The Coupling Test Function of a Square-Integrable Plan

In the setting of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation, write $[0,\infty)=\{t\in\mathbb{R}:0\le t\}$. For $\varpi\in\Sigma^{2}_{2d}$ and $\lambda\in\Sigma^{2}_{d}$, $\mathcal{C}_{\varpi}(\lambda)\subseteq\Sigma^{2}_{3d}$ is the nonempty set of couplings of $\varpi$ with $\lambda$, defined through the affine data $B$ and $C$ of Couplings of a Square-Integrable Plan with a Law: Displacement and Momentum Pairing, with $B(X,P,X')=(X,P)$ and $C(X,P,X')=X'$; for $\gamma\in\Sigma^{2}_{3d}$, $s(\gamma)$ and $p(\gamma)$ are the displacement and momentum pairing of $\gamma$, computed from any realisation $(X,P,X')$ of $\gamma$ as $s(\gamma)=\lVert X'-X\rVert_{2}$ and $p(\gamma)=\langle P,X'-X\rangle_{2}$. The marginal data $\mathrm{pr}^{1},\mathrm{pr}^{2}$ and their push-forwards are those of Square-Integrable Noncommutative Laws: Standing Notation §affine, and the second moment $\widehat{M}$ is that of Square-Integrable Noncommutative Laws: Standing Notation §moments. Differences, the pairing $\langle\cdot,\cdot\rangle_{2}$ and the $L^{2}$ norm $\lVert\cdot\rVert_{2}$ of $L^{2}$ $d$-tuples are those of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §pairing, and laws are those of Tracial W*-Probability Spaces and Square-Integrable Tuples: Standing Notation §tuples.

Definition.¶ Let $h:[0,\infty)\to[0,\infty)$ be a function, let $\theta\ge0$ be real, and let $\varpi\in\Sigma^{2}_{2d}$. For $\lambda\in\Sigma^{2}_{d}$ and $\gamma\in\mathcal{C}_{\varpi}(\lambda)$ with a realisation $(X,P,X')$, one has $\mathrm{law}(X,P)=B_{\#}\gamma=\varpi$ and $\mathrm{law}(X')=C_{\#}\gamma=\lambda$ by Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §push-forward, so that $\lVert X\rVert_{2}^{2}=\widehat{M}(\mathrm{pr}^{1}_{\#}\varpi)$, $\lVert P\rVert_{2}^{2}=\widehat{M}(\mathrm{pr}^{2}_{\#}\varpi)$ and $\lVert X'\rVert_{2}^{2}=\widehat{M}(\lambda)$ by Every Square-Integrable Noncommutative Law is the Law of a Square-Integrable Tuple; Realisation of Couplings and of Almost Optimal Pairs §coupling and Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §moments; these norms depend only on $\varpi$ and $\lambda$. By the Cauchy--Schwarz and triangle inequalities in the complex Hilbert space of Sums, Real Multiples and the Pairing of Square-Integrable Tuples in a Tracial W*-Probability Space §pairing (in force by Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §background), $p(\gamma)\le\lVert P\rVert_{2}\,s(\gamma)$ and $s(\gamma)\le\lVert X\rVert_{2}+\lVert X'\rVert_{2}$; since $h\ge0$ and $\theta\ge0$,

$$p(\gamma)-h\bigl(s(\gamma)\bigr)-\theta\,s(\gamma)^{2}\le\lVert P\rVert_{2}\bigl(\lVert X\rVert_{2}+\lVert X'\rVert_{2}\bigr).$$

Hence the set $S^{h,\theta}_{\varpi}(\lambda)=\{p(\gamma)-h(s(\gamma))-\theta\,s(\gamma)^{2}:\ \gamma\in\mathcal{C}_{\varpi}(\lambda)\}$ is nonempty by Couplings of a Square-Integrable Plan with a Law: Displacement and Momentum Pairing §couplings and bounded above, and it has a least upper bound. The coupling test function of $\varpi$ with gauge $h$ and weight $\theta$ is

$$\Phi^{h,\theta}_{\varpi}:\Sigma^{2}_{d}\to\mathbb{R},\qquad\Phi^{h,\theta}_{\varpi}(\lambda)=\sup S^{h,\theta}_{\varpi}(\lambda).$$