Ten equivalent definitions of the fractional laplace operator

By Mateusz Kwaśnicki. 2015

696 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/c8c3d83feb4299664f527812b03e000b8b66c7b9

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In Applied Mathematics

1. Mikko Salo
2. Angkana Rüland
3. Minhao Cai
4. Jesse Railo
5. Keijo Monkkonen
6. Giovanni Covi
7. Changpin Li
8. Zhiping Mao

Citations in the corpus

• On Riesz derivative (2019, background): “([13, 22]) Let (−Δ)s with 0 < s < 1 be the fractional Laplacian defined by the singular integral definition (2.”
• Unique continuation property and Poincaré inequality for higher order fractional Laplacians with applications in inverse problems (2020, background): “If s ∈ (0, 1), the fractional Laplacian can be defined in several equivalent ways [46].”
• What is the fractional Laplacian? A comparative review with new results (2018, background): “The fractional Laplacian can be defined in Rd in many equivalent ways [5]; however, when these definitions are restricted to bounded domains, the associated boundary conditions lead to distinct operators.”
• Global uniqueness for the fractional semilinear Schrödinger equation (2017, neutral): “For the other equivalent definitions of the fractional Laplacian (−∆), we refer readers to [11].”
• Quantitative approximation properties for the fractional heat equation (2017, methodology): “We refer to [23] for further equivalent definitions of the fractional Laplacian.”
• Numerical approximation of fractional powers of elliptic operators (2013, background): “For a detailed discussion about this setting and other possible ways to define fractional powers of the Laplacian (and more general elliptic operators) we refer to [4, 14, 16].”
• Fractional Laplace Operator and Meijer G-function (2015, background): “24 in [24] in the setting of L spaces, see also [16, 25].”