Multidimensional ultrametric pseudodifferential equations

TL;DR

This paper develops a framework for analyzing wavelets and pseudodifferential operators on multidimensional ultrametric spaces, introducing new bases, function spaces, and solving ultrametric Cauchy problems.

Contribution

It introduces bases of wavelets, generalized function spaces, and establishes existence and uniqueness theorems for ultrametric pseudodifferential equations.

Findings

Established bases of wavelets on multidimensional ultrametric spaces

Defined and analyzed pseudodifferential operators in this setting

Proved a theorem on existence and uniqueness of solutions for ultrametric Cauchy problems

Abstract

We develop an analysis of wavelets and pseudodifferential operators on multidimensional ultrametric spaces which are defined as products of locally compact ultrametric spaces. We introduce bases of wavelets, spaces of generalized functions and Lizorkin generalized functions on multidimensional ultrametric spaces. We also consider some family of pseudodifferential operators on multidimensional ultrametric spaces. The notions of Cauchy problem for ultrametric pseudodifferential equations and of ultrametric characteristics are introduced. A theorem about existence and uniqueness of the solution for the Cauchy problem (the analogue of the Kovalevskaya theorem) is proven.