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

TL;DR

This paper develops a p-adic and F_q((t)) analogue of the van der Corput Lemma, replacing smoothness with convergent power series, enabling analysis of singular Fourier transforms and restriction theorems over non-archimedean fields.

Contribution

It introduces a new van der Corput Lemma for non-archimedean fields based on convergent power series, extending harmonic analysis tools to these settings.

Findings

Established a p-adic and F_q((t)) van der Corput Lemma.

Applied the lemma to study singular Fourier transforms on curved manifolds.

Proved restriction theorems for Fourier transforms over non-archimedean fields.

Abstract

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 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 L^p functions to suitably curved analytic manifolds over non-archimedean local fields, similar to the real restriction result by E. Stein and C. Fefferman.