Quantization of the affine group of a local field

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 Calderón-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.

to appear in JFG