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

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 ap...