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

2010/01/12 by Raf Cluckers, Cluckers, Raf
Mathematics · #11S80 #42A99 #42B20 #42B25 #Advanced Harmonic Analysis Research #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Analysis and Transform Methods #Number Theory (math.NT) #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.1001.2013

openalex publication_date 2010/01/12 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

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.

Citations

• The Multiplier Problem for the Ball

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