On the Schauder Estimates for Non-local Equations with Drift: The Supercritical Case

Abstract

We establish a Schauder estimate for a nonlocal Cauchy problem with drift. The leading operator is the generator of a non-degenerate $\alpha$-stable process with $\alpha \in (0,1)$, and the drift is $\beta$-Hölder continuous with $\beta \in (1-\alpha ,1)$. The proof relies on a refined conic Littlewood--Paley decomposition and a one-dimensional one-sided dyadic maximum principle.

10 pages