Some results on powers of the Laplacian between 1 and 2

We will review some results --obtained in collaboration with Sven Jarohs (Frankfurt) and Alberto Saldaña (Karlsruhe)-- concerning higher-order fractional Laplacians, which are nonlocal operators of non-integer order larger than 2. These pertain the validity (or rather the failure) of maximum principles, the extension of Boggio's formula for the Green function, the structure and the boundary behaviour of corresponding harmonics, the formulation of suitable natural boundary conditions, and the pointwise evaluation of the operators.