Extensions of regularity for a Lévy process

We obtain necessary and sufficient conditions for the finiteness of certain moment functions of the random variable $T_0^-$, which is the first passage time of a Lévy process $(X_t)_{t\ge 0}$ below zero, and the position $X_{T_0^-}$ of the process at this time. Our results generalize classical results of Rogozin and Bertoin on the regularity of $X$, and extend earlier results of Blumenthal and Getoor on the regularity index.