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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.

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.