Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 105 / 2 / b

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- Solution b

Solution

The one-dimensional Sobolev representative of

u

is absolutely continuous, and its zero trace gives

u (x) = \int_{0}^{x} u^{'} (t) d t .

Consequently

u (x) / x = A (u^{'}) (x)

for the Hardy operator. Applying the Hardy averaging inequality to

u^{'}

gives the Hardy inequality on an interval:

\frac{u}{x}_{L^{p} (0, 1)} \leq \frac{p}{p - 1} ∥ u^{'} ∥_{L^{p} (0, 1)} .

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