The Torus Displacement Pairing: Linearity, the Cost Bound, and Vanishing of a Field with First-Order Small Pairings

In the setting of Optimal Transport on the Flat Torus: Standing Notation, let $\mu\in\mathcal{P}(\mathbb{T}^{d})$, let $L^{2}(\mu;\mathbb{R}^{d})$ and $\lVert\cdot\rVert_{\mu}$ be as in Optimal Transport on the Flat Torus: Standing Notation §fields, and let $\mathcal{J}_{\mathbb{T}}$ be the torus displacement pairing. Then the following hold.

1. (Linearity)¶ For $\nu\in\mathcal{P}(\mathbb{T}^{d})$, $\gamma\in\Pi(\mu,\nu)$, $\eta,\eta'\in L^{2}(\mu;\mathbb{R}^{d})$ and real $a,b$: $\mathcal{J}_{\mathbb{T}}(a\eta+b\eta',\gamma)=a\,\mathcal{J}_{\mathbb{T}}(\eta,\gamma)+b\,\mathcal{J}_{\mathbb{T}}(\eta',\gamma)$.

2. (Cost bound)¶ For $\nu\in\mathcal{P}(\mathbb{T}^{d})$, $\gamma\in\Pi(\mu,\nu)$ and $\eta\in L^{2}(\mu;\mathbb{R}^{d})$: $|\mathcal{J}_{\mathbb{T}}(\eta,\gamma)|\le\lVert\eta\rVert_{\mu}\sqrt{I_{\mathbb{T}}(\gamma)}$.

3. (Vanishing)¶ Let $\eta\in L^{2}(\mu;\mathbb{R}^{d})$, and suppose that for every real $\varepsilon>0$ there is a real $\theta>0$ such that $|\mathcal{J}_{\mathbb{T}}(\eta,\gamma)|\le\varepsilon\sqrt{I_{\mathbb{T}}(\gamma)}$ for every $\nu\in\mathcal{P}(\mathbb{T}^{d})$ and every $\gamma\in\Pi(\mu,\nu)$ with $I_{\mathbb{T}}(\gamma)<\theta^{2}$. Then $\eta$ is the zero element of $L^{2}(\mu;\mathbb{R}^{d})$.