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Ten equivalent definitions of the fractional Laplace operator. (English) Zbl 1375.47038
Summary: This article discusses several definitions of the fractional Laplace operator \(L=-(-\Delta)^{\alpha/2}\) in \(\mathbb{R}^d\), also known as the Riesz fractional derivative operator; here \(\alpha\in (0,2)\) and \(d\geq 1\). 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 \(\mathcal L^p\) (with \(p\in [1,\infty))\), on the space \(\mathcal C_0\) of continuous functions vanishing at infinity and on the space \(\mathcal C_{bu}\) 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:
47G30 Pseudodifferential operators
35S05 Pseudodifferential operators as generalizations of partial differential operators
60J35 Transition functions, generators and resolvents
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