  Under mild assumptions, we establish a Liouville theorem for the "Laplace"
equation $Au=0$ associated with the infinitesimal generator $A$ of a L\'evy
process: If $u$ is a weak solution to $Au=0$ which is at most of (suitable)
polynomial growth, then $u$ is a polynomial. As a by-product, we obtain new
regularity estimates for semigroups associated with L\'evy processes.
