First, we concentrate on the asymptotic behaviour of the first hitting time of arbitrary compact sets which contain the origin (Theorem \[gtw\] and Corollary \[wgtw\]). The obtained asymptotics hold true if ${\operatorname{Re}}\psi$ varies regularly with parameter $\alpha \in (1,2]$ and ${\operatorname{Im}}\psi$ displays a similar behaviour. We devote Section \[sec:regvar\] to discussion about situations in which such condition holds true (see Theorem \[thm:Re\_asymp\_Im\]). In particular, that turns out to be true if the Lévy measure is of the special form $$\nu(dx) = C_d {{\bf 1}}_{x<0}\nu_0(dx)+C_u {{\bf 1}}_{x>0}\nu_0(dx).$$ An important class of processes which clearly exhibit such behaviour are spectrally one-sided Lévy processes.

Next, we turn our attention to derivation of sharp two-sided estimates on the first hitting time. To that end, we prove the global scale invariant Harnack inequality under weak lower scaling condition on the real part of the characteristic exponent. That result seems to be of some value in and of itself, as Harnack inequality is very strong theoretic tool in the potential theory. For instance, its easy consequences are estimates on the derivative of the renewal function for the ladder height process, which in turn entail estimates on the density of the distribution of the supremum process (see 