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

Abstract

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.

The Vladimirov–Taibleson operator: inequalities, Dirichlet problem, boundary Hölder regularity

“…Let L be an unramified extension of degree n of a local field K. Taking into account (2.1), we see the expansion with respect to a canonical basis in L defines an isometric linear isomorphism between L and K n . In various applications (see, for example, [20]), it is convenient to reduce problems for multi-dimensional operators acting on functions K n → C, to one-dimensional operators on functions L → C where L/K is an unramified extension of degree n.…”

“…A number of important results on spectral properties of D α , its perturbations and generalizations, as well as the theory of related partial (pseudo-) differential equations are covered by the monographs [1,16,34,30,36,14] and many recent papers, such as [4,2,3,17,19,20,37,33] and others.…”

“…The structure of this paper is as follows. In Section 2, we collect necessary preliminaries about local fields K, the structure of open sets in K n , the representation of K n in terms of the unramified extension of K [31,16,20], Sobolev spaces of complex-valued functions on local fields [11,9,10]. In Section 3, we prove analogs of classical inequalities.…”

The Vladimirov-Taibleson Operator: Inequalities, Dirichlet Problem, Boundary Hölder Regularity

“…A number of important results on spectral properties of D α , its perturbations and generalizations, as well as the theory of related partial (pseudo-) differential equations are covered by the monographs [1,16,34,30,36,14] and many recent papers, such as [4,2,3,17,19,20,37,33] and others.…”

“…The structure of this paper is as follows. In Section 2, we collect necessary preliminaries about local fields K, the structure of open sets in K n , the representation of K n in terms of the unramified extension of K [31,16,20], Sobolev spaces of complex-valued functions on local fields [11,9,10]. In Section 3, we prove analogs of classical inequalities.…”