Fractional operators with singular drift: smoothing properties and Morrey–Campanato spaces

Abstract

We investigate some smoothness properties for a linear transport-diffusion equation involving a class of non-degenerate Lévy type operators with singular drift. Our main argument is based on a duality method using the molecular decomposition of Hardy spaces through which we derive some Hölder continuity for the associated parabolic PDE. This property will be fulfilled as far as the singular drift belongs to a suitable Morrey–Campanato space for which the regularizing properties of the Lévy operator suffice to obtain global Hölder continuity.