389 research outputs found

    Sharp Morrey-Sobolev inequalities on complete Riemannian Manifolds

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    Two Morrey-Sobolev inequalities (with support-bound and L1−L^1-bound, respectively) are investigated on complete Riemannian manifolds with their sharp constants in Rn\mathbb R^n. We prove the following results in both cases: ∙\bullet If (M,g)(M,g) is a {\it Cartan-Hadamard manifold} which verifies the n−n-dimensional Cartan-Hadamard conjecture, sharp Morrey-Sobolev inequalities hold on (M,g)(M,g). Moreover, extremals exist if and only if (M,g)(M,g) is isometric to the standard Euclidean space (Rn,e)(\mathbb R^n,e). ∙\bullet If (M,g)(M,g) has {\it non-negative Ricci curvature}, (M,g)(M,g) supports the sharp Morrey-Sobolev inequalities if and only if (M,g)(M,g) is isometric to (Rn,e)(\mathbb R^n,e).Comment: 15 pages, in press (Potential Analysis, 2014

    Endpoint resolvent estimates for compact Riemannian manifolds

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    We prove Lp→Lp′L^p\to L^{p'} bounds for the resolvent of the Laplace-Beltrami operator on a compact Riemannian manifold of dimension nn in the endpoint case p=2(n+1)/(n+3)p=2(n+1)/(n+3). It has the same behavior with respect to the spectral parameter zz as its Euclidean analogue, due to Kenig-Ruiz-Sogge, provided a parabolic neighborhood of the positive half-line is removed. This is region is optimal, for instance, in the case of a sphere.Comment: 14 page

    Smoothing effects for the porous medium equation on Cartan-Hadamard manifolds

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    We prove three sharp bounds for solutions to the porous medium equation posed on Riemannian manifolds, or for weighted versions of such equation. Firstly we prove a smoothing effect for solutions which is valid on any Cartan–Hadamard manifold whose sectional curvatures are bounded above by a strictly negative constant. This bound includes as a special case the sharp smoothing effect recently proved by V´azquez on the hyperbolic space in V´azquez (2015), which is similar to the absolute bound valid in the case of bounded Euclidean domains but has a logarithmic correction. Secondly we prove a bound which interpolates between such smoothing effect and the Euclidean one, supposing a suitable quantitative Sobolev inequality holds, showing that it is sharp by means of explicit examples. Finally, assuming a stronger functional inequality of sub-Poincar´e type, we prove that the above mentioned (sharp) absolute bound holds, and provide examples of weighted porous media equations on manifolds of infinite volume in which it holds, in contrast with the nonweighted Euclidean situation. It is also shown that sub-Poincar´e inequalities cannot hold on Cartan–Hadamard manifolds

    Sharp Uncertainty Principles on Riemannian Manifolds: the influence of curvature

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    We present a rigidity scenario for complete Riemannian manifolds supporting the Heisenberg-Pauli-Weyl uncertainty principle with the sharp constant in Rn\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)(M,g) has {\it non-positive sectional curvature}, the sharp HPW principle holds on (M,g)(M,g). However, {\it positive extremals exist} in the sharp HPW principle if and only if (M,g)(M,g) is isometric to Rn\mathbb R^n, n=dim(M)n={\rm dim}(M). (b) When (M,g)(M,g) has {\it non-negative Ricci curvature}, the sharp HPW principle holds on (M,g)(M,g) if and only if (M,g)(M,g) is isometric to Rn\mathbb R^n. Since the sharp HPW principle and the Hardy-Poincar\'e inequality are endpoints of the Caffarelli-Kohn-Nirenberg interpolation inequality, we establish further quantitative results for the latter inequalities in terms of the curvature on Cartan-Hadamard manifolds
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