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.

中文翻译：

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p-Adic 整数紧群上的伪微分算子的 Hörmander 类

本文的目的是为 $p$-adic 整数的紧群上的伪微分算子引入 H\"Ormander 类的新定义。我们的定义具有符号演算、渐近展开和参数化，以及一个有趣的关系与 K. Gr\"ochenig 和 S. Jaffard 研究的无限矩阵代数。此外，我们还展示了我们对 H\"ormander 类的定义如何与 M. Ruzhansky 和 ​​V. Turunen 在环形情况下给出的定义相关。为了展示我们定义的特殊性质，在以后的工作中，我们将根据此处定义的 H\" 序类中的伪微分算子的符号研究几个谱特性。