= Solution
Denote the <damped free-transport evolution> from time $s$ to time $t$ by
$$
(U_a(t,s)g)(x,v)=
e^{-\int_s^t a(r,x-v(t-r),v)\,dr}g(x-v(t-s),v).
$$
By the argument in (b), $\|U_a(t,s)g\|_2\leq\|g\|_2$. The integral operator is the <Boltzmann Volterra operator>
$$
\tau f(t)=\int_0^tU_a(t,s)K_sf(s)\,ds.
$$
The <Minkowski integral inequality> and (a) give the useful stronger pointwise bound
$$
\|\tau f(t)\|_2\leq C\int_0^t\|f(s)\|_2\,ds.
$$
Therefore \b[the requested estimate is]
$$
\boxed{\|\tau f(t)\|_2\leq Ct\,\sup_{0\leq s\leq t}\|f(s)\|_2.}
$$
For strongly measurable bounded $L^2$-valued $f$, these are <Bochner integrals>; their finite norm bounds establish existence of the integrals.
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