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

TL;DR

This paper studies the regularity of solutions to a transport-diffusion PDE with Lévy operators and singular drift, showing that under certain conditions, solutions are globally Hölder continuous.

Contribution

It introduces a duality approach using Hardy space decomposition to establish Hölder continuity for PDEs with singular drifts in Morrey-Campanato spaces.

Findings

Hölder continuity of solutions under specified conditions

Global regularity results for Lévy-type operators with singular drift

Application of duality and molecular decomposition methods

Abstract

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.