Hörmander Classes of Pseudo-Differential Operators over the Compact Group of p-Adic Integers

The purpose of this paper is to introduce new definitions of Hörmander classes for pseudo-differential operators over the compact group of p-adic integers. Our definitions possesses a symbolic calculus, asymptotic expansions and parametrices, together with an interesting relation with the infinite matrices algebras studied by K. Gröchenig and S. Jaffard. Also, we show how our definition of Hörmander classes is related to the definition given in the toroidal case by M. Ruzhansky and V. Turunen. In order to show the special properties of our definition, in a later work, we will study several spectral properties in terms of the symbol for pseudo-differential operators in the Hörmander classes here defined.