The Laplacian of a Laplacian Test Function on the Torus Wasserstein Space

In the setting of Optimal Transport on the Flat Torus: Standing Notation, let $\varphi$ be a Laplacian test function and let $g_{\varphi}$ be its gradient field, which is unique by Laplacian Test Functions on the Torus: Uniqueness of the Gradient Field, a Criterion, and Linearity of the Test Classes §unique. For $\mu\in\mathcal{P}(\mathbb{T}^{d})$ each $\partial_{i}g_{\varphi,i}(\mu,\cdot)$ belongs to $C_{\mathrm{per}}$ by Gradient Fields and Laplacian Test Functions on the Torus Wasserstein Space §regular and Elementary Properties of Lattice-Periodic Functions §derivative, so the sum $x\mapsto\sum_{i=1}^{d}\partial_{i}g_{\varphi,i}(\mu,x)$ belongs to $C_{\mathrm{per}}$ by Finite Linear Combinations of Continuous Periodic Functions and Their Classes on the Torus §member and is integrable with respect to $\mu$ by The Torus Wasserstein Distance: Comparison with the Euclidean Distance, Wrapping, and Integrals of Periodic Functions §integrals.

¶The Laplacian of $\varphi$ is the function $\mathcal{L}\varphi:\mathcal{P}(\mathbb{T}^{d})\to\mathbb{R}$,

$$\mathcal{L}\varphi(\mu)=\int\sum_{i=1}^{d}\partial_{i}g_{\varphi,i}(\mu,x)\,\mu(dx).$$