WF-holonomicity of C-exp-class distributions on non-archimedean local fields

TL;DR

This paper proves that all $C^{ ext{exp}}$-class distributions on non-archimedean local fields are WF-holonomic, establishing a stable framework for analyzing these distributions and their Fourier transforms.

Contribution

It demonstrates that $C^{ ext{exp}}$-class distributions are WF-holonomic and introduces a regularization method within this class, linking zero and smooth loci for these distributions.

Findings

All $C^{ ext{exp}}$-class distributions are WF-holonomic.

Distributions can be regularized without leaving the $C^{ ext{exp}}$-class.

Established a connection between zero loci and smooth loci for $C^{ ext{exp}}$-class functions and distributions.

Abstract

In the context of geometry and analysis on non-archimedean local fields, we study two recent notions, $C^{\mathrm exp}$-class distributions from [11] and WF-holonomicity from [1], and we show that any distribution of $C^{\mathrm exp}$-class is WF-holonomic. Thus we answer a question from [1] by providing a framework of WF-holonomic distributions for non-archimedean local fields which is stable under taking Fourier transforms and which contains many natural distributions, in particular, the distributions studied in [1]. We further show that one can regularize distributions without leaving the $C^{\mathrm exp}$-class. Finally, we show a close link between zero loci and smooth loci for functions and distributions of $C^{\mathrm exp}$-class, by proving a converse to a result of [11]. A key ingredient is a new resolution result for subanalytic functions (by alterations), based on embedded resolution for analytic functions and model theory.