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$L^\infty_\mathrm{loc}$ assumption in global existence for Boltzmann equation

I thought if we replace $1+\int f^n \, d\xi$ by $1+A_n * f_n$, then the rest of the argument (passing to the limit etc.) will be just fine. If we go this way, we don't need to assume $A\in L^{\infty}( …
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