= Free transport dispersion
{title2=$\|f(t)\|_{L_x^\infty L_v^1}\leq|t|^{-d}\|f_0\|_{L_x^1L_v^\infty}$}
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.
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