Interior Schauder estimates for elliptic equations associated with L\'evy operators

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 L'evy 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 L'evy generators, including generators of stable L'evy processes and subordinated Brownian motions.