# Distributions and wave front sets in the uniform non-archimedean setting

## TL;DR

This paper develops a framework for distributions in non-archimedean fields, introducing a new class called ${\mathscr C}^{\mathrm{exp}}$-class, and studies their wave front sets, Fourier transforms, and behavior under various operations.

## Contribution

It introduces the ${\mathscr C}^{\mathrm{exp}}$-class of distributions in non-archimedean settings, generalizing previous results and analyzing wave front sets using model theoretic methods.

## Key findings

- The wave front set equals the complement of the zero locus of a ${\mathscr C}^{\mathrm{exp}}$-class function.
- The class of distributions is stable under Fourier transform.
- Results extend and generalize Heifetz's work on $p$-adic wave front sets.

## 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 ${\mathscr C}^{\mathrm{exp}}$-class and which is based on the notion of ${\mathscr C}^{\mathrm{exp}}$-class functions from [6]. 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 ${\mathscr C}^{\mathrm{exp}}$-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 H\"ormander. In the final section, we study sizes of neighborhoods of local constancy of Schwartz-Bruhat functions and their push forwards in relation to discriminants.