Restriction of the Fourier transform with applications to the Schrödinger and wave equations
In 1967, Stein proved that the Fourier transform of functions in L^p could be meaningfully restricted to the sphere for certain p>1. The restriction conjecture, which asserts the maximal range of such p, was solved by Fefferman in two dimensions, but the conjecture remains open in higher dimensions. Strichartz considered the same question but with the sphere replaced by the paraboloid or the cone, and a great deal of progress has been made in the last two decades by Bourgain, Wolff and Tao, among others. Due to the fact that the adjoint operators of the restriction operators to the paraboloid and cone correspond to the Schrödinger and wave evolution operators, respectively, this work has been hugely influential. The main goal of this proposal is to improve the state of the art for the mixed norm analogues of these conjectures.