Ten equivalent definitions of the fractional Laplace operator

TL;DR

This paper reviews and unifies multiple definitions of the fractional Laplace operator across various function spaces, proving their equivalence and common domain, thus clarifying foundational aspects of fractional calculus.

Contribution

It systematically compares different definitions of the fractional Laplace operator and proves their equivalence on multiple function spaces, extending known results.

Findings

All definitions agree on their common domain.

Operators coincide on the common domain.

Results extend previous known equivalences.

Abstract

This article reviews several definitions of the fractional Laplace operator (-Delta)^{alpha/2} (0 < alpha < 2) in R^d, also known as the Riesz fractional derivative operator, as an operator on Lebesgue spaces L^p, on the space C_0 of continuous functions vanishing at infinity and on the space C_{bu} of bounded uniformly continuous functions. Among these definitions are ones involving singular integrals, semigroups of operators, Bochner's subordination and harmonic extensions. We collect and extend known results in order to prove that all these definitions agree: on each of the function spaces considered, the corresponding operators have common domain and they coincide on that common domain.