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- W4300548115 abstract "We prove emph{optimal} improvements of the Hardy inequality on the hyperbolic space. Here, optimal means that the resulting operator is critical in the sense of [J.Funct.Anal. 266 (2014), pp. 4422-89], namely the associated inequality cannot be further improved. Such inequalities arise from more general, emph{optimal} ones valid for the operator $ P_{lambda}:= -Delta_{mathbb{H}^N} - lambda$ where $0 leq lambda leq lambda_{1}(mathbb{H}^N)$ and $lambda_{1}(mathbb{H}^N)$ is the bottom of the $L^2$ spectrum of $-Delta_{mathbb{H}^N} $, a problem that had been studied in [J.Funct.Anal. 272 (2017), pp. 1661-1703 ] only for the operator $P_{lambda_{1}(mathbb{H}^N)}$. A different, critical and new inequality on $mathbb{H}^N$, locally of Hardy type, is also shown. Such results have in fact greater generality since there are shown on general Cartan-Hadamard manifolds under curvature assumptions, possibly depending on the point. Existence/nonexistence of extremals for the related Hardy-Poincar'e 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 $P_lambda.$" @default.
- W4300548115 created "2022-10-03" @default.
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- W4300548115 date "2017-11-22" @default.
- W4300548115 modified "2023-09-29" @default.
- W4300548115 title "An optimal improvement for the Hardy inequality on the hyperbolic space and related manifolds" @default.
- W4300548115 doi "https://doi.org/10.48550/arxiv.1711.08423" @default.
- W4300548115 hasPublicationYear "2017" @default.
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