Fundamental Solution for a New Class of Non-Archimedean Pseudo-Differential Equations

TL;DR

This paper introduces a new class of pseudo-differential equations on functions of two p-adic variables, proving the existence and uniqueness of solutions and analyzing their properties such as finite dependence and L^1-estimates.

Contribution

It develops a novel framework for pseudo-differential equations in two p-adic variables, establishing fundamental solution properties and solution estimates.

Findings

Unique solutions to the new class of equations are proved.

Solutions exhibit finite dependence property.

An L^1-estimate for solutions is established.

Abstract

In this paper we study a new class of pseudo-differential equations on functions of two $p$-adic variables. It is proved that the correspondent Cauchy problem has a unique solution. Some properties of this solution are studied, in particular, the finite dependence property and an $L^1$-estimate.