Linear Boltzmann collision operator
= Linear Boltzmann collision operator
{title2=$K_tg(x,v)=\int k(t,x,v,v_*)g(x,v_*)\,dv_*$}
The gain term integrates over the incoming velocity. If $C^2=\int\sup_{t,x}|k(t,x,v,v_*)|^2\,dv\,dv_*<\infty$, the <Hilbert-Schmidt kernel bound> at each $x$ gives $\|K_tg\|_{L^2_{x,v}}\leq C\|g\|_{L^2_{x,v}}$, uniformly in time. The existence proof uses boundedness; kernel positivity is not needed for that estimate.