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Take the positive integer $n\ge 1$ and put $m=n+1\ge 2$. From the preceding estimate and the absent zero Fourier mode, for $t\ne 0$,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. HenceThis 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.