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We give the p-adic and F_q((t)) analogue of the real van der Corput Lemma,where the real condition of sufficient smoothness for the phase is replaced bythe condition that the phase is a convergent power series. This van der Corputstyle result allows us, in analogy to the real situation, to study singularFourier transforms on suitably curved (analytic) manifolds and opens the wayfor further applications. As one such further application we give therestriction theorem for Fourier transforms of L^p functions to suitably curvedanalytic manifolds over non-archimedean local fields, similar to the realrestriction result by E. Stein and C. Fefferman.

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