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 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.