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.

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