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Ten equivalent definitions of the fractional laplace operator

Authors
  • Kwaśnicki, Mateusz
Type
Published Article
Journal
Fractional Calculus and Applied Analysis
Publisher
De Gruyter
Publication Date
Feb 01, 2017
Volume
20
Issue
1
Pages
7–51
Identifiers
DOI: 10.1515/fca-2017-0002
Source
De Gruyter
Keywords
License
Yellow

Abstract

This article discusses several definitions of the fractional Laplace operator L = — (—Δ)α/2 in Rd , 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 ℒp (with p ∈ [1,∞)), on the space 𝒞0 of continuous functions vanishing at infinity and on the space 𝒞bu 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.

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