= Solution
Interpret the given velocity regularity in the stated <Sobolev space> sense. Repeated <integration by parts>, or the <Fourier transform of a derivative> in distributions, gives
$$
(2\pi i\xi)^{n+1}\widehat f_0(k,\xi)
=\widehat{\partial_v^{n+1}f_0}(k,\xi).
$$
The usual one-dimensional <one-dimensional Sobolev representative> or a smooth approximation justifies this identity without imposing extra decay of classical derivatives at specific boundary points. The transform of an integrable function has absolute value at most its $L^1$ <norm>. Therefore
$$
|\widehat f_0(k,\xi)|\,|\xi|^{n+1}
\leq\frac{\|\partial_v^{n+1}f_0\|_1}{(2\pi)^{n+1}}
\leq\frac{\|f_0\|_{L_x^1W_v^{n+1,1}}}{(2\pi)^{n+1}}.
$$
Setting $\xi=kt$ and taking the supremum gives the required \b[uniform weighted bound]. The $k=0$ term on the left is zero; there is no division by that frequency in this argument.
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