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

Abstract

Let \(D^{\alpha }\) 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.