Free-transport phase mixing (${\hat{f}}_t(k,\xi )={\hat{f}}_0(k,\xi +kt)$)

On the unit-period spatial circle, free streaming has solution $f_t(x,v)=f_0(x-tv,v)$. With Fourier phase $e^{-2\pi i(kx+\xi v)}$, its mixed Fourier transform is ${\hat{f}}_t(k,\xi )={\hat{f}}_0(k,\xi +kt)$. A nonzero spatial mode of the velocity-integrated density therefore samples increasingly large velocity frequencies. The conserved spatial zero mode remains, while extra velocity regularity supplies quantitative decay of the other modes.

For $m\ge 2$ integrable weak velocity derivatives on a unit-period spatial circle, the Fourier transform of a derivative gives $|{\hat{f}}_0(k,kt)|\le \Vert {\partial }_v^m{f}_0{\Vert }_1/(2\pi |kt|{)}^m$ for $k\ne 0$. Summing these modes uses ${\sum }_{k\ne 0}|k{|}^{-m}=2\zeta (m)$. The resulting absolute Fourier summability gives the displayed uniform bound for the continuous representative of the density. For $m=1$ the same summation diverges, so this argument needs additional input.