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

Anatoly N. Kochubei

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

Functional Analysis 2021-07-05 v1 Classical Analysis and ODEs

Authors: Anatoly N. Kochubei

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.

Keywords

non-archimedean local field vladimirov-taibleson operator fractional differentiation l^p convergence truncated integrals compact support

Cite

@article{arxiv.2107.00889,
  title  = {$L^p$ properties of non-Archimedean fractional differentiation operators},
  author = {Anatoly N. Kochubei},
  journal= {arXiv preprint arXiv:2107.00889},
  year   = {2021}
}