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Combine the damped free-transport evolution and the linear Boltzmann collision operator in a time-ordered Bochner integral. Its norm obeys $\Vert \tau f(t){\Vert }_2\le C{\int }_0^t\Vert f(s){\Vert }_{2}ds$. This reduces the characteristic integral equation to $(I-\tau )f=U_a(t,0)f_0$.

Fix $t,x$. The Cauchy-Schwarz inequality in the integration variable $v_{\ast }$ givesIntegrate first in $v$, then in $x$. The uniform kernel hypothesis bounds the first factor after velocity integration by $C^2$, independently of $t,x$. HenceThe linear Boltzmann collision operator is linear by its integral definition, so this proves it is a bounded linear operator on $L_{x,v}^2$. The estimate is the Hilbert-Schmidt kernel bound applied at each spatial point. The printed real-kernel square is $|k{|}^2$ for a complex kernel. Measurability of the coefficients is understood so that the displayed integrals are defined.