Non-Archimedean Radial Calculus: Volterra Operator and Laplace Transform

TL;DR

This paper explores a non-Archimedean analog of classical calculus, focusing on the Volterra operator and Laplace transform within the framework of pseudo-differential operators, extending previous work on fractional differentiation.

Contribution

It introduces an operator-theoretic analysis of the inverse fractional differentiation operator and develops a non-Archimedean Laplace transform analog.

Findings

Established properties of the inverse operator $I^\alpha$

Developed a non-Archimedean Laplace transform framework

Linked fractional equations to integral equations resembling Volterra equations

Abstract

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.