only when $lambda$ is the bottom of the spactrum of the Laplacian. 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 shifted Laplacian.
2020
Extremals
Hyperbolic space
Optimal Hardy inequality
File in questo prodotto:
File Dimensione Formato  
1711.08423.pdf

Open Access dal 02/02/2021

Descrizione: file del lavoro accettato
: Post-Print (DRAFT o Author’s Accepted Manuscript-AAM)
Dimensione 404.4 kB
Formato Adobe PDF
404.4 kB Adobe PDF Visualizza/Apri

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/11311/1157518
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 17
  • ???jsp.display-item.citation.isi??? 14
social impact