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. 
Chang-Song Deng and Ren\'e L. Schilling
math.PR math.FA