Transfer principles for bounds of motivic exponential functions

We study transfer principles for upper bounds of motivic exponential functions and for linear combinations of such functions, directly generalizing the transfer principles from Cluckers and Loeser (Ann Math 171:1011–1065, 2010) and Shin and Templier (Invent Math, 2015, Appendix B). 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.