Solution
= Solution
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 $q$.