Uniform analysis on local fields and applications to orbital integrals

TL;DR

This paper develops a field-independent analytical framework for functions on local fields, establishing transfer principles and bounds relevant to harmonic analysis and representation theory.

Contribution

It introduces a uniform approach to analysis on local fields, enabling transfer principles and bounds for orbital integrals in a broad, field-independent setting.

Findings

Established transfer principles for local constancy and continuity.

Proved that Fourier transforms of motivic exponential functions remain within the same class.

Derived uniform bounds for orbital integrals on reductive p-adic groups.

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 the realm of representation theory, we prove uniform bounds for the normalized by the discriminant Fourier transforms of orbital integrals on connected reductive $p$-adic groups.