On the energy-critical fractional schrödinger equation in the radial case

We consider the Cauchy problem for the energy-critical nonlinear Schrödinger equation with fractional Laplacian (fNLS) in the radial case. We obtain global well-posedness and scattering in the energy space in the defo-cusing case, and in the focusing case with energy below the ground state. The main feature of the present work is the nonlocality of the operator. This does not allow us to use standard computations for the rigidity part of the theorem. Instead we develop a commutator argument which has its own interest for problems with nonlocal operators.