Distributions and wave front sets in the uniform non‐archimedean setting

Abstract

We study some constructions on distributions in a uniform p ‐adic context, and also in large positive characteristic, using model theoretic methods. We introduce a class of distributions which we call distributions of C exp ‐class and which is based on the notion of C exp ‐class functions from Cluckers and Halupczok [J. Ecole Polytechnique (JEP) 5 (2018) 45–78]. This class of distributions is stable under Fourier transformation and has various forms of uniform behavior across non‐archimedean local fields. We study wave front sets, pull‐backs and push‐forwards of distributions of this class. In particular, we show that the wave front set is always equal to the complement of the zero locus of a C exp ‐class function. We first revise and generalize some of the results of Heifetz that he developed in the p ‐adic context by analogy to results about real wave front sets by Hörmander. In the final section, we study sizes of neighborhoods of local constancy of Schwartz–Bruhat functions and their push‐forwards in relation to discriminants.