Radial Solutions of Non-Archimedean Pseudo-Differential Equations

TL;DR

This paper studies radial solutions to non-Archimedean pseudo-differential equations involving fractional derivatives, introducing an inverse operator to simplify the equations to integral form similar to classical Volterra equations.

Contribution

It introduces a right inverse to the fractional differentiation operator for radial functions on non-Archimedean fields, simplifying the analysis of such equations.

Findings

Reduction of fractional differential equations to integral equations

Simplification of the behavior of $D^eta$ on radial functions

New tools for solving non-Archimedean pseudo-differential equations

Abstract

We consider a class of equations with the fractional differentiation operator $D^\alpha$, $\alpha >0$, for complex-valued functions $x\mapsto f(|x|_K)$ on a non-Archimedean local field $K$ depending only on the absolute value $|\cdot |_K$. We introduce a right inverse $I^\alpha$ to $D^\alpha$, such that the change of an unknown function $u=I^\alpha v$ reduces the Cauchy problem for an equation with $D^\alpha$ (for radial functions) to an integral equation whose properties resemble those of classical Volterra equations. This contrasts much more complicated behavior of $D^\alpha$ on other classes of functions.