Schauder estimates for equations associated with Lévy generators

Abstract

We study the regularity of solutions to the integro-differential equation $Af-\lambda f=g$ 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 subordinate Brownian motions. Moreover, we obtain new insights on the (domain of the) infinitesimal generator of a Lévy 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.

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