Sharp Morrey-Sobolev inequalities on complete Riemannian Manifolds

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

Endpoint resolvent estimates for compact Riemannian manifolds

We prove $L^p\to L^{p'}$ bounds for the resolvent of the Laplace-Beltrami operator on a compact Riemannian manifold of dimension $n$ in the endpoint case $p=2(n+1)/(n+3)$. It has the same behavior with respect to the spectral parameter $z$ 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

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

We present a rigidity scenario for complete Riemannian manifolds supporting the Heisenberg-Pauli-Weyl 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 non-positive 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 non-negative 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 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