Radial Solutions of Non-Archimedean Pseudo-Differential Equations

We consider a class of equations with the fractional differentiation operator D , > 0, for complex-valued functions x7! f.jxjK/ on a non-Archimedean local field K depending only on the absolute valuejj K . We introduce a right inverse I to D , such that the change of an unknown function uD I v reduces the Cauchy problem for an equation with D (for radial functions) to an integral equation whose properties resemble those of classical Volterra equations. This contrasts much more complicated behavior of D on other classes of functions.