  We study the asymptotic behaviour of the tail of the distribution of the
first passage time of a L\'evy process over a one-sided moving boundary. Our
main result states that if the boundary behaves as $t^{\gamma}$ for large $t$
for some $\gamma<1/2$ then the probability that the process stays below the
boundary behaves asymptotically as in the case of a constant boundary. We do
not have to assume Spitzer's condition in contrast to all previously known
results. Both positive ($+t^\gamma$) and negative ($-t^\gamma$) boundaries are
considered. These results extend the findings of Greenwood and Novikov (1986)
and are also motivated by results in the case of Brownian motion, for which the
above result was proved in Uchiyama (1980).
