# Quantization of the affine group of a local field

## TL;DR

This paper develops a pseudo-differential calculus for non-Archimedean local fields, extending classical analysis tools to a new algebraic setting using Wigner functions and coherent states.

## Contribution

It introduces a covariant pseudo-differential calculus on quotient groups of the affine group over certain local fields, generalizing the Calderón-Vaillancourt estimate in this context.

## Key findings

- Construction of a covariant pseudo-differential calculus for non-Archimedean fields
- Extension of Calderón-Vaillancourt estimate to this setting
- Framework based on Wigner functions and coherent states

## Abstract

For a non Archimedean local field which is not of characteristic $2$, nor an extension of $\mathbb Q_2$, we construct a pseudo-differential calculus covariant under a unimodular subgroup of the affine group of the field. Our phase space is a quotient group of the covariance group. Our main result is a generalisation on that context of the Calder\'on-Vaillancourt estimate. Our construction can be thought as the non Archimedean version of Unterberger's Fuchs calculus and our methods are mainly based on Wigner functions and on coherent states transform.