= Rank-one gain on bounded continuous functions
{title2=$(Kf)(t,x,v)=k_1(t,x,v)\int k_2(w)f(t,x,w)\,dw$}
A bounded continuous factor $k_1$ and an integrable factor $k_2$ give a bounded velocity-integral gain on <bounded continuous functions>, with bound $\|Kf\|_\infty\leq\|k_1\|_\infty\|k_2\|_1\|f\|_\infty$. 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.
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