Kristály, Alexandru (2017) Sharp Uncertainty Principles on Riemannian Manifolds: the influence of curvature. Journal de Mathématiques Pures et Appliquées. ISSN 00217824 (In Press)

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Abstract
We present a rigidity scenario for complete Riemannian manifolds supporting the HeisenbergPauliWeyl uncertainty principle with the sharp constant in $\mathbb R^n$ (shortly, {\it sharp HPW principle}). Our results deeply depend on the curvature of the Riemannian manifold which can be roughly formulated as follows: (a) When $(M,g)$ has {\it nonpositive sectional curvature}, the sharp HPW principle holds on $(M,g)$. However, {\it positive extremals exist} in the sharp HPW principle if and only if $(M,g)$ is isometric to $\mathbb R^n$, $n={\rm dim}(M)$. (b) When $(M,g)$ has {\it nonnegative Ricci curvature}, the sharp HPW principle holds on $(M,g)$ if and only if $(M,g)$ is isometric to $\mathbb R^n$. Since the sharp HPW principle and the HardyPoincar\'e inequality are endpoints of the CaffarelliKohnNirenberg interpolation inequality, we establish further quantitative results for the latter inequalities in terms of the curvature on CartanHadamard manifolds.
Item Type:  Article 

Subjects:  Q Science / természettudomány > QA Mathematics / matematika > QA73 Geometry / geometria Q Science / természettudomány > QA Mathematics / matematika > QA74 Analysis / analízis 
Depositing User:  Dr. Alexandru Kristaly 
Date Deposited:  07 Nov 2017 08:52 
Last Modified:  07 Nov 2017 08:52 
URI:  http://real.mtak.hu/id/eprint/67174 
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