of the first hitting time of a point or a bounded interval, as well as its asymptotic behaviour. In the case of symmetric processes, it can be described by the compensated potential kernel (see [@GR17]), which is given by $$K(x) = \int_0^{\infty} \big( p(s,0)-p(s,x) \big)\,ds,$$ where $p(s,\cdot)$ is the transition density of the process (which exists due to the integral condition). However, in our setting one of substantial difficulties one has to overcome is the fact that we do not a priori know if $K$ exists, and even if it does exist, it may vanish on the whole half-line. For symmetric Lévy processes its existence is an easy consequence of the monotone convergence theorem, but in our case we are, in general, forced to adopt a different method. Instead, we propose an approach based on the symmetrized compensated potential kernel $$H(x) = \frac{1}{\pi} \int_0^{\infty} (1-\cos xs) {\operatorname{Re}}\Bigg[ \frac{1}{\psi(s)} \Bigg]\,ds,$$ which turns out to be the proper object for description of the behaviour of the first hitting time. Its huge advantage is the fact that the integrability condition we assume in the whole paper ensures that $H$ is well-defined and therefore can serve our purpose.

Let us briefly describe our results. 