Let $g:\mathbb R\to\mathbb R$ be continuously differentiable. Applying the Poincaré inequality for $p$ to $g\circ\phi$ and using the <chain rule> gives
$$
\begin{aligned}
\operatorname{Var}_q(g(Y))
&=\operatorname{Var}_p(g(\phi(X)))\\
&\leq c^2\mathbb E_p\lVert g'(\phi(X))\nabla\phi(X)\rVert^2\\
&\leq c^2L^2\mathbb E_q[g'(Y)^2].
\end{aligned}
$$
Thus the <Pushforward of a Poincaré inequality by a Lipschitz function> gives a $cL$-Poincaré inequality for $q$.

Solution

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

{c}

If $\mu$ has Poincaré constant $C$ and $\phi$ is $L$-Lipschitz, then the <pushforward measure> $\phi_*\mu$ has Poincaré constant at most $CL^2$. Apply the original inequality to $g\circ\phi$ and use the <chain rule>.

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 Incoming links: Pushforward of a Poincaré inequality by a Lipschitz function