Proof of Linear and Quadratic Kernel Functionals of a Measure Are Test Functions on the Wasserstein Space

Each result cited is universally quantified over the data in its own statement. Throughout, a function of class $C^{1}$ on $\mathbb{R}^{d}$ and its partial derivatives are continuous (clause 1 of Functions of Class Cᵏ on an Open Subset of ℝⁿ, claim 3 of Euclidean Space Is Open in Itself), hence Borel (claim 3 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets); a map $\mathbb{R}^{n}\to\mathbb{R}^{d}$ is Borel when its components are, and compositions of Borel maps are Borel (Probability Measures on Euclidean Space and Random Vectors: Standing Notation (ii)); a bounded Borel function is integrable with respect to every probability measure (Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iii)); the integral of a constant against a probability measure is that constant (Simple Functions, Nonnegative Simple Functions, and the Simple Integral); "change of variables" refers to Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iv); $\lVert x\rVert^{2}=\sum_{i}x_{i}^{2}$ and $|x_{i}|\le\lVert x\rVert$ (claims 1 and 4 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square); and for $u,w\in\mathbb{R}^{d}$, $|u\cdot w|\le\lVert u\rVert\lVert w\rVert$ and $\lVert u-w\rVert^{2}\le2\lVert u\rVert^{2}+2\lVert w\rVert^{2}$ (Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (iv)). For a function $g$ of class $C^{1}$ on $\mathbb{R}^{d}$ with $|\partial_{i}g|\le C$ for all $i$, part (i) of Multivariate Taylor Expansion with Uniform Second-Order Remainder (the segment between any two points lies in $\mathbb{R}^{d}$, and the Euclidean distance of $u$ and $w$ is $\lVert u-w\rVert$ by claim 2 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square) gives

$$|g(u)-g(w)|\le\sqrt{d}\,C\,\lVert u-w\rVert\qquad(u,w\in\mathbb{R}^{d}),\tag{Lip}$$

and if $g$ is of class $C^{2}$ with $|\partial_{j}\partial_{i}g|\le C$, part (ii) gives, with $Dg(u)\cdot w=\sum_{i}\partial_{i}g(u)w_{i}$,

$$|g(u+w)-g(u)-Dg(u)\cdot w|\le\tfrac12\,d\,C\,\lVert w\rVert^{2}\qquad(u,w\in\mathbb{R}^{d}).\tag{Tay}$$

For a Borel map $\Gamma:\mathbb{R}^{d}\to\mathbb{R}^{d}$ with $|\Gamma_{i}|\le C$ one has $\lVert\Gamma\rVert^{2}\le dC^{2}$ (claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, claim 1 of Comparison and Absolute Value Bounds for Finite Sums of Real Numbers), so its class lies in $L^{2}(\mu;\mathbb{R}^{d})$ for every $\mu\in\mathcal{P}(\mathbb{R}^{d})$; for $X\in L^{2}(\Omega;\mathbb{R}^{d})$ with $\mathcal{L}(X)=\mu$, $\Gamma\circ X$ is the class of Composition of a Square-Integrable Vector Field with a Random Vector, and the Lifted Score: Isometry, Norm, Weak Identity and Second-Moment Identity (i), and by that clause and Random Vectors: Coordinates, Functions, Pairs and Almost-Sure Equality (iii), $\lVert\Gamma\circ X\rVert_{L^{2}}^{2}=\int\lVert\Gamma\rVert^{2}d\mu$. Finally, for $V\in L^{2}(\Omega;\mathbb{R}^{d})$ with a representative,

$$\mathbb{E}[\lVert V\rVert]\le\lVert V\rVert_{L^{2}},\tag{H}$$

by Hölder's Inequality (i) with $p=q=2$ (conjugate exponents in the sense of Conjugate Exponents and Young's Inequality (i), as $\tfrac12+\tfrac12=1$) applied to $\lVert V\rVert$ and the constant $1$: $\mathbb{E}[\lVert V\rVert]\le(\mathbb{E}[\lVert V\rVert^{2}])^{1/2}=\lVert V\rVert_{L^{2}}$ (The Space of Square-Integrable Random Vectors (iii)).

Iterated integrals (FB). Let $p,r\in\mathbb{N}$ with $1\le p,r$, $\alpha\in\mathcal{P}(\mathbb{R}^{p})$, $\beta\in\mathcal{P}(\mathbb{R}^{r})$, and let $F:\mathbb{R}^{p+r}\to\mathbb{R}$ be Borel with $|F(\iota(x,y))|\le g(x)+h(y)$ for all $x,y$, where $g:\mathbb{R}^{p}\to[0,\infty)$ and $h:\mathbb{R}^{r}\to[0,\infty)$ are Borel and integrable with respect to $\alpha$ and $\beta$. Then: for every $x$ the function $y\mapsto F(\iota(x,y))$ is Borel and $\beta$-integrable; the function $x\mapsto\int F(\iota(x,y))\beta(dy)$ is Borel and $\alpha$-integrable; symmetrically in the other order; $F$ is integrable with respect to $\alpha\boxtimes\beta$; and

$$\int_{\mathbb{R}^{p+r}}F\,d(\alpha\boxtimes\beta)=\int\Bigl(\int F(\iota(x,y))\,\beta(dy)\Bigr)\alpha(dx)=\int\Bigl(\int F(\iota(x,y))\,\alpha(dx)\Bigr)\beta(dy).$$

Indeed, let $F^{+}=\max(F,0)$ and $F^{-}=\max(-F,0)$, nonnegative Borel functions (claim 4 of Arithmetic of Measurable Real- and [0,∞]-Valued Maps) with $F=F^{+}-F^{-}$ and $F^{\pm}\le|F|$. By Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (iii), $F^{\pm}\circ\iota$ is measurable for the product $\sigma$-algebra and $\int F^{\pm}d(\alpha\boxtimes\beta)=\int F^{\pm}\circ\iota\,d(\alpha\otimes\beta)$; by Tonelli's and Fubini's Theorems (Sections and Tonelli), the sections of $F^{\pm}\circ\iota$ are measurable, the functions $x\mapsto\int F^{\pm}(\iota(x,y))\beta(dy)$ and $y\mapsto\int F^{\pm}(\iota(x,y))\alpha(dx)$ are measurable, and the three integrals of $F^{\pm}$ (product, and the two iterated orders) coincide in $[0,\infty]$. By the domination and claim 1 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion, $\int F^{\pm}(\iota(x,y))\beta(dy)\le g(x)+\int h\,d\beta<\infty$ for every $x$, and the common value of the three integrals is at most $\int g\,d\alpha+\int h\,d\beta<\infty$. Hence every section is integrable, the functions $x\mapsto\int F(\iota(x,y))\beta(dy)=\int F^{+}(\iota(x,y))\beta(dy)-\int F^{-}(\iota(x,y))\beta(dy)$ are real-valued, measurable (claim 2 of Arithmetic of Measurable Real- and [0,∞]-Valued Maps) and integrable, $F$ is $\alpha\boxtimes\beta$-integrable (Integrable Functions and Their Lebesgue Integral), and the displayed identities follow by subtracting the identities for $F^{-}$ from those for $F^{+}$, using claim 2 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion and the definition of the integral of an integrable function as the difference of the integrals of its positive and negative parts. A bounded Borel $F$ with $|F|\le C$ satisfies the hypothesis with $g=C$, $h=0$.

Proof of claim 1. Let $f$ be as in claim 1 and $U$ the lift of $u_{f}$. Apply The Lift of a Linear Functional of the Measure: Integrability, L-Gradient, Lipschitz Gradient Map and Translation Laplacian with $M_{1}=M_{2}=M$.

(a) $u_{f}$ is continuously $L$-differentiable by The Lift of a Linear Functional of the Measure: Integrability, L-Gradient, Lipschitz Gradient Map and Translation Laplacian (iii), i.e. $U\in C^{1}(L^{2}(\Omega;\mathbb{R}^{d}))$ (L-Differentiability of a Function on the Wasserstein Space via the Fréchet Derivative of Its Lift (ii)).

(b) By The Lift of a Linear Functional of the Measure: Integrability, L-Gradient, Lipschitz Gradient Map and Translation Laplacian (ii), $DU(X)=Df\circ X$ for every $X$, the class of the composition of $Df$ with a representative of $X$, which is the class $Df\circ X$ of Composition of a Square-Integrable Vector Field with a Random Vector, and the Lifted Score: Isometry, Norm, Weak Identity and Second-Moment Identity (i) for the class $Df\in L^{2}(\mathcal{L}(X);\mathbb{R}^{d})$. For $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ the class $Df$ lies in $T_{\mu}$ by Gradients of Functions with Bounded Derivatives Belong to the Tangent Space; Constants Are Tangent; the Score Identity; the Score Has Mean Zero; Translation of Tangent Fields (i), so property (b) holds with $\eta=Df$, and $\nabla u_{f}(\mu)$ is the class of $x\mapsto Df(x)$ (Test Functions on the Wasserstein Space: the Intrinsic Gradient and the Translation Hessian (ii)).

(c) $U$ is twice continuously differentiable along translations by The Lift of a Linear Functional of the Measure: Integrability, L-Gradient, Lipschitz Gradient Map and Translation Laplacian (iv).

(d) and the Lipschitz constant. Let $X,Y\in L^{2}(\Omega;\mathbb{R}^{d})$ with representatives. By The Lift of a Linear Functional of the Measure: Integrability, L-Gradient, Lipschitz Gradient Map and Translation Laplacian (i), $U(X)-U(Y)=\mathbb{E}[f\circ X]-\mathbb{E}[f\circ Y]=\mathbb{E}[f\circ X-f\circ Y]$, and by (Lip), $|f(X(\omega))-f(Y(\omega))|\le\sqrt{d}\,M\lVert X(\omega)-Y(\omega)\rVert$ for every $\omega$, so claim 2 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion and (H) give $|U(X)-U(Y)|\le\sqrt{d}\,M\,\mathbb{E}[\lVert X-Y\rVert]\le\sqrt{d}\,M\lVert X-Y\rVert_{L^{2}}$. Thus $U$ is Lipschitz with constant $\sqrt{d}\,M$ (Lipschitz Map Between Metric Spaces, with the distance $d_{L^{2}}$ of The Wasserstein Space and Its Lift to Square-Integrable Random Vectors: Standing Notation (iii)), so $u_{f}$ is Lipschitz with constant $\sqrt{d}\,M$ by Basic Properties of the Lift: Law Invariance, the Correspondence on a Rich Space, and Transfer of Boundedness, Lipschitz Constants and Continuity (vi), the space being rich, and continuous by A Lipschitz Map is Uniformly Continuous and Continuous. Hence $u_{f}$ is a test function.

The translation Hessian. Let $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, $X$ with $\mathcal{L}(X)=\mu$ (The Wasserstein Distance and the Mean-Square Distance of Random Vectors (iii)), and $\phi_{X}(a)=U(X+c_{a})$. By The Wasserstein Space and Its Lift to Square-Integrable Random Vectors: Standing Notation (vi), The Lift of a Linear Functional of the Measure: Integrability, L-Gradient, Lipschitz Gradient Map and Translation Laplacian (i), Random Vectors: Coordinates, Functions, Pairs and Almost-Sure Equality (iii) and change of variables, $\phi_{X}(a)=\int f\,d(\tau_{a})_{\#}\mu=\int f(x+a)\,\mu(dx)$. Fix $a$ and $i\in[d]$. For $t\in(-1,1)$ and $x\in\mathbb{R}^{d}$ put $F(t,x)=f(x+a+te_{i})$. Each $x\mapsto F(t,x)$ is integrable (The Lift of a Linear Functional of the Measure: Integrability, L-Gradient, Lipschitz Gradient Map and Translation Laplacian (i) for $(\tau_{a+te_{i}})_{\#}\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, Gradients of Functions with Bounded Derivatives Belong to the Tangent Space; Constants Are Tangent; the Score Identity; the Score Has Mean Zero; Translation of Tangent Fields (v), and change of variables); for each $x$, $t\mapsto F(t,x)$ is differentiable at every $t$ with derivative $\partial_{i}f(x+a+te_{i})$ by Chain Rule Along an Affine Path and Continuous Partial Derivatives Imply Differentiability; and $|\partial_{i}f|\le M$, a constant integrable against $\mu$. By Differentiation under the Integral Sign, $t\mapsto\phi_{X}(a+te_{i})$ is differentiable at $0$ with derivative $\int\partial_{i}f(x+a)\mu(dx)$; its difference quotients at $0$ are those of Partial Derivative of a Real-Valued Function on an Open Subset of $\mathbb{R}^n$ for $\phi_{X}$ at $a$, so $\partial_{i}\phi_{X}(a)=\int\partial_{i}f(x+a)\,\mu(dx)$. Let $\check{\mu}=(-\mathrm{id})_{\#}\mu$, the push-forward under the map $x\mapsto-x$, which is Borel by the componentwise criterion and claim 2 of Arithmetic of Measurable Real- and [0,∞]-Valued Maps; change of variables gives $\partial_{i}\phi_{X}(a)=\int\partial_{i}f(a-y)\,\check{\mu}(dy)=((\partial_{i}f)*\check{\mu})(a)$ in the notation of The Convolution of a Bounded Continuous Function with a Probability Measure on Euclidean Space: Continuity, Boundedness and Differentiation Under the Integral. Since $\partial_{i}f$ is of class $C^{1}$ with $|\partial_{i}f|\le M$ and $|\partial_{j}\partial_{i}f|\le M$, The Convolution of a Bounded Continuous Function with a Probability Measure on Euclidean Space: Continuity, Boundedness and Differentiation Under the Integral (ii) shows that $\partial_{i}\phi_{X}$ is of class $C^{1}$ on $\mathbb{R}^{d}$ with $\partial_{j}\partial_{i}\phi_{X}(a)=((\partial_{j}\partial_{i}f)*\check{\mu})(a)=\int\partial_{j}\partial_{i}f(x+a)\,\mu(dx)$ (change of variables back). Also $\phi_{X}$ is continuous, since $|\phi_{X}(a)-\phi_{X}(a')|\le\int|f(x+a)-f(x+a')|\mu(dx)\le\sqrt{d}\,M\lVert a-a'\rVert$ by (Lip) and claim 2 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion, so $\phi_{X}$ is of class $C^{2}$ (Functions of Class Cᵏ on an Open Subset of ℝⁿ), and by The Hessian Matrix of a Function of Class C² and Test Functions on the Wasserstein Space: the Intrinsic Gradient and the Translation Hessian (iii), $H_{u_{f}}(\mu)=D^{2}\phi_{X}(0_{\mathbb{R}^{d}})$ has entry $\partial_{i}\partial_{j}\phi_{X}(0_{\mathbb{R}^{d}})=\int\partial_{i}\partial_{j}f\,d\mu$ in row $i$ and column $j$; since this matrix belongs to $\mathcal{S}(d)$ by Test Functions on the Wasserstein Space: the Intrinsic Gradient and the Translation Hessian (iii), its entry in row $i$ and column $j$ equals its entry in row $j$ and column $i$, namely $\int\partial_{j}\partial_{i}f\,d\mu$, as stated; each $\partial_{j}\partial_{i}f$ is bounded and Borel as noted.

Proof of claim 2. Let $K$ be as in claim 2.

Oddness of $DK$. Let $x\in\mathbb{R}^{d}$, $i\in[d]$. For $h\ne0$, evenness gives $\bigl(K(x+he_{i})-K(x)\bigr)/h=\bigl(K(-x+(-h)e_{i})-K(-x)\bigr)/h=-\bigl(K(-x+(-h)e_{i})-K(-x)\bigr)/(-h)$. Given $\varepsilon>0$, let $\delta$ and $\delta'$ be the radii of Partial Derivative of a Real-Valued Function on an Open Subset of $\mathbb{R}^n$ for $K$ at $-x$ and at $x$ with $\varepsilon/2$ in place of $\varepsilon$ (claim 8 of Elementary Order Arithmetic in an Ordered Field), and let $h$ satisfy $0<|h|<\min(\delta,\delta')$ (claim 9 there; then $0<|-h|<\delta$ by claim 2 of Properties of the Absolute Value in an Ordered Field). The left side lies within $\varepsilon/2$ of $\partial_{i}K(x)$ and the right side within $\varepsilon/2$ of $-\partial_{i}K(-x)$, so $|\partial_{i}K(x)+\partial_{i}K(-x)|<\varepsilon$ by claim 5 of Properties of the Absolute Value in an Ordered Field; as $\varepsilon>0$ was arbitrary, $\partial_{i}K(x)=-\partial_{i}K(-x)$ by Comparison of Real Numbers with Arbitrary Positive Slack (iii).

Borel functions and the potential. $K$, $\partial_{i}K$ and $\partial_{j}\partial_{i}K$ are continuous, hence Borel. The map $z\mapsto\mathrm{pr}_{1}(z)-\mathrm{pr}_{2}(z)$ on $\mathbb{R}^{d+d}$ is Borel (its components are differences of the Borel components of the projections, Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (i), claim 2 of Arithmetic of Measurable Real- and [0,∞]-Valued Maps), so $z\mapsto K(\mathrm{pr}_{1}(z)-\mathrm{pr}_{2}(z))$ is Borel, and bounded by $M$; likewise $y\mapsto K(x-y)$ for fixed $x$. Let $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$. In the notation of The Convolution of a Bounded Continuous Function with a Probability Measure on Euclidean Space: Continuity, Boundedness and Differentiation Under the Integral (dimension $q=d$), $K*\mu$ is the function $(H*\mu)$ of that lemma with $H=K$. By The Convolution of a Bounded Continuous Function with a Probability Measure on Euclidean Space: Continuity, Boundedness and Differentiation Under the Integral (ii) with $G=K$ (of class $C^{1}$, $|K|\le M$, $|\partial_{i}K|\le M$), $K*\mu$ is of class $C^{1}$ with $\partial_{i}(K*\mu)=(\partial_{i}K)*\mu$; by the same clause with $G=\partial_{i}K$ (of class $C^{1}$ since $K$ is of class $C^{2}$, clause 2 of Functions of Class Cᵏ on an Open Subset of ℝⁿ, with $|\partial_{i}K|\le M$, $|\partial_{j}\partial_{i}K|\le M$), $(\partial_{i}K)*\mu$ is of class $C^{1}$ with $\partial_{j}((\partial_{i}K)*\mu)=(\partial_{j}\partial_{i}K)*\mu$. Hence $K*\mu$ is of class $C^{2}$ on $\mathbb{R}^{d}$ with the displayed formulas for $\partial_{i}(K*\mu)$ and $\partial_{j}\partial_{i}(K*\mu)$, and the three bounds by $M$ follow from The Convolution of a Bounded Continuous Function with a Probability Measure on Euclidean Space: Continuity, Boundedness and Differentiation Under the Integral (i). Write $\Gamma_{\mu}=D(K*\mu)$, a Borel map with $|(\Gamma_{\mu})_{i}|\le M$, and $\Gamma_{\mu}(x)=\int DK(x-y)\mu(dy)$ coordinatewise.

The two expressions for $\mathcal{K}_{K}$. By (FB) with $p=r=d$, $\alpha=\beta=\mu$ and the bounded Borel $F(z)=K(\mathrm{pr}_{1}(z)-\mathrm{pr}_{2}(z))$, for which $F(\iota(x,y))=K(x-y)$ (Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (i)),

$$\int_{\mathbb{R}^{d+d}}F\,d(\mu\boxtimes\mu)=\int\Bigl(\int K(x-y)\,\mu(dy)\Bigr)\mu(dx)=\int K*\mu\,d\mu ,$$

which is the identity in the statement. More generally, for $\alpha,\beta\in\mathcal{P}(\mathbb{R}^{d})$ write $\mathcal{K}(\alpha,\beta)=\int(\int K(x-y)\beta(dy))\alpha(dx)$; by (FB) it also equals $\int(\int K(x-y)\alpha(dx))\beta(dy)$, and $\mathcal{K}_{K}(\mu)=\mathcal{K}(\mu,\mu)$.

Translation invariance. Let $a\in\mathbb{R}^{d}$ and $\mu_{a}=(\tau_{a})_{\#}\mu$, an element of $\mathcal{P}_{2}(\mathbb{R}^{d})$ by Gradients of Functions with Bounded Derivatives Belong to the Tangent Space; Constants Are Tangent; the Score Identity; the Score Has Mean Zero; Translation of Tangent Fields (v). For every $x$, change of variables gives $(K*\mu_{a})(x)=\int K(x-y-a)\mu(dy)=(K*\mu)(x-a)$, and then $\mathcal{K}_{K}(\mu_{a})=\int(K*\mu)(x-a)\,\mu_{a}(dx)=\int(K*\mu)(x+a-a)\,\mu(dx)=\mathcal{K}_{K}(\mu)$, the function $K*\mu$ being bounded and continuous, hence Borel and integrable.

Reduction to the law of a pair. Let $X,H\in L^{2}(\Omega;\mathbb{R}^{d})$ with representatives, $\mu=\mathcal{L}(X)$, and let $\pi=\mathcal{L}((X,H))\in\mathcal{P}(\mathbb{R}^{d+d})$, the law of the pairing, a coupling of $\mu$ and $\mathcal{L}(H)$ by Random Vectors: Coordinates, Functions, Pairs and Almost-Sure Equality (v), so that $(\mathrm{pr}_{1})_{\#}\pi=\mu$. Let $\theta:\mathbb{R}^{d+d}\to\mathbb{R}^{d}$, $\theta(z)=\mathrm{pr}_{1}(z)+\mathrm{pr}_{2}(z)$, a Borel map; since $X+H=\theta\circ(X,H)$ pointwise, Random Vectors: Coordinates, Functions, Pairs and Almost-Sure Equality (ii) gives $\mathcal{L}(X+H)=\theta_{\#}\pi$. For a bounded Borel $G:\mathbb{R}^{d}\to\mathbb{R}$ and Borel $S:\mathbb{R}^{d+d}\to\mathbb{R}^{d}$, change of variables gives $\int G\,d(S_{\#}\pi)=\int G\circ S\,d\pi$. Applying this twice (inside, for fixed $x$, to $y\mapsto K(x-y)$, then outside, to the bounded Borel function $x\mapsto\int K(x-S(z'))\pi(dz')$, Borel by (FB) for the bounded Borel function $(x,z')\mapsto K(x-S(z'))$ on $\mathbb{R}^{d+(d+d)}$),

$$\mathcal{K}_{K}(S_{\#}\pi)=\int\Bigl(\int K\bigl(S(z)-S(z')\bigr)\,\pi(dz')\Bigr)\pi(dz)\qquad\text{for }S\in\{\mathrm{pr}_{1},\theta\}.\tag{R}$$

Thus, with $\Phi$ the lift of $\mathcal{K}_{K}$, $\Phi(X)=\mathcal{K}_{K}(\mu)$ is (R) with $S=\mathrm{pr}_{1}$ and $\Phi(X+H)=\mathcal{K}_{K}(\theta_{\#}\pi)$ is (R) with $S=\theta$.

(a): differentiability. Keep $X,H,\pi$ as above and write, for $z,z'\in\mathbb{R}^{d+d}$, $u=\mathrm{pr}_{1}(z)-\mathrm{pr}_{1}(z')$ and $w=\mathrm{pr}_{2}(z)-\mathrm{pr}_{2}(z')$, so $\theta(z)-\theta(z')=u+w$. Consider the Borel functions on $\mathbb{R}^{(d+d)+(d+d)}$ given at $\iota(z,z')$ by $F_{1}=K(u+w)$, $F_{2}=K(u)$ and $F_{3}=DK(u)\cdot w$ (Borel by the preamble and claims 2 and 3 of Arithmetic of Measurable Real- and [0,∞]-Valued Maps). $F_{1},F_{2}$ are bounded by $M$, and $|F_{3}|\le\sqrt{d}\,M\lVert w\rVert\le\sqrt{d}\,M(\lVert\mathrm{pr}_{2}(z)\rVert+\lVert\mathrm{pr}_{2}(z')\rVert)$ (claims 5 and 6 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square), where $z\mapsto\lVert\mathrm{pr}_{2}(z)\rVert$ is Borel and $\pi$-integrable with $\int\lVert\mathrm{pr}_{2}(z)\rVert\pi(dz)=\mathbb{E}[\lVert H\rVert]\le\lVert H\rVert_{L^{2}}$ (change of variables, (H)); so (FB) applies to $F_{1},F_{2},F_{3}$ and to $F_{1}-F_{2}-F_{3}$ with $\alpha=\beta=\pi$. By (Tay), $|F_{1}-F_{2}-F_{3}|\le\tfrac12dM\lVert w\rVert^{2}\le dM(\lVert\mathrm{pr}_{2}(z)\rVert^{2}+\lVert\mathrm{pr}_{2}(z')\rVert^{2})$, and $\int\lVert\mathrm{pr}_{2}(z)\rVert^{2}\pi(dz)=\mathbb{E}[\lVert H\rVert^{2}]=\lVert H\rVert_{L^{2}}^{2}$; hence, by (FB), claim 2 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion and Tonelli for the dominating function,

$$\Bigl|\int(F_{1}-F_{2}-F_{3})\,d(\pi\boxtimes\pi)\Bigr|\le2dM\lVert H\rVert_{L^{2}}^{2}.$$

By (R) and (FB), $\int F_{1}d(\pi\boxtimes\pi)=\Phi(X+H)$ and $\int F_{2}\,d(\pi\boxtimes\pi)=\Phi(X)$. For $F_{3}$, write $F_{3}=F_{3}'-F_{3}''$ with $F_{3}'=DK(u)\cdot\mathrm{pr}_{2}(z)$ and $F_{3}''=DK(u)\cdot\mathrm{pr}_{2}(z')$, both dominated as $F_{3}$ is. Integrating $F_{3}'$ first in $z'$: for fixed $z$, $\int DK(\mathrm{pr}_{1}(z)-\mathrm{pr}_{1}(z'))\cdot\mathrm{pr}_{2}(z)\,\pi(dz')=\Bigl(\int DK(\mathrm{pr}_{1}(z)-y)\,\mu(dy)\Bigr)\cdot\mathrm{pr}_{2}(z)=\Gamma_{\mu}(\mathrm{pr}_{1}(z))\cdot\mathrm{pr}_{2}(z)$ (coordinatewise, claim 2 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion and change of variables with $(\mathrm{pr}_{1})_{\#}\pi=\mu$), and then $\int\Gamma_{\mu}(\mathrm{pr}_{1}(z))\cdot\mathrm{pr}_{2}(z)\,\pi(dz)=\mathbb{E}[\Gamma_{\mu}(X)\cdot H]=\langle\Gamma_{\mu}\circ X,H\rangle_{L^{2}}$ (Random Vectors: Coordinates, Functions, Pairs and Almost-Sure Equality (iii) for the pairing $(X,H)$, The Space of Square-Integrable Random Vectors (iii)). Integrating $F_{3}''$ first in $z$ (the other order in (FB)): for fixed $z'$, $\int DK(\mathrm{pr}_{1}(z)-\mathrm{pr}_{1}(z'))\,\pi(dz)=\int DK(x-\mathrm{pr}_{1}(z'))\,\mu(dx)=-\int DK(\mathrm{pr}_{1}(z')-x)\,\mu(dx)=-\Gamma_{\mu}(\mathrm{pr}_{1}(z'))$ by the oddness of $DK$, so $\int F_{3}''\,d(\pi\boxtimes\pi)=-\int\Gamma_{\mu}(\mathrm{pr}_{1}(z'))\cdot\mathrm{pr}_{2}(z')\,\pi(dz')=-\langle\Gamma_{\mu}\circ X,H\rangle_{L^{2}}$. Therefore $\int F_{3}\,d(\pi\boxtimes\pi)=2\langle\Gamma_{\mu}\circ X,H\rangle_{L^{2}}=\langle2\Gamma_{\mu}\circ X,H\rangle_{L^{2}}$, and altogether

$$\bigl|\Phi(X+H)-\Phi(X)-\langle2\,\Gamma_{\mu}\circ X,H\rangle_{L^{2}}\bigr|\le2dM\lVert H\rVert_{L^{2}}^{2}\qquad(H\in L^{2}(\Omega;\mathbb{R}^{d})).$$

Given $\varepsilon>0$, put $\delta=\varepsilon/(2dM+1)$; for $\lVert H\rVert_{L^{2}}<\delta$ the right side is at most $\varepsilon\lVert H\rVert_{L^{2}}$ (claim 5 of Elementary Arithmetic in an Ordered Field). Hence $\Phi$ is differentiable at $X$ with gradient $D\Phi(X)=2\,\Gamma_{\mu}\circ X$ (Fréchet Differentiability and the Gradient on an Open Subset of a Real Inner Product Space (i), Fréchet Differentiability and the Gradient on an Open Subset of a Real Inner Product Space (ii)), for every $X$ with $\mathcal{L}(X)=\mu$.

(a): continuity of the gradient map. Let $X,Y\in L^{2}(\Omega;\mathbb{R}^{d})$ with laws $\mu,\nu$. By The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity (i) and the linearity of composition (Composition of a Square-Integrable Vector Field with a Random Vector, and the Lifted Score: Isometry, Norm, Weak Identity and Second-Moment Identity (i)), $\lVert D\Phi(X)-D\Phi(Y)\rVert_{L^{2}}\le2\lVert\Gamma_{\mu}\circ X-\Gamma_{\mu}\circ Y\rVert_{L^{2}}+2\lVert(\Gamma_{\mu}-\Gamma_{\nu})\circ Y\rVert_{L^{2}}$. Each $\partial_{i}(K*\mu)$ is of class $C^{1}$ with partials bounded by $M$, so (Lip) gives $|(\Gamma_{\mu})_{i}(x)-(\Gamma_{\mu})_{i}(y)|\le\sqrt{d}\,M\lVert x-y\rVert$ and hence $\lVert\Gamma_{\mu}(x)-\Gamma_{\mu}(y)\rVert\le dM\lVert x-y\rVert$; thus $\lVert\Gamma_{\mu}\circ X-\Gamma_{\mu}\circ Y\rVert_{L^{2}}^{2}=\mathbb{E}[\lVert\Gamma_{\mu}(X)-\Gamma_{\mu}(Y)\rVert^{2}]\le d^{2}M^{2}\lVert X-Y\rVert_{L^{2}}^{2}$. For every $y\in\mathbb{R}^{d}$ and $i$, change of variables and Random Vectors: Coordinates, Functions, Pairs and Almost-Sure Equality (iii) give $(\Gamma_{\mu})_{i}(y)-(\Gamma_{\nu})_{i}(y)=\mathbb{E}[\partial_{i}K(y-X)-\partial_{i}K(y-Y)]$, whose absolute value is at most $\sqrt{d}\,M\,\mathbb{E}[\lVert X-Y\rVert]\le\sqrt{d}\,M\lVert X-Y\rVert_{L^{2}}$ by (Lip) for $\partial_{i}K$, claim 2 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion and (H); so $\lVert\Gamma_{\mu}(y)-\Gamma_{\nu}(y)\rVert\le dM\lVert X-Y\rVert_{L^{2}}$ for every $y$, and $\lVert(\Gamma_{\mu}-\Gamma_{\nu})\circ Y\rVert_{L^{2}}\le dM\lVert X-Y\rVert_{L^{2}}$. Altogether $\lVert D\Phi(X)-D\Phi(Y)\rVert_{L^{2}}\le4dM\lVert X-Y\rVert_{L^{2}}$, so the gradient map is Lipschitz, hence continuous (A Lipschitz Map is Uniformly Continuous and Continuous), and $\Phi\in C^{1}(L^{2}(\Omega;\mathbb{R}^{d}))$.

(b). For $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ the class of $2\Gamma_{\mu}=2D(K*\mu)$ lies in $T_{\mu}$: $K*\mu$ is of class $C^{1}$ with $|\partial_{i}(K*\mu)|\le M$, so $D(K*\mu)\in T_{\mu}$ by Gradients of Functions with Bounded Derivatives Belong to the Tangent Space; Constants Are Tangent; the Score Identity; the Score Has Mean Zero; Translation of Tangent Fields (i), and $T_{\mu}$ is a linear subspace (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)). By the differentiability paragraph, $D\Phi(X)=(2\Gamma_{\mu})\circ X$ for every $X$ with $\mathcal{L}(X)=\mu$ (linearity of composition, Composition of a Square-Integrable Vector Field with a Random Vector, and the Lifted Score: Isometry, Norm, Weak Identity and Second-Moment Identity (i)). So (b) holds, and $\nabla\mathcal{K}_{K}(\mu)$ is the class of $x\mapsto2D(K*\mu)(x)$ (Test Functions on the Wasserstein Space: the Intrinsic Gradient and the Translation Hessian (ii)).

(c) and the translation Hessian. For $X$ with law $\mu$, $\phi_{X}(a)=\Phi(X+c_{a})=\mathcal{K}_{K}((\tau_{a})_{\#}\mu)=\mathcal{K}_{K}(\mu)$ by The Wasserstein Space and Its Lift to Square-Integrable Random Vectors: Standing Notation (vi) and translation invariance; a constant function is smooth (claim 2 of Constants, Affine Functions, Sums, Products and Quotients of Cᵏ Functions) and its partial derivatives, and theirs, vanish identically (the difference quotients of Partial Derivative of a Real-Valued Function on an Open Subset of $\mathbb{R}^n$ are $0$). Hence $\Phi$ is twice continuously differentiable along translations at every $X$, and $H_{\mathcal{K}_{K}}(\mu)=D^{2}\phi_{X}(0_{\mathbb{R}^{d}})=0_{d}$ (The Hessian Matrix of a Function of Class C², Test Functions on the Wasserstein Space: the Intrinsic Gradient and the Translation Hessian (iii)).

(d) and the Lipschitz constant. Let $X,Y\in L^{2}(\Omega;\mathbb{R}^{d})$ with laws $\mu,\nu$ and let $\pi=\mathcal{L}((X,Y))$, a coupling of $\mu$ and $\nu$ (Random Vectors: Coordinates, Functions, Pairs and Almost-Sure Equality (v)). By (R) with $S=\mathrm{pr}_{1}$ and with $S=\mathrm{pr}_{2}$ (the same argument, $(\mathrm{pr}_{2})_{\#}\pi=\nu$), and (FB),

$$\Phi(X)-\Phi(Y)=\int\Bigl(\int\bigl[K(\mathrm{pr}_{1}(z)-\mathrm{pr}_{1}(z'))-K(\mathrm{pr}_{2}(z)-\mathrm{pr}_{2}(z'))\bigr]\pi(dz')\Bigr)\pi(dz).$$

By (Lip) and claims 5 and 6 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square, the integrand is bounded in absolute value by $\sqrt{d}\,M\bigl(\lVert\mathrm{pr}_{1}(z)-\mathrm{pr}_{2}(z)\rVert+\lVert\mathrm{pr}_{1}(z')-\mathrm{pr}_{2}(z')\rVert\bigr)$, and $\int\lVert\mathrm{pr}_{1}(z)-\mathrm{pr}_{2}(z)\rVert\pi(dz)=\mathbb{E}[\lVert X-Y\rVert]\le\lVert X-Y\rVert_{L^{2}}$ (change of variables, (H)). Integrating twice with claim 2 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion, $|\Phi(X)-\Phi(Y)|\le2\sqrt{d}\,M\lVert X-Y\rVert_{L^{2}}$. Thus $\Phi$ is Lipschitz with constant $2\sqrt{d}\,M$, so $\mathcal{K}_{K}$ is Lipschitz with constant $2\sqrt{d}\,M$ by Basic Properties of the Lift: Law Invariance, the Correspondence on a Rich Space, and Transfer of Boundedness, Lipschitz Constants and Continuity (vi) and continuous by A Lipschitz Map is Uniformly Continuous and Continuous. Hence $\mathcal{K}_{K}$ is a test function with the stated gradient and translation Hessian.