H\"ormander classes of pseudo-differential operators over the compact group of $p$-adic integers

TL;DR

This paper introduces new H"ormander classes for pseudo-differential operators on the compact group of p-adic integers, establishing symbolic calculus and relations to matrix algebras and toroidal cases.

Contribution

It defines novel H"ormander classes over p-adic integers with symbolic calculus, asymptotic expansions, and links to existing frameworks.

Findings

Defined new H"ormander classes with symbolic calculus

Established relation to infinite matrix algebras

Connected to toroidal case by Ruzhansky and Turunen

Abstract

The purpose of this paper is to introduce new definitions of H\"ormander classes for pseudo-differential operators over the compact group of $p$-adic integers. Our definitions possess a symbolic calculus, asymptotic expansions and parametrices, together with an interesting relation with the infinite matrices algebras studied by K. Gr\"ochenig and S. Jaffard. Also, we show how our definition of H\"ormander 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 H\"ormander classes here defined.