相关论文: Resolvent bounds imply observability from measurab…

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 $z$…

We characterize the observability property (and, by duality, the controllability and the stabilization) of the wave equation on a Riemannian manifold $\Omega,$ with or without boundary, where the observation (or control) domain is…

We prove uniform $L^p$ estimates for resolvents of higher order elliptic self-adjoint differential operators on compact manifolds without boundary, generalizing a corresponding resul of [3] in the case of Laplace-- Beltrami operators on…

Solutions of a system of wave equations are constructed for both homogeneous and inhomogeneous Dirichlet boundary conditions at every regularity level. We prove that boundary observability, and thus boundary exact controllability, at some…

We prove semi-classical resolvent estimates for the Schr{\"o}dinger operator with a real-valued L $\infty$ potential on non-compact, connected Riemannian manifolds which may have a compact smooth boundary. We show that the resolvent bound…

We prove $L^p-L^{p^\prime}$ boundedness of spectral projections and the resolvent of the Laplace-Beltrami operator on Damek-Ricci spaces with the explicit norms in terms of the spectral parameter. To prove these results we established…

Given a closed product Riemannian manifold N = M x M equipped with the product Riemannian metric g = h + h , we explore the observability properties for the generalized Schr{\"o}dinger equation i$\partial$ t u = F (g)u, where g is the…

We prove resolvent estimates in Schatten spaces for Laplace-Beltrami operators on compact manifolds at the critical exponent. Our proof only uses known bounds for the Hadamard parametrix.

The purpose of this paper is to study the property of the resolvent of the Laplace-Beltrami operator on a noncompact complete Riemannian manifold with various ends each of which has a different limit of the growth rate of the Riemannian…

We prove that the Riemannian metric on a compact manifold of dimension $n\geq 3$ with smooth boundary can be uniquely determined, up to an isometry fixing the boundary, by the Dirichlet-to-Neumann map associated to the Laplace-Beltrami…

On a fairly general class of Riemannian manifolds M, we prove lower estimates in terms of the Ricci curvature for the spectral bound (when M has infinite volume) and for the spectral gap (when M has finite volume) for the Laplace-Beltrami…

It is shown that eigenvalues of Laplace-Beltrami operators on compact Riemannian manifolds can be determined as limits of eigenvalues of certain finite-dimensional operators in spaces of polyharmonic functions with singularities. In…

It is well known that uniform resolvent estimates imply spectral cluster estimates. We show that the converse is also true in some cases. In particular, the universal spectral cluster estimates of Sogge \cite{MR930395} for the…

We establish observability inequalities for various problems involving fractional Schr\"odinger operators $(-\Delta)^{\alpha/2}+V$, $\alpha>0$, on a compact Riemannian manifold. Observability from an open set for the corresponding…

Upper bounds for the eigenvalues of the Laplace-Beltrami operator on a hypersurface bounding a domain in some ambient Riemannian manifold are given in terms of the isoperimetric ratio of the domain. These results are applied to the…

We establish uncertainty principles on compact Riemannian manifolds without boundary by combining restriction estimates for orthonormal systems with spectral projection bounds for Laplace-Beltrami and Schr\"odinger operators. Our results…

We address an interesting question raised by Dos Santos Ferreira, Kenig and Salo about regions ${\mathcal R}_g\subset {\mathbb C}$ for which there can be uniform $L^{\frac{2n}{n+2}}\to L^{\frac{2n}{n-2}}$ resolvent estimates for…

The purpose of this note is to use the results and methods of our previous work with Bourgain to obtain control and observability by rough functions and sets on rectangular 2-tori. We show that any Lebesgue measurable set of positive…

We study Laplace-type operators on hybrid manifolds, i.e. on configurations consisting of closed two-dimensional manifolds and one-dimensional segments. Such an operator can be constructed by using the Laplace-Beltrami operators on each…

On tori of arbitrary dimensions, Schr\"odinger propagators with bounded potentials are conjectured to be observable from space-time domains of positive Lebesgue measure. We reduce this conjecture to certain integrability bounds for free…