$L^p$ properties of non-Archimedean fractional differentiation operators

TL;DR

This paper investigates the $L^p$ properties of non-Archimedean fractional differentiation operators, extending known identities to broader function classes and comparing real and non-Archimedean cases.

Contribution

It extends the $L^p$-convergence of fractional differentiation identities for non-Archimedean operators beyond compact support functions.

Findings

Extended the identity $D^eta D^{-eta}f=f$ to $L^p$ functions

Compared non-Archimedean and real fractional Laplacian properties

Discussed differences between real and non-Archimedean cases

Abstract

Let $D^\alpha, \alpha>0$, be the Vladimirov-Taibleson fractional differentiation operator acting on complex-valued functions on a non-Archimedean local field. The identity $D^\alpha D^{-\alpha}f=f$ was known only for the case where $f$ has a compact support. Following a result by Samko about the fractional Laplacian of real analysis, we extend the above identity in terms of $L^p$-convergence of truncated integrals. Differences between real and non-Archimedean cases are discussed.