Fractional operators with singular drift: Smoothing properties and Morrey-Campanato spaces

We investigate some smoothness properties for a transport-diffusion equation involving a class of non-degerate L{\'e}vy 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 H{\"o}lder 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 L{\'e}vy operator suffice to obtain global H{\"o}lder continuity.