Moment estimates for L\'{e}vy Processes

For real L\'{e}vy processes $(X\_t)\_{t \geq 0}$ having no Brownian component with Blumenthal-Getoor index $\beta$, the estimate $\E \sup\_{s \leq t} | X\_s - a\_p s |^p \leq C\_p t$ for every $t \in [0,1]$ and suitable $a\_p \in \R$ has been established by Millar \cite{MILL} for $\beta < p \leq 2$ provided $X\_1 \in L^p$. We derive extensions of these estimates to the cases $p > 2$ and $p \leq\beta$.