Ten Equivalent Definitions of the Fractional Laplace Operator

• Published in: Fractional calculus & applied analysis Vol. 20; no. 1; pp. 7 - 51
• Main Author: Kwasnicki, Mateusz
• Format: Journal Article
• Language: English
• Published: Warsaw Versita 01.02.2017; De Gruyter; Nature Publishing Group
• Subjects: 35S05; 47G30; 60J35; Abstract Harmonic Analysis; Analysis; Balakrishnan’s formula; Bochner’s subordination; Continuity (mathematics); Derivatives; Dynkin’s characteristic operator; Equivalence; extension technique; fractional Laplacian; Functional Analysis; Functions (mathematics); Integral Transforms; Mathematical analysis; Mathematics; Operational Calculus; Operators; Operators (mathematics); Riesz potential; Scripts; singular integral; Survey Paper; weak definition; Dynkin’s characteristic operator; extension technique; 47G30; fractional Laplacian; singular integral; Bochner’s subordination; 60J35; Balakrishnan’s formula; 35S05; Riesz potential; weak definition
• ISSN: 1311-0454; 1314-2224
• DOI: 10.1515/fca-2017-0002

This article discusses several definitions of the fractional Laplace operator L = — (—Δ) α /2 in R d , also known as the Riesz fractional derivative operator; here α ∈ (0,2) and d ≥ 1. This is a core example of a nonlocal pseudo-differential operator, appearing in various areas of theoretical and applied mathematics. As an operator on Lebesgue spaces ℒ p (with p ∈ [1,∞)), on the space 𝒞 0 of continuous functions vanishing at infinity and on the space 𝒞 bu of bounded uniformly continuous functions, L can be defined, among others, as a singular integral operator, as the generator of an appropriate semigroup of operators, by Bochner’s subordination, or using harmonic extensions. It is relatively easy to see that all these definitions agree on the space of appropriately smooth functions. We collect and extend known results in order to prove that in fact all these definitions are completely equivalent: on each of the above function spaces, the corresponding operators have a common domain and they coincide on that common domain.