Oscillatory phenomena for higher-order fractional Laplacians

We collect some peculiarities of higher-order fractional Laplacians (-Delta)(s), s > 1, with special attention to the range s is an element of (1, 2), which show their oscillatory nature. These include the failure of the polarization and Polya-Szego inequalities and the explicit example of a domain with sign-changing first eigenfunction. In spite of these fluctuating behaviours, we prove how the Faber-Krahn inequality still holds for any s > 1 in dimension one.