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In this paper we consider the inverse problem of determining on a compact Riemannian mani-fold the electric potential or the magnetic field in a Schrödinger equation with Dirichlet data from measured Neumann boundary observations. This information is enclosed in the dynamical Dirichlet-to-Neumann map associated to the magnetic Schrödinger equation. We prove in dimension n ě 2 that the knowledge of the Dirichlet-to-Neumann map for the Schrödinger equation uniquely determines the magnetic field and the electric potential and we establish Hölder-type stability.

We prove weighted local smoothing estimates for the resolvent of the Laplacian in three dimensions with weights belonging to the Kerman-Sawyer class. This class contains the well-known global Kato and Rollnik classes. We go on to discuss dispersive and Strichartz estimates for perturbations of the Laplacian by small potentials, and apply our results and observations to the well-posedness in L 2 of the Cauchy problem for some linear and semilinear Schrödinger equations.

We prove optimal radially weighted L 2 -norm inequalities for the Fourier extension operator associated to the unit sphere in R n . Such inequalities valid at all scales are well understood. The purpose of this short paper is to establish certain more delicate single-scale versions of these.2000 Mathematics subject classification: 42B10.

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