LAUSR

We study the regularity of solutions to the integro-differential equation $$Af-\lambda f=g$$Af-λf=g associated with the infinitesimal generator A of a Lévy 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 Lévy generators, including generators of stable Lévy processes and subordinated Brownian motions. Moreover, we obtain new insights on the (domain of the) infinitesimal generator of a Lévy process whose characteristic exponent $$\psi $$ψ satisfies $$\text {Re} \, \psi (\xi ) \asymp |\xi |^{\alpha }$$Reψ(ξ)≍|ξ|α for large $$|\xi |$$|ξ|. We discuss the optimality of our results by studying in detail the domain of the infinitesimal generator of the Cauchy process.