\ni x \mapsto x^{-\alpha}f(x)$$ is almost increasing. Clearly, the condition ${\operatorname{Re}}\psi \in {{\rm WLSC}(\alpha,c)}$ for some $\alpha>1$ and $c \in (0,1]$, implies . Moreover, by [@GS2019 Theorem 3.1], for all $x \in {\mathbb{R}}$, $$\label{eq:2}
    {\operatorname{Re}}\psi(x) \approx \psi^*(|x|) \approx h(1/|x|).$$ We note that under assumptions of the global scaling property of the real part of the characteristic exponent, its control over the imaginary part (see [@TG19 Lemma 12]) and vanishing of the first moment, i.e. ${\mathbb{E}}X_1=0$, one can show that $\int_0^1 {\operatorname{Re}}(1/\psi(\xi))d\xi=\infty$ and, in view of [@Bertoin96 Theorem I.17], conclude that $X_t$ is recurrent, and consequently, $u(x)=\infty$ for all $x \in {\mathbb{R}}$. As the processes we study in this paper often satisfy such assumptions, we are in dire need of some object alternative to the (infinite) potential kernel. The symmetrized compensated potential kernel $H$ can be of usage here, but it appears that the (ordinary) compensated kernel $K$ is more appropriate for description of hitting probability behaviour. The only problem is that in general we are not able to determine whether it exists. We devote Sections \[sec:asymptotics\] and \[sec:estimates\] to the detailed discussion on the subject.

We note one simple yet crucial observation.

\[prop:1\] Suppose that ${\mathbb{E}}X_1=0$ and ${\operatorname{Re}}\psi \in {{\rm 