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For standard Gaussian measure $\gamma$, the sharp Poincaré constant is one:

Apply the Poincaré inequality in probability theory to $g=e^{\lambda f/2}$. Since $\Vert \nabla f\Vert \le 1$,HenceIterating this estimate $m$ times yieldsvalid under the stated bound on $\lambda$.

A probability distribution $\mu$ has Poincaré constant $C_P$ when every sufficiently regular $f$ satisfies

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.

If each probability measure ${\mu }_i$ has Poincaré constant $C_i$, then the product measure ${⨂}_i{\mu }_i$ has Poincaré constant at most ${\max }_i{C}_i$. This follows by iterating the law of total variance and applying Jensen inequality to derivatives of conditional expectations.