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

We study the local regularity of solutions $f$ to the integro-differential equation $$Af=g \quad \text{in $U$}$$ associated with the infinitesimal generator $A$ of a L\'evy process $(X_t)_{t \geq 0}$. Under the assumption that the transition density of $(X_t)_{t \geq 0}$ 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.