Free transport dispersion ($\Vert f(t){\Vert }_{L_x^{\infty }{L}_v^1}\le |t{|}^{-d}\Vert f_0{\Vert }_{L_x^1{L}_v^{\infty }}$)

For the free transport equation, substitute $y=x-tv$ into $\int |f_0(x-tv,v)|dv$. The resulting factor $|t{|}^{-d}$ and the bound by ${\sup }_v|f_0(y,v)|$ prove this mixed Lebesgue norm estimate. The right side contains initial data. The same-time mixed norm is not a conserved replacement: prescribing a product $\psi (x)\chi (v)$ at a later time and dilating $\chi$ disproves such a claim.