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 generalization 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.