Laplacian Test Functions on the Torus: Uniqueness of the Gradient Field, a Criterion, and Linearity of the Test Classes

In the setting of Optimal Transport on the Flat Torus: Standing Notation, let intrinsic test functions on a subset of $\mathcal{P}(\mathbb{T}^{d})$ and their gradients along couplings $\nabla\varphi(\mu)$ be as in that definition, let gradient fields and Laplacian test functions be as in Gradient Fields and Laplacian Test Functions on the Torus Wasserstein Space, and let $T_{\mu}$ be the tangent space at $\mu$. Then the following hold.

1. (Uniqueness of the gradient field)¶ Every $\varphi:\mathcal{P}(\mathbb{T}^{d})\to\mathbb{R}$ has at most one gradient field.

2. (Criterion)¶ Let $\varphi:\mathcal{P}(\mathbb{T}^{d})\to\mathbb{R}$ and let $g$ be a gradient field of $\varphi$ such that the class of $g(\mu,\cdot)$ in $L^{2}(\mu;\mathbb{R}^{d})$ belongs to $T_{\mu}$ for every $\mu\in\mathcal{P}(\mathbb{T}^{d})$. Then $\varphi$ is a Laplacian test function.

3. (Linearity)¶ Let $\mathcal{K}\subseteq\mathcal{P}(\mathbb{T}^{d})$, let $\varphi$ and $\psi$ be intrinsic test functions on $\mathcal{K}$, and let $a,b$ be real numbers. Then $a\varphi+b\psi$ is an intrinsic test function on $\mathcal{K}$ with $\nabla(a\varphi+b\psi)(\mu)=a\nabla\varphi(\mu)+b\nabla\psi(\mu)$ for every $\mu\in\mathcal{K}$. If $\varphi$ and $\psi$ are Laplacian test functions with gradient fields $g$ and $h$, then $a\varphi+b\psi$ is a Laplacian test function with gradient field $ag+bh$.