Lp properties of non-Archimedean fractional differentiation operators

Let D α be the Vladimirov–Taibleson fractional differentiation operator acting on complex-valued functions on a non-Archimedean local field. The identity D α D - α 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.