WLSC}(\alpha,\chi)}$ for some $\alpha>1$ and $\chi \in (0,1]$. There is $c {\geqslant}1$ such that for all $r>0$, $$c^{-1}\frac{1}{rh(r)} {\leqslant}H(r) {\leqslant}c \frac{1}{rh(r)}.$$ In particular, $H \in {{\rm WLSC}(\alpha-1,\tilde{\chi})}$ for some $\tilde{\chi} \in (0,1]$.

Observe that by [@TG19 Lemma 12], for any $r>0$, $$H(r) \approx \int_0^{\infty} (1-\cos rs) \frac{1}{{\operatorname{Re}}\psi(s)}\,ds.$$ Now, the claim follows by [@TG19 Lemma 13] and [@GS2019 Lemma 2.3].

Apart from *free processes*, for an open set $D$ one can consider *the process killed when exiting $D$*, denoted by $X_t^D$. Namely, for any measurable function $f$, $${\mathbb{E}}^x f \big( X_t^D \big) = {\mathbb{E}}^x \big[t<{\tau_D};\,f(X_t)\big], \quad t>0,\,x \in {\mathbb{R}}.$$ By analogy, by $p^D(t;\cdot,\cdot)$ we denote its transition density. It is then known that $$p^D(t;x,y) = p(t,y-x)-{\mathbb{E}}^x \big[ t>{\tau_D};\,p\big({\tau_D}-t,y-X_{{\tau_D}}\big) \big], \quad t>0,\,x,y \in {\mathbb{R}}.$$ By analogy to the free process, for any $x \in {\mathbb{R}}$ we define *the $\lambda$-potential measure* of $X_t^D$ as $$G^{\lambda}_D(x,A) = \int_0^{\infty} e^{-\lambda t} {\mathbb{P}}^x \big( X_{{\tau_D}} \in A \big) \,dt, \quad A \in {\mathcal{B}}_{{\mathbb{R}}}.$$ Since $p^D(t;\cdot,\cdot)$ exists for all $t>0$, the $\lambda$-potential measures are absolutely continuous with the density $G_D^{\lambda}$ given by $$G_D^{\lambda}(x,y) = \int_0^{\infty} e^{-\lambda t} p^D(t;x,y)\,dt, \quad x,y \in D.$$ The $0$-Green function is simply called *the Green function* and denoted by $G_D(x,y)$.

Next, we 