[@JM19], the ideas from [@MK12spectral] were extended to non-symmetric Lévy processes. A different approach was proposed in [@GR17], where the authors prove and apply the global Harnack inequality in order to obtain sharp estimates on the tail probability of the first hitting time of points and bounded intervals for symmetric processes under global lower scaling condition imposed on the characteristic exponent. In the present paper we generalize these ideas to non-symmetric Lévy processes, which, to our best knowledge, has not been investigated in such generality before.

The article is organized as follows. In Section \[sec:prelims\] we introduce our setting, basic objects and tools exploited in the paper and prove some auxiliary results. In Section \[sec:regvar\] we prove that some specific form of the Lévy measure and regular variation of ${\operatorname{Re}}\psi$ implies that ${\operatorname{Im}}\psi$ is in fact, comparable to the real part; we apply that result in Section \[sec:asymptotics\], where asymptotic behaviour of the tail of the distribution of the first hitting time is obtained. Here, Theorem \[thm:Re\_asymp\_Im\] provides an important example. Section \[sec:harmonic\] is devoted to the proof of the Harnack inequality, some of its consequences and boundary behaviour of harmonic functions. We apply those results in Section \[sec:estimates\], where 