A pointwise inequality for fractional laplacians

The fractional laplacian is an operator appearing in several evolution models where diffusion coming from a Lévy process is present but also in the analysis of fluid interphases. We provide an extension of a pointwise inequality that plays a rôle in their study. We begin recalling two scenarios where it has been used. After stating the results, for fractional Laplace–Beltrami and Dirichlet–Neumann operators, we provide a sketch of their proofs, unravelling the underlying principle to such inequalities.