Its Laplace transform $$\label{eq:2}
\tilde J({\lambda}):=\int_0^{\infty}e^{-{\lambda}t}J(t)dt= \int_{{{\mathbb T}}}
\frac{\lambda}{\lambda^2 + \omega^2(k)} dk,\quad {\lambda}\in \mathbb C_+:=[z:\,{\rm Re}\,z>0].$$ One can easily see that ${\rm Re}\,\tilde J({\lambda})>0$ for ${\lambda}\in
\mathbb C_+$, therefore we can define the function $$\label{tg}
\tilde g(\lambda) := ( 1 + \gamma_1 \tilde J(\lambda))^{-1},\quad {\lambda}\in \mathbb C_+.$$ We have $$\label{012410}
|\tilde g(\lambda)|\le 1,\quad {\lambda}\in \mathbb C_+.$$ The function $\tilde g(\cdot)$ is analytic on $ \mathbb C_+$ so, by the Fatou theorem, see e.g. p. 107 of [@koosis], we know that $$\label{nu}
\nu(k) :=\lim_{{\epsilon}\to0+}\tilde g({\epsilon}-i{\omega}(k))$$ exists a.e. in ${{\mathbb T}}$ and in any $L^p({{\mathbb T}})$ for $p\in[1,\infty)$.

Let us introduce $$\label{033110}
\wp(k):=\frac{{\gamma}_1 \nu(k)}{2|\bar{\omega}'(k)|},\quad \fgeeszett(k):=\frac{{\gamma}_1|\nu(k)|^2}{|\bar{\omega}'(k)|},\quad p_+(k):=\left|1-\wp(k)\right|^2 ,\quad 
p_-(k):=|\wp(k)|^2  ,$$ where $
\bar{\omega}'(k):={\omega}'(k)/(2\pi)$. We have shown in [@kors] that $$\label{feb1402}
{\rm Re}\,\nu(k)=\left(1+\frac{{\gamma}_1}{2|\bar{\omega}'(k)|}\right)|\nu(k)|^2.$$ The functions $p_\pm(\cdot)$ and $\fgeeszett(\cdot)$ are even. Thanks to we have $$\label{012304}
p_+(k)+p_-(k)+\fgeeszett(k)=1.$$

### Linear kinetic equation with an interface

Let $L$ be the operator given by $$\label{L}
LF(k):=  2\int_{{{\mathbb T}}}R(k,k')
\left[F\left(k'\right) - F\left(k\right)\right]dk',\quad k\in{{\mathbb T}},$$ [for $F\in L^1({{\mathbb T}})$]{} and $$\begin{aligned}
\label{R}
&
R(k,k'):=\frac12\left\{r^2\left(k,k-
    k'\right)+r^2\left(k,k+k'\right)\right\}\\
&
=8{\frak s}^2(k){\frak s}^2(k')\left\{{\frak s}^2(k){\frak c}^2(k')+{\frak s}^2(k'){\frak c}^2(k)\right\},\quad k,k'\in{{\mathbb T}}.\nonumber\end{aligned}$$ Note that (cf ) the total scattering kernel equals $$\label{Rk}
 R(k):=\int_{{{\mathbb T}}}R(k,k')dk'=\frac{\hat\theta(k)}{4}.$$

\[df012603-19\] Given $T\in{{\mathbb R}}$, let ${\cal C}_T$ be a subclass of $C_b({{\mathbb R}}_*\times{{\mathbb 