Ten Equivalent Definitions of the Fractional Laplace Operator

Abstract

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. MSC 2010 : 47G30, 35S05, 60J35

Fractional Laplace Operator and Meijer G-function

“…Indeed, let U be a region with compact closure contained in the set of admissible parameters of f under Condition A. Using the definition (16), with an appropriately modified contour of integration in the general case, one can prove that the constants in the asymptotic estimates (18) for the Meijer G-function can be chosen continuously with respect to the parameters; we omit the details. It follows that the supremum of |f (y)| taken over all parameters from U has, for some ε > 0, the following properties: (i) it is locally bounded in y = 0; (ii) it is O(|y| −d+ε ) as y → 0; (iii) it is O(|y| −α−ε ) as |y| → ∞.…”

“…This is made precise in Theorem 3.24 in [24] in the setting of L p spaces, see also [16,25]. We will need the following pointwise version of this result.…”

“…Under Condition A, formula (16) in fact defines the Meijer G-function as an analytic function in a sector | arg r| < (m + n − (p + q)/2)π. When condition B holds true (with arbitrary ν), the Meijer G-function extends to an analytic function in the unit disk |r| < 1 and in the complement of the unit disk |r| > 1.…”

On the Well-Posedness for a Class of Pseudo-Differential Parabolic Equations

“…We refer the reader to [17] for a recent review on the most common ones and the equivalence between them. For more details on the following decomposition we refer to [12].…”

Models of space-fractional diffusion: A critical review

“…For Ω = R, many of the different definitions coincide [11]. We mention three of them, which are linked to the operators in this section.…”