Gaussian logarithmic Sobolev inequality on $\mathbb{R}^\iota$: $\mathrm{Ent}_{\gamma_\iota}(g^2)\le 2\int\|\nabla g\|_2^2\,d\gamma_\iota$ for $C^1$ functions $g$

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Gaussian logarithmic Sobolev inequality on $\mathbb{R}^\iota$: $\mathrm{Ent}_{\gamma_\iota}(g^2)\le 2\int\|\nabla g\|_2^2\,d\gamma_\iota$ for $C^1$ functions $g$ · Prove2Me