On Shift Harnack Inequalities for Subordinate Semigroups and Moment Estimates for L\'evy Processes

We show that shift Harnack type inequalities (in the sense of F.-Y.~Wang \cite{Wan14}) are preserved under Bochner's subordination. The proofs are based on two types of moment estimates for subordinators. As a by-product we establish moment estimates for general L\'evy processes.
Comment: to appear in Stochastic Process. Appl