Schauder Estimates for Equations Associated with Levy Generators

摘要

We study the regularity of solutions to the integro-differential equation Af-f=g associated with the infinitesimal generator A of a Levy 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 Levy generators, including generators of stable Levy processes and subordinated Brownian motions. Moreover, we obtain new insights on the (domain of the) infinitesimal generator of a Levy process whose characteristic exponent satisfies. We discuss the optimality of our results by studying in detail the domain of the infinitesimal generator of the Cauchy process.