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

Abstract

Let > 3 be a prime number. In this note, we calculate explicitly the unitary dual and the matrix coefficients of the Engel group over the -adic integers  4 (ℤ ). We use this information to calculate explicitly the spectrum of the Vladimirov sub-Laplacian, and show how it defines a globally hypoelliptic operator on  4 .

Hörmander classes of pseudo-differential operators over the compact group of $p$-adic integers

“…With that purpose we state the characterization of compact operators in Hörmander classes. The proof of this statement is given in [37]…”

“…For example the simplicity of the composition formula in this setting allows one to provide necessary and sufficient conditions for belonging to Schatten-Von Neumann classes. Also in [37] the following version of the Weyl law is proved. We will use the special case when = 1.…”

“…More applications of the Hörmander classes and the symbolic calculus developed in this work are presented in [37]. There we treat mostly spectral properties of pseudo-differential operators and its relation with the associated symbol.…”

“…It is because many of the properties here studied rely mostly on the Hilbert space structure of 2 (ℤ ) and the known Fourier analysis on compact abelian groups. Anyway in the last three sections we will explain the implications of the non-archimidean structure and we will use them in a later work to study important spectral properties of pseudo-differential operators [37].…”

“…The most important applications of the theory that we intend to develop here are, first, to prove the regularity of solutions to certain pseudo-differential equations, and second, to put in terms of the symbol important properties such as -boundedness, compactness, belonging to Schatten classes and nuclearity, Riesz spectral theory, Fredholmness, ellipticity and Gohber's lemma, among others. We refer the reader to the work [37] where we study several spectral properties of pseudo-differential operators in the Hörmander classes here defined.…”