 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, givesThe 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. ThereforeSetting $\xi =kt$ and taking the supremum gives the required uniform weighted bound. The $k=0$ term on the left is zero; there is no division by that frequency in this argument.