Non-local diffusion equations with Lévy-type operators and divergence free drift

We are interested in some properties related to the solutions of non-local diffusion equations with divergence free drift. Existence, maximum principle and a positivity principle are proved. In order to study Holder regularity, we apply a method that relies in the Holder-Hardy spaces duality and in the molecular characterisation of local Hardy spaces. In these equations, the diffusion is given by Lévy-type operators with an associated Lévy measure satisfying some upper and lower bounds.