Uniform phase-mixing bound from velocity derivatives ($\Vert {\rho }_t-{\rho }_{\infty }{\Vert }_{\infty }\le 2\zeta (m)\Vert {\partial }_v^m{f}_0{\Vert }_1/(2\pi |t|{)}^m$)

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.