The restriction of the Fourier transform to surfaces: the hyperbolic case

The problem of restriction of the Fourier transform to hypersurfaces was posed by Stein in the seventies. This operator, in its adjoint form, gives the solution of dispersive equations (Schrödinger, wave, etc) in terms of the Fourier transform of the initial data. There are many open problems about dispersive equations for which it can be a powerful tool. Also, the restriction operator can be thought as a model case for more complicated oscillatory integral operators, such as, for instance, the operators of spherical summation of the Fourier transform. We will make a review of the problem, which is still open. We will present some new results for the case of surfaces with negative curvature. This part is joint work with Detlef Müller and Stefan Buschenhenke.