Schauder estimates for Poisson equations associated with non-local Feller generators

We show how H\"older estimates for Feller semigroups can be used to obtain regularity results for solutions to the Poisson equation $Af=g$ associated with the (extended) infinitesimal generator $A$ of a Feller process. The regularity of $f$ is described in terms of H\"older-Zygmund spaces of variable order and, moreover, we establish Schauder estimates. Since H\"{o}lder estimates for Feller semigroups have been intensively studied in the last years, our results apply to a wide class of Feller processes, e.g. random time changes of L\'evy processes and solutions to L\'evy-driven stochastic differential equations. Most prominently, we establish Schauder estimates for the Poisson equation associated with the fractional Laplacian of variable order. As a by-product, we obtain new regularity estimates for semigroups associated with stable-like processes.