  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.
