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

الأصل · EN

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