Estimates and structure of $\alpha$-harmonic functions
  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 $D^c\cup \{\infty\}$ satisfying an integrability condition. The corresponding Martin boundary of $D$ is a subset of the Euclidean boundary determined by an integral test. 
Krzysztof Bogdan, Tadeusz Kulczycki, Mateusz Kwa\'snicki