 Solution

The Fubini's theorem and integral triangle inequality giveDefine the normalized velocity-reset collision operator using the normalized velocity-reset projection $Pg(x,v)=\rho (g)(x)M(v)$ and $B=P-I$. Since $M\ge 0$ and $\int M=1$,Also $P^2=P$; the collision gain replaces the velocity distribution by $M$ while preserving the spatial mass. In Bochner integral notation the printed, undamped source operator isUsing the transport isometry and the integral triangle inequality provesThese estimates hold for measurable, locally time-bounded $E$-valued functions. If the displayed supremum is infinite, the numerical bound is interpreted in the extended sense; the construction below works in a space where it is finite.