  In an earlier paper (A. N. Kochubei, {\it Pacif. J. Math.} 269 (2014),
355--369), the author considered a restriction of Vladimirov's fractional
differentiation operator $D^\alpha$, $\alpha >0$, to radial functions on a
non-Archimedean field. In particular, it was found to possess such a right
inverse $I^\alpha$ that the appropriate change of variables reduces equations
with $D^\alpha$ (for radial functions) to integral equations whose properties
resemble those of classical Volterra equations. In other words, we found, in
the framework of non-Archimedean pseudo-differential operators, a counterpart
of ordinary differential equations. In the present paper, we begin an
operator-theoretic investigation of the operator $I^\alpha$, and study a
related analog of the Laplace transform.
