Interior Schauder estimates for elliptic equations associated with Lévy operators

Abstract

We study the local regularity of solutions $f$ to the integro-differential equation $$Af=g\text{in U}$$ associated with the infinitesimal generator $A$ of a Lévy process $(X_t{)}_{t\ge 0}$. Under the assumption that the transition density of $(X_t{)}_{t\ge 0}$ satisfies a certain gradient estimate, we establish interior Schauder estimates for both pointwise and weak solutions $f$. Our results apply for a wide class of Lévy generators, including generators of stable Lévy processes and subordinated Brownian motions.