Solution
= Solution
Take the positive integer $n\geq1$ and put $m=n+1\geq2$. From the preceding estimate and the absent zero <Fourier mode>, for $t\ne0$,
$$
\begin{aligned}
\sum_{k\in\mathbb Z}|\widehat r_t(k)|
&\leq\frac{\|\partial_v^m f_0\|_1}{(2\pi)^m|t|^m}
\sum_{k\ne0}\frac1{|k|^m}\\
&=\frac{2\zeta(m)\|\partial_v^m f_0\|_1}{(2\pi)^m|t|^m}.
\end{aligned}
$$
Here $\zeta$ is the <Riemann zeta function>, and its displayed series is finite because $m>1$. One may replace the last derivative <norm> by the given full mixed <Sobolev norm> to obtain the requested constant depending only on $n$ and $f_0$.
An absolutely summable sequence of <Fourier coefficients> gives a uniformly and absolutely convergent <Fourier series>, with supremum bounded by the sum of their absolute values. Its sum agrees almost everywhere with $r_t$ by uniqueness of <Fourier coefficients> for integrable periodic functions. Hence
$$
\boxed{\|\rho_t-\rho_\infty\|_{L^\infty(\mathbb T)}
\leq\frac{2\zeta(n+1)}{(2\pi)^{n+1}}
\frac{\|\partial_v^{n+1}f_0\|_1}{|t|^{n+1}}
\longrightarrow0.}
$$
This is the <uniform phase-mixing bound from velocity derivatives>: uniform convergence for the continuous representative of the density, with rate $O(|t|^{-n-1})$. No uniform decay of the full phase-space distribution is asserted. The <free-transport phase mixing> acts by shifting nonzero spatial modes to large velocity frequency. If a convention allows $0\in\mathbb N$, that endpoint needs separate assumptions or an argument: the harmonic series in this proof would diverge.