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

The one-dimensional Sobolev representative of $u$ is absolutely continuous, and its zero trace givesConsequently $u(x)/x=A(u^{\prime })(x)$ for the Hardy operator. Applying the Hardy averaging inequality to $u^{\prime }$ gives the Hardy inequality on an interval: