  We study the asymptotic tail behaviour of the first-passage time over a
moving boundary for asymptotically $\alpha$-stable L\'evy processes with
$\alpha<1$.
  Our main result states that if the left tail of the L\'evy measure is
regularly varying with index $- \alpha$ and the moving boundary is equal to $1
- t^{\gamma}$ for some $\gamma<1/\alpha$, then the probability that the process
stays below the moving boundary has the same asymptotic polynomial order as in
the case of a constant boundary. The same is true for the increasing boundary
$1 + t^{\gamma}$ with $\gamma<1/\alpha$ under the assumption of a regularly
varying right tail with index $- \alpha$.
