Schauder estimates for equations associated with L\'evy generators

We study the regularity of solutions to the integro-differential equation $Af-\lambda f=g$ associated with the infinitesimal generator $A$ of a L\'evy process. We show that gradient estimates for the transition density can be used to derive Schauder estimates for $f$. Our main result allows us to establish Schauder estimates for a wide class of L\'evy generators, including generators of stable L\'evy processes and subordinate Brownian motions. Moreover, we obtain new insights on the (domain of the) infinitesimal generator of a L\'evy process whose characteristic exponent $\psi$ satisfies $\text{Re} \, \psi(\xi) \asymp |\xi|^{\alpha}$ for large $|\xi|$. We discuss the optimality of our results by studying in detail the domain of the infinitesimal generator of the Cauchy process.