  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.
