Transfer principles for Bounds of motivic exponential functions

TL;DR

This paper develops transfer principles for bounding motivic exponential functions, which are derived from oscillatory integrals on local fields and are useful in analyzing Fourier transforms of orbital integrals.

Contribution

It generalizes existing transfer principles to bounds of motivic exponential functions, expanding their applicability to more complex oscillatory integrals.

Findings

Established transfer principles for bounds of motivic exponential functions.

Reduced complex oscillatory integrals to simpler functions with residue field oscillation.

Extended previous transfer principles to broader classes of functions.

Abstract

We study transfer principles for upper bounds of motivic exponential functions and for linear combinations of such functions, directly generalizing the transfer principles from [7] by Cluckers-Loeser and [13, Appendix B] by Shin-Templier (appendix B by Cluckers-Gordon-Halupczok). These functions come from rather general oscillatory integrals on local fields, and can be used to describe e.g. Fourier transforms of orbital integrals. One of our techniques consists in reducing to simpler functions where the oscillation only comes from the residue field.