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Analytic van der Corput lemma for \(p\)-adic and \(\mathbf F_q((t))\) oscillatory integrals, singular Fourier transforms, and restriction theorems. (English) Zbl 1231.42011

The author gives a non-archimedean analogue of the van der Corput Lemma on oscillatory 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 results allows, in analogy to the real situation, the study of singular Fourier transforms on suitably curved (\(p\)-adic analytic) manifolds. As an application the author gives a restriction theorem for Fourier transforms of \(L^{q}\) functions to suitably curved analytic manifolds over non-archimedean local fields.

MSC:

42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
43A32 Other transforms and operators of Fourier type
11L07 Estimates on exponential sums
11L40 Estimates on character sums
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