Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 208 / 2 / c / Solution

Let $g:\mathbb{R}\to \mathbb{R}$ be continuously differentiable. Applying the Poincaré inequality for $p$ to $g\circ \varphi$ and using the chain rule givesThus the Pushforward of a Poincaré inequality by a Lipschitz function gives a $cL$-Poincaré inequality for $q$.