Uniform analysis on local fields and applications to orbital integrals

Cluckers, Raf; Gordon, Julia; Halupczok, Immanuel

doi:10.1090/btran/25

Uniform analysis on local fields and applications to orbital integrals
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Trans. Amer. Math. Soc. Ser. B 5 (2018), 125-166

Abstract:

We study upper bounds, approximations, and limits for functions of motivic exponential class uniformly in non-Archimedean local fields whose characteristic is $0$ or sufficiently large. Our results together form a flexible framework for doing analysis over local fields in a field-independent way. As corollaries, we obtain many new transfer principles, for example, for local constancy, continuity, and existence of various kinds of limits. Moreover, we show that the Fourier transform of an $L^2$-function of motivic exponential class is again of motivic exponential class. As an application in representation theory, we prove uniform bounds for the Fourier transforms of orbital integrals on connected reductive $p$-adic groups.

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