Analytic van der Corput Lemma for pp-adic and Fq((t)) oscillatory integrals, singular Fourier transforms, and restriction theorems

We give a non-archimedean analogue of the van der Corput Lemma on oscillating integrals, where the condition of sufficient smoothness for the phase in the real case is replaced by the condition that the phase is a convergent power series. This result allows us, in analogy to the real situation, to study singular Fourier transforms on suitably curved (pp-adic analytic) manifolds. As an application we give a restriction theorem for Fourier transforms of LqLq functions to suitably curved analytic manifolds over non-archimedean local fields, similar to a real restriction result by E.M. Stein. Several analogues of the van der Corput Lemma were already known when the phase is a polynomial.