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

DOI10.1007/S11868-021-00428-5arXiv2107.00889MaRDI QIDQ6371795

Anatoly N. Kochubei

Publication date: 2 July 2021

Abstract: Let $D^{a}lpha,alpha>0$, be the Vladimirov-Taibleson fractional differentiation operator acting on complex-valued functions on a non-Archimedean local field. The identity $D^{a}lpha{D}^{-alpha}f=f$ 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.

Mathematics Subject Classification ID

Pseudodifferential operators as generalizations of partial differential operators (35S05) Other analytic theory (analogues of beta and gamma functions, (p)-adic integration, etc.) (11S80) Pseudodifferential operators (47G30) Fractional ordinary differential equations (34A08)

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