Restrictions, Sums, Real Multiples and Differences of Intrinsic Test Functions on the Wasserstein Space

In the setting of The Intrinsic Calculus on the Wasserstein Space: Standing Notation, let $Q\subseteq\mathcal{P}_{2}(\mathbb{R}^{d})$. Intrinsic test functions on a subset of $\mathcal{P}_{2}(\mathbb{R}^{d})$ and their translation Hessians are those of that definition, and $\nabla\varphi(\mu)$ is the gradient along couplings of $\varphi$ at $\mu$. For $\varphi,\chi:\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}$ and $a,b\in\mathbb{R}$, the function $a\varphi+b\chi$ on $\mathcal{P}_{2}(\mathbb{R}^{d})$ takes the value $a\,\varphi(\mu)+b\,\chi(\mu)$ at $\mu$, and $\varphi-\chi$ and $-\varphi$ take the values $\varphi(\mu)-\chi(\mu)$ and $-\varphi(\mu)$.

Let $\varphi$ and $\chi$ be intrinsic test functions on $Q$. Then the following hold.

1. (Restriction)(i) For every $Q'\subseteq Q$ the function $\varphi$ is an intrinsic test function on $Q'$. Its gradient along couplings at a point of $Q'$ and its translation Hessian at a point of $\mathcal{P}_{2}(\mathbb{R}^{d})$ do not depend on whether $\varphi$ is regarded as an intrinsic test function on $Q$ or on $Q'$.

2. (Linear combinations)(ii) For all $a,b\in\mathbb{R}$ the function $a\varphi+b\chi$ is an intrinsic test function on $Q$, and

$$\nabla(a\varphi+b\chi)(\mu)=a\,\nabla\varphi(\mu)+b\,\nabla\chi(\mu)\quad(\mu\in Q),\qquad H_{a\varphi+b\chi}(\mu)=a\,H_{\varphi}(\mu)+b\,H_{\chi}(\mu)\quad\bigl(\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})\bigr).$$

3. (Differences)(iii) The functions $\varphi-\chi$ and $-\varphi$ are intrinsic test functions on $Q$; for $\mu\in Q$,

$$\nabla(\varphi-\chi)(\mu)=\nabla\varphi(\mu)-\nabla\chi(\mu),\qquad\nabla(-\varphi)(\mu)=-\nabla\varphi(\mu),$$

and for $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$,

$$H_{\varphi-\chi}(\mu)=H_{\varphi}(\mu)-H_{\chi}(\mu),\qquad H_{-\varphi}(\mu)=-H_{\varphi}(\mu).$$