Asymptotic properties of the \(p\)-adic fractional integration operator. (English) Zbl 1389.11140

Kochubei, Anatoly N.; Soskin, Daniel S.

Methods Funct. Anal. Topol. 23, No. 2, 155-163 (2017).

A. N. Kochubei [Pac. J. Math. 269, No. 2, 355–369 (2014; Zbl 1396.35070)] introduced a right inverse \(I^\alpha\) of Vladimirov’s fractional differentiation operator \(D^\alpha\) (\(\alpha >0\)) acting on complex-valued functions on the field of \(p\)-adic numbers. \(I^\alpha\) can be seen as a \(p\)-adic counterpart of the Riemann-Liouville fractional integral of real analysis. In this paper, the authors study asymptotic properties of \(I^\alpha f\), given the asymptotics of a function \(f\). See S. G. Samko et al. [Fractional integrals and derivatives: theory and applications. New York, NY: Gordon and Breach. xxxvi, 976 p. (1993; Zbl 0818.26003)] for results of this kind for the Riemann-Liouville integration operator.

Reviewer: Anatoly N. Kochubei (Kyïv)

MSC:

11S80	Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
26A33	Fractional derivatives and integrals

Keywords:

fractional differentiation operator; fractional integration operator; \(p\)-adic numbers

Citations:

Zbl 0818.26003; Zbl 1396.35070