[proofplan] We use the Hamilton-Jacobi form of the Otto-Villani argument. The logarithmic Sobolev inequality and the stated approximation hypothesis give the Bobkov-Gentil-Ledoux infimum-convolution estimate for bounded Lipschitz functions. The entropy variational inequality converts that exponential estimate into a dual inequality. The bounded-Lipschitz form of quadratic [Kantorovich duality](/theorems/6799) then identifies the supremum of those dual quantities with $\frac{1}{2}W_2^2(\nu,\mu)$. [/proofplan]

[step:Invoke the Hamilton-Jacobi consequence of logarithmic Sobolev]Let $F:\mathbb{R}^n\to\mathbb{R}$ be bounded and Lipschitz. For $t>0$, define the map $Q_tF:\mathbb{R}^n\to\mathbb{R}$ by \begin{align*} Q_tF(x):=\inf_{y\in\mathbb{R}^n}\left\{F(y)+\frac{|x-y|^2}{2t}\right\}. \end{align*} We use the following standard Bobkov-Gentil-Ledoux theorem in the normalization matching the statement. If a probability measure $\mu$ on $\mathbb{R}^n$ satisfies the logarithmic Sobolev inequality with constant $C>0$ for $C_c^\infty(\mathbb{R}^n)$ test functions, and if bounded Lipschitz functions admit the approximation stated in the theorem, then every bounded Lipschitz function $F:\mathbb{R}^n\to\mathbb{R}$ satisfies \begin{align*} \int_{\mathbb{R}^n}\exp\left(\frac{1}{C}Q_1F(x)\right)\,d\mu(x) \leq \exp\left(\frac{1}{C}\int_{\mathbb{R}^n}F(x)\,d\mu(x)\right). \end{align*} The hypotheses of this theorem are exactly the hypotheses imposed on $\mu$ in the formal statement, so the estimate applies to the present function $F$. The logarithmic Sobolev normalization used here is \begin{align*} \operatorname{Ent}_{\mu}(f^2) \leq 2C\int_{\mathbb{R}^n}|\nabla f|^2\,d\mu \end{align*} for $f\in C_c^\infty(\mathbb{R}^n)$, where \begin{align*} \operatorname{Ent}_{\mu}(u) := \int_{\mathbb{R}^n}u\log u\,d\mu - \left(\int_{\mathbb{R}^n}u\,d\mu\right) \log\left(\int_{\mathbb{R}^n}u\,d\mu\right). \end{align*}[/step]

[guided]The first task is to set up the Hamilton-Jacobi transform with the same quadratic normalization that appears in the transport cost. Start with a bounded Lipschitz function $F:\mathbb{R}^n\to\mathbb{R}$. For $t>0$, define $Q_tF:\mathbb{R}^n\to\mathbb{R}$ by \begin{align*} Q_tF(x):=\inf_{y\in\mathbb{R}^n}\left\{F(y)+\frac{|x-y|^2}{2t}\right\}. \end{align*} This is the Hopf-Lax infimum convolution for the cost $|x-y|^2/(2t)$. The Bobkov-Gentil-Ledoux theorem says, in the normalization used here, that a logarithmic Sobolev inequality with constant $C>0$ implies the time-one exponential estimate with coefficient $1/C$. The extra approximation hypothesis in the formal statement is precisely what extends the estimate from $C_c^\infty(\mathbb{R}^n)$ test functions to bounded Lipschitz functions. Therefore, for the bounded Lipschitz function $F:\mathbb{R}^n\to\mathbb{R}$ chosen above, \begin{align*} \int_{\mathbb{R}^n}\exp\left(\frac{1}{C}Q_1F(x)\right)\,d\mu(x) \leq \exp\left(\frac{1}{C}\int_{\mathbb{R}^n}F(x)\,d\mu(x)\right). \end{align*} This display is the only analytic consequence of logarithmic Sobolev used later. The logarithmic Sobolev normalization behind it is \begin{align*} \operatorname{Ent}_{\mu}(f^2) \leq 2C\int_{\mathbb{R}^n}|\nabla f|^2\,d\mu \end{align*} for $f\in C_c^\infty(\mathbb{R}^n)$, with \begin{align*} \operatorname{Ent}_{\mu}(u) := \int_{\mathbb{R}^n}u\log u\,d\mu - \left(\int_{\mathbb{R}^n}u\,d\mu\right) \log\left(\int_{\mathbb{R}^n}u\,d\mu\right). \end{align*}[/guided]

[step:Convert the exponential estimate into a dual transport inequality]Let $\nu\in\mathcal{P}_2(\mathbb{R}^n)$ and first assume that $H(\nu\mid\mu)<+\infty$. Then $\nu\ll\mu$. The entropy variational inequality states that every bounded measurable function $G:\mathbb{R}^n\to\mathbb{R}$ satisfies \begin{align*} \int_{\mathbb{R}^n}G(x)\,d\nu(x) \leq H(\nu\mid\mu)+\log\int_{\mathbb{R}^n}e^{G(x)}\,d\mu(x). \end{align*} Apply this inequality to the bounded measurable function $G:\mathbb{R}^n\to\mathbb{R}$ defined by \begin{align*} G(x):=\frac{1}{C}Q_1F(x). \end{align*} Using the estimate from the previous step gives \begin{align*} \frac{1}{C}\int_{\mathbb{R}^n}Q_1F(x)\,d\nu(x) \leq H(\nu\mid\mu)+\frac{1}{C}\int_{\mathbb{R}^n}F(x)\,d\mu(x). \end{align*} Multiplying by $C$ yields \begin{align*} \int_{\mathbb{R}^n}Q_1F(x)\,d\nu(x) - \int_{\mathbb{R}^n}F(x)\,d\mu(x) \leq C H(\nu\mid\mu). \end{align*} This bound holds for every bounded Lipschitz function $F:\mathbb{R}^n\to\mathbb{R}$.[/step]

[guided]Let $\nu\in\mathcal{P}_2(\mathbb{R}^n)$, and first assume that \begin{align*} H(\nu\mid\mu)<+\infty. \end{align*} Then $\nu\ll\mu$. We use the entropy variational inequality: every bounded measurable function $G:\mathbb{R}^n\to\mathbb{R}$ satisfies \begin{align*} \int_{\mathbb{R}^n}G(x)\,d\nu(x) \leq H(\nu\mid\mu)+\log\int_{\mathbb{R}^n}e^{G(x)}\,d\mu(x). \end{align*} Choose the bounded measurable function $G:\mathbb{R}^n\to\mathbb{R}$ by \begin{align*} G(x):=\frac{1}{C}Q_1F(x). \end{align*} Substituting this choice and then using the estimate from the previous step gives \begin{align*} \frac{1}{C}\int_{\mathbb{R}^n}Q_1F(x)\,d\nu(x) \leq H(\nu\mid\mu)+\frac{1}{C}\int_{\mathbb{R}^n}F(x)\,d\mu(x). \end{align*} Multiplying by $C$ yields \begin{align*} \int_{\mathbb{R}^n}Q_1F(x)\,d\nu(x) - \int_{\mathbb{R}^n}F(x)\,d\mu(x) \leq C H(\nu\mid\mu). \end{align*} Thus the same dual bound holds for every bounded Lipschitz function $F:\mathbb{R}^n\to\mathbb{R}$.[/guided]

[step:Take the Kantorovich supremum]Define the quadratic cost $c:\mathbb{R}^n\times\mathbb{R}^n\to[0,\infty)$ by \begin{align*} c(x,y):=\frac{1}{2}|x-y|^2. \end{align*} Since $\nu\in\mathcal{P}_2(\mathbb{R}^n)$ and $\mu\in\mathcal{P}_2(\mathbb{R}^n)$, the standard quadratic Kantorovich duality theorem applies. In the bounded-Lipschitz form compatible with the preceding step, it says that \begin{align*} \frac{1}{2}W_2^2(\nu,\mu) = \sup_F\left\{ \int_{\mathbb{R}^n}Q_1F(x)\,d\nu(x) - \int_{\mathbb{R}^n}F(x)\,d\mu(x) \right\}, \end{align*} where the supremum is taken over bounded Lipschitz functions $F:\mathbb{R}^n\to\mathbb{R}$ and \begin{align*} Q_1F(x)=\inf_{y\in\mathbb{R}^n}\{F(y)+c(x,y)\}. \end{align*} The estimate from the previous step bounds every term in this supremum by $C H(\nu\mid\mu)$. Hence \begin{align*} \frac{1}{2}W_2^2(\nu,\mu)\leq C H(\nu\mid\mu). \end{align*} Multiplying by $2$ gives \begin{align*} W_2^2(\nu,\mu)\leq 2C H(\nu\mid\mu). \end{align*}[/step]

[guided]Define $c:\mathbb{R}^n\times\mathbb{R}^n\to[0,\infty)$ by \begin{align*} c(x,y):=\frac{1}{2}|x-y|^2. \end{align*} Both measures have finite second moment: $\nu\in\mathcal{P}_2(\mathbb{R}^n)$ in this step and $\mu\in\mathcal{P}_2(\mathbb{R}^n)$ by the formal statement. The standard quadratic Kantorovich duality theorem therefore applies. Its bounded-Lipschitz form follows from the usual lower-semicontinuous-cost duality for measures in $\mathcal{P}_2(\mathbb{R}^n)$ by the standard truncation and infimum-convolution density of bounded Lipschitz $c$-potentials. In that form, \begin{align*} \frac{1}{2}W_2^2(\nu,\mu) = \sup_F\left\{ \int_{\mathbb{R}^n}Q_1F(x)\,d\nu(x) - \int_{\mathbb{R}^n}F(x)\,d\mu(x) \right\}, \end{align*} where $F:\mathbb{R}^n\to\mathbb{R}$ ranges over bounded Lipschitz functions and \begin{align*} Q_1F(x)=\inf_{y\in\mathbb{R}^n}\{F(y)+c(x,y)\}. \end{align*} The previous step proves that every expression inside this supremum is at most $C H(\nu\mid\mu)$. Taking the supremum gives \begin{align*} \frac{1}{2}W_2^2(\nu,\mu)\leq C H(\nu\mid\mu). \end{align*} Multiplying by $2$ gives \begin{align*} W_2^2(\nu,\mu)\leq 2C H(\nu\mid\mu). \end{align*}[/guided]

[step:Handle infinite entropy]If $H(\nu\mid\mu)=+\infty$, then \begin{align*} W_2^2(\nu,\mu)\leq +\infty, \end{align*} which is true in the extended-real order. Combining this case with the finite-entropy argument proves the claim for every $\nu\in\mathcal{P}_2(\mathbb{R}^n)$.[/step]

[guided]It remains to include the case excluded in the finite-entropy part of the proof. If \begin{align*} H(\nu\mid\mu)=+\infty, \end{align*} then the desired estimate is \begin{align*} W_2^2(\nu,\mu)\leq +\infty, \end{align*} which is automatic in the extended-real order. Together with the finite-entropy argument, this proves the result for every $\nu\in\mathcal{P}_2(\mathbb{R}^n)$.[/guided]