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

Two-term Weyl-type asymptotic law for the eigenvalues of one-dimensional fractional Laplace operator (−∆) α/2 (α ∈ (0, 2)) in the interval (−1, 1) is given: the n-th eigenvalue is equal to (nπ/2 − (2 − α)π/8) α + O(1/n). Simplicity of eigenvalues is proved for α ∈ [1, 2). L 2 and L ∞ properties of eigenfunctions are studied. We also give precise numerical bounds for the first few eigenvalues.

We prove a uniform boundary Harnack inequality for nonnegative harmonic functions of the fractional Laplacian on arbitrary open set D. This yields a unique representation of such functions as integrals against measures on D c ∪{∞} satisfying an integrability condition. The corresponding Martin boundary of D is a subset of the Euclidean boundary determined by an integral test.