Radial Solutions of 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.