Radial Solutions of Non-Archimedean Pseudo-Differential Equations

We consider a class of equations with the fractional differentiation operator $D^{\alpha }$, $\alpha >0$, for complex-valued functions $x\mapsto f(\mathrm{\mid }x{\mathrm{\mid }}_K)$ on a non-Archimedean local field $K$ depending only on the absolute value $|\cdot |_K$\mathrm{.}Weintroducearightinverse$I^\alpha$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.