Linear Boltzmann collision operator ($K_{t}g(x,v)=\int k(t,x,v,v_{\ast })g(x,v_{\ast })d{v}_{\ast }$)

The gain term integrates over the incoming velocity. If $C^2=\int {\sup }_{t,x}|k(t,x,v,v_{\ast }){|}^{2}dvd{v}_{\ast }<\infty$, the Hilbert-Schmidt kernel bound at each $x$ gives $\Vert K_{t}g{\Vert }_{L_{x,v}^2}\le C\Vert g{\Vert }_{L_{x,v}^2}$, uniformly in time. The existence proof uses boundedness; kernel positivity is not needed for that estimate.

A bounded continuous factor $k_1$ and an integrable factor $k_2$ give a bounded velocity-integral gain on bounded continuous functions, with bound $\Vert Kf{\Vert }_{\infty }\le \Vert k_1{\Vert }_{\infty }\Vert k_2{\Vert }_1\Vert f{\Vert }_{\infty }$. The dominated convergence theorem proves joint continuity of the velocity integral. Combined with damped transport, the factorial bound for a Volterra iterate proves existence on arbitrary finite time slabs.