Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 7 / 2 / d / 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 πi ξ)^{n + 1} f_{0} (k, ξ) = \partial_{v}^{n + 1} f_{0} (k, ξ) .

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

∣ f_{0} (k, ξ) ∣ ∣ ξ ∣^{n + 1} \leq \frac{∥ \partial _{v}^{n + 1} f _{0} ∥ _{1}}{( 2 π ) ^{n + 1}} \leq \frac{∥ f _{0} ∥ _{L_{x}^{1} W_{v}^{n + 1, 1}}}{( 2 π ) ^{n + 1}} .

Setting

ξ = k t

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.

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