= Solution
For the <free transport equation>, the <mixed Lebesgue norm> dispersion estimate is
$$
\boxed{\|f(t)\|_{L_x^\infty L_v^1}
\leq |t|^{-d}\|f_{\mathrm{in}}\|_{L_x^1L_v^\infty},\qquad t\ne0.}
$$
Indeed, use the <method of characteristics> and then the <change of variables> $y=x-tv$:
$$
\int |f_{\mathrm{in}}(x-tv,v)|\,dv
=|t|^{-d}\int
\left|f_{\mathrm{in}}\left(y,\frac{x-y}{t}\right)\right|dy
\leq |t|^{-d}\int\sup_w|f_{\mathrm{in}}(y,w)|\,dy.
$$
Taking the <essential supremum> over $x$ proves the result. This decay measures the spreading of a velocity average; the total <phase space> <Lp norm> is conserved rather than decaying. The datum on the right is the initial datum, not the solution at the same time.
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