 Pushforward of a Poincaré inequality by a Lipschitz function

If $\mu$ has Poincaré constant $C$ and $\varphi$ is $L$-Lipschitz, then the pushforward measure ${\varphi }_{\ast }\mu$ has Poincaré constant at most $C{L}^2$. Apply the original inequality to $g\circ \varphi$ and use the chain rule.