[@MR3531705]). We also provide estimates on the boundary behaviour of harmonic functions. Meanwhile, we derive estimates on the Green function of a bounded interval and a half-line (Lemma \[lem:1\] and Corollary \[cor:green\_0inf\]), which also seem to be interesting on its own. For example, the first one together with some regularity assumptions on the Lévy masure imply the boundary Harnack inequality, which is also an important potential theoretic tool (see e.g. [@MR3271268]).

Eventually, we apply those results to obtain estimates on the tail of the first hitting time of points and intervals. They are derived under assumptions of global lower scaling property, zero mean and control of $K^{\lambda}$ from below by $H$ for small $\lambda$. The estimates are expressed in terms of symmetrized compensated potential kernel $H$ and renewal function for the dual process $\widehat{V}$, and therefore, in particular, do not require the existence of the compensated potential kernel $K$. Nevertheless, in Section \[sec:asymptotics\] we provide conditions which assure that $K$ is well-defined. We remark here that if the process is symmetric, then our assumptions reduce to those obtained in [@GR17]. Since the third assumption is not a priori obvious for general non-symmetric processes, in Subsection \[subsec:ex\] we present an example 