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

Each result cited is universally quantified over the data in its own statement. For $z\in\mathbb{R}^{d+d}$ we write $x=\mathrm{pr}_{1}(z)$ and $y=\mathrm{pr}_{2}(z)$, as in The Intrinsic Calculus on the Wasserstein Space: Standing Notation (i). For $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ let $\phi_{\mu}$ and $\chi_{\mu}$ be the functions on $\mathbb{R}^{d}$ attached to $\varphi$ and to $\chi$ at $\mu$ by property (d) of Intrinsic Test Functions on the Wasserstein Space and Their Translation Hessians (i), with values $\varphi((\tau_{e})_{\#}\mu)$ and $\chi((\tau_{e})_{\#}\mu)$ at $e\in\mathbb{R}^{d}$.

Claim 1. Let $Q'\subseteq Q$. Properties (a) and (d) of Intrinsic Test Functions on the Wasserstein Space and Their Translation Hessians (i) do not involve the set. Property (b) on $Q'$ asserts at each point of $Q'$ what property (b) on $Q$ asserts at that point, which lies in $Q$. In property (c) on $Q'$, a point $\mu\in Q'$, a sequence in $Q'$ and a sequence of couplings as there are also admissible data in property (c) on $Q$, which yields the required convergence. So $\varphi$ is an intrinsic test function on $Q'$. At $\mu\in Q'$ its gradient along couplings is, by Differentiability of a Function on the Wasserstein Space Along Couplings, and Its Gradient (ii), the unique field having the property of Differentiability of a Function on the Wasserstein Space Along Couplings, and Its Gradient (i), which involves only $\varphi$ and $\mu$; and its translation Hessian at $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ is $D^{2}\phi_{\mu}(0_{\mathbb{R}^{d}})$ by Intrinsic Test Functions on the Wasserstein Space and Their Translation Hessians (ii), which again involves only $\varphi$ and $\mu$.

Claim 2. Let $a,b\in\mathbb{R}$, put $\psi=a\varphi+b\chi$ and $c=|a|+|b|+1$. The numbers $|a|$ and $|b|$ are nonnegative by claim 1 of Properties of the Absolute Value in an Ordered Field and $0<1$ by claim 6 of Elementary Order Arithmetic in an Ordered Field, so $0\le|a|+|b|$ by claim 2 of Elementary Arithmetic in an Ordered Field and then $c$ is positive by claim 3 of Elementary Order Arithmetic in an Ordered Field, and $|a|+|b|\le c$ by the compatibility of the order with addition, an axiom of Ordered Field. We verify the four properties of Intrinsic Test Functions on the Wasserstein Space and Their Translation Hessians (i) for $\psi$ on $Q$.

(a) The functions $\varphi$ and $\chi$ are continuous on $\mathcal{P}_{2}(\mathbb{R}^{d})$ by property (a). By claim 5 of Continuity of Sums and Products of Real-Valued Functions on a Metric Space, read with $A$ the whole space $\mathcal{P}_{2}(\mathbb{R}^{d})$, the functions $a\varphi$ and $b\chi$ are continuous there, and then so is their sum $\psi$.

(b) Let $\mu\in Q$ and put $\eta=a\,\nabla\varphi(\mu)+b\,\nabla\chi(\mu)\in L^{2}(\mu;\mathbb{R}^{d})$. Let $\varepsilon\in\mathbb{R}$ be positive; then $\varepsilon c^{-1}$ is positive by claims 7 and 5 of Elementary Order Arithmetic in an Ordered Field. By property (b) for $\varphi$ and for $\chi$, Differentiability of a Function on the Wasserstein Space Along Couplings, and Its Gradient (i), read with $\varepsilon c^{-1}$ in place of its $\varepsilon$, provides positive $\theta_{1}$ for $\varphi$ and $\theta_{2}$ for $\chi$. Let $\theta=\min\{\theta_{1},\theta_{2}\}$ be their minimum, positive because it equals one of them by claim 2 of Elementary Properties of the Minimum of Two Elements. Let $\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and $\pi\in\Pi(\mu,\nu)$ satisfy $I(\pi)<\theta^{2}$. Since $\theta\le\theta_{1}$ and $\theta\le\theta_{2}$ by claim 1 of that lemma, all four numbers being nonnegative, claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field gives $\theta^{2}\le\theta_{1}^{2}$ and $\theta^{2}\le\theta_{2}^{2}$, whence $I(\pi)<\theta_{1}^{2}$ and $I(\pi)<\theta_{2}^{2}$ by claim 2 of Elementary Order Arithmetic in an Ordered Field. Put

$$A=\varphi(\nu)-\varphi(\mu)-\mathcal{J}\bigl(\nabla\varphi(\mu),\pi\bigr),\qquad B=\chi(\nu)-\chi(\mu)-\mathcal{J}\bigl(\nabla\chi(\mu),\pi\bigr),$$

so that $|A|\le\varepsilon c^{-1}\sqrt{I(\pi)}$ and $|B|\le\varepsilon c^{-1}\sqrt{I(\pi)}$. By The Displacement Pairing of a Square-Integrable Vector Field Along a Coupling (iii), $\mathcal{J}(\eta,\pi)=a\,\mathcal{J}(\nabla\varphi(\mu),\pi)+b\,\mathcal{J}(\nabla\chi(\mu),\pi)$, so the field axioms of Field give

$$\psi(\nu)-\psi(\mu)-\mathcal{J}(\eta,\pi)=aA+bB .$$

By claims 5 and 4 of Properties of the Absolute Value in an Ordered Field and claim 5 of Elementary Arithmetic in an Ordered Field,

$$|aA+bB|\le|a|\,|A|+|b|\,|B|\le\bigl(|a|+|b|\bigr)\,\varepsilon c^{-1}\sqrt{I(\pi)}\le\varepsilon\sqrt{I(\pi)},$$

the last step because $(|a|+|b|)c^{-1}\le c\,c^{-1}=1$, by claim 5 of Elementary Arithmetic in an Ordered Field applied to $|a|+|b|\le c$ with the nonnegative multiplier $c^{-1}$, and $\varepsilon\sqrt{I(\pi)}$ is nonnegative. Hence $\psi$ is differentiable along couplings at $\mu$ with gradient $\eta$, and $\nabla\psi(\mu)=\eta$ by Differentiability of a Function on the Wasserstein Space Along Couplings, and Its Gradient (ii). The fields $\nabla\varphi(\mu)$ and $\nabla\chi(\mu)$ lie in $T_{\mu}$ by property (b), and $T_{\mu}$ is a linear subspace of $L^{2}(\mu;\mathbb{R}^{d})$ by Basic Properties of the Tangent Space: Closed Subspace, the Identity Map Belongs to It, Second-Moment Limits, and Representation of Bounded Functionals on Gradients (i), so $\eta\in T_{\mu}$.

(c) Let $\mu$, $(\mu_{n})_{n\in\mathbb{N}}$ and $(\pi_{n})_{n\in\mathbb{N}}$ be as in property (c). For $n\in\mathbb{N}$ let $A_{n}$, $B_{n}$ and $C_{n}$ be the discrepancies along $\pi_{n}$ of $\nabla\varphi(\mu_{n})$ and $\nabla\varphi(\mu)$, of $\nabla\chi(\mu_{n})$ and $\nabla\chi(\mu)$, and of $\nabla\psi(\mu_{n})$ and $\nabla\psi(\mu)$. The sequences $(A_{n})$ and $(B_{n})$ converge to $0$ by property (c) for $\varphi$ and for $\chi$. By (b), $\nabla\psi(\mu_{n})=a\,\nabla\varphi(\mu_{n})+b\,\nabla\chi(\mu_{n})$ and $\nabla\psi(\mu)=a\,\nabla\varphi(\mu)+b\,\nabla\chi(\mu)$. Fix representatives $f_{n}$, $f$, $g_{n}$, $g$ of $\nabla\varphi(\mu_{n})$, $\nabla\varphi(\mu)$, $\nabla\chi(\mu_{n})$, $\nabla\chi(\mu)$; then $af_{n}+bg_{n}$ and $af+bg$, formed pointwise, represent $\nabla\psi(\mu_{n})$ and $\nabla\psi(\mu)$ by Square-Integrable Random Vectors: Coordinates, Operations, Almost Sure Equality and the Mean-Square Form (v), and the discrepancies do not depend on the representatives by The Discrepancy of Two Square-Integrable Vector Fields Along a Coupling of Their Base Measures (i). For $z\in\mathbb{R}^{d+d}$ put $U=f_{n}(x)-f(y)$ and $V=g_{n}(x)-g(y)$; then

$$\bigl\lVert\bigl(af_{n}(x)+bg_{n}(x)\bigr)-\bigl(af(y)+bg(y)\bigr)\bigr\rVert^{2}=\lVert aU+bV\rVert^{2}\le2\lVert aU\rVert^{2}+2\lVert bV\rVert^{2}=2|a|^{2}\lVert U\rVert^{2}+2|b|^{2}\lVert V\rVert^{2},$$

by the inequality $\lVert s-t\rVert^{2}\le2\lVert s\rVert^{2}+2\lVert t\rVert^{2}$ of Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (iv), read with $s=aU$ and $t=-bV$, and by claim 5 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square, which together with $|-b|=|b|$ (claim 2 of Properties of the Absolute Value in an Ordered Field) also gives $\lVert -bV\rVert=\lVert bV\rVert$. Integrating against $\pi_{n}$, all functions involved being nonnegative and Borel, claim 1 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion gives

$$0\le C_{n}\le2|a|^{2}A_{n}+2|b|^{2}B_{n}\qquad(n\in\mathbb{N}).$$

The right-hand side converges to $0$ by claims 1 and 3 of Arithmetic of Limits of Real Sequences, and the constant sequence $0$ converges to $0$, so $(C_{n})$ converges to $0$ by the squeeze principle, claim 2 of Order Properties of Limits of Real Sequences.

(d) Let $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$. The function attached to $\psi$ at $\mu$ by property (d) is $a\phi_{\mu}+b\chi_{\mu}$, of class $C^{2}$ on $\mathbb{R}^{d}$ by claim 3 of Constants, Affine Functions, Sums, Products and Quotients of Cᵏ Functions, since $\phi_{\mu}$ and $\chi_{\mu}$ are.

Therefore $\psi$ is an intrinsic test function on $Q$, and $\nabla\psi(\mu)=a\,\nabla\varphi(\mu)+b\,\nabla\chi(\mu)$ for $\mu\in Q$ by (b). For the translation Hessians let $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$. The partial derivatives of $\phi_{\mu}$ and $\chi_{\mu}$ exist at every point of $\mathbb{R}^{d}$, so by claim 1 of Constants, Affine Functions, Sums, Products and Quotients of Cᵏ Functions, $\partial_{j}(a\phi_{\mu}+b\chi_{\mu})=a\,\partial_{j}\phi_{\mu}+b\,\partial_{j}\chi_{\mu}$ on $\mathbb{R}^{d}$ for each index $j$; applying that claim once more at $0_{\mathbb{R}^{d}}$ gives

$$\partial_{i}\partial_{j}(a\phi_{\mu}+b\chi_{\mu})(0_{\mathbb{R}^{d}})=a\,\partial_{i}\partial_{j}\phi_{\mu}(0_{\mathbb{R}^{d}})+b\,\partial_{i}\partial_{j}\chi_{\mu}(0_{\mathbb{R}^{d}})$$

for all indices $i$ and $j$. The entries of a Hessian matrix are exactly these second partial derivatives, and matrices with the same entries are equal, so by Sum of Real Matrices, Scalar Multiple of a Real Matrix and Intrinsic Test Functions on the Wasserstein Space and Their Translation Hessians (ii), $H_{\psi}(\mu)=a\,H_{\varphi}(\mu)+b\,H_{\chi}(\mu)$.

Claim 3. In $\mathbb{R}$ one has $1s+(-1)t=s-t$, $(-1)s+0t=-s$ by the multiplicative identity axiom of the field $\mathbb{R}$ and claims 1 and 2 of Zero Products and Elementary Identities in a Field, so $\varphi-\chi$ is the function of claim 2 with $(a,b)=(1,-1)$ and $-\varphi$ the function of claim 2 with $(a,b)=(-1,0)$. Both are therefore intrinsic test functions on $Q$, with

$$\nabla(\varphi-\chi)(\mu)=1\,\nabla\varphi(\mu)+(-1)\nabla\chi(\mu),\qquad\nabla(-\varphi)(\mu)=(-1)\nabla\varphi(\mu)+0\,\nabla\chi(\mu)\qquad(\mu\in Q),$$

and the corresponding identities for the translation Hessians at every $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$. In the real vector space $L^{2}(\mu;\mathbb{R}^{d})$ one has $1\zeta=\zeta$ by the axioms of a vector space, $0\zeta=0$ and $(-1)\zeta=-\zeta$ by claims 3 and 5 of Elementary Identities in a Vector Space, and $\zeta+(-\omega)=\zeta-\omega$ by claim 2 of that lemma; for matrices the same identities hold entrywise by Scalar Multiple of a Real Matrix, Sum of Real Matrices, Difference of Real Matrices and claims 1 and 2 of Zero Products and Elementary Identities in a Field. Rewriting the displayed identities accordingly gives the assertion.