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 C^k Maps on a Euclidean Open Set, claim 3 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous), 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 §borel-maps); a bounded Borel function is integrable with respect to every probability measure (Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures); the integral of a constant against a probability measure is that constant (Simple Function and Its Integral); "change of variables" refers to Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward; $\lVert x\rVert^{2}=\sum_{i}x_{i}^{2}$ and $|x_{i}|\le\lVert x\rVert$ (claims 1 and 4 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$); 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 §functions). 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 Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$) gives

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

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 §composition, and by that clause and Basic Properties of Random Vectors: Coordinates, Borel Images and Arithmetic, Change of Variables, Almost Sure Equality and Pairs §expectation, $\lVert\Gamma\circ X\rVert_{L^{2}}^{2}=\int\lVert\Gamma\rVert^{2}d\mu$. Finally, for $V\in L^{2}(\Omega;\mathbb{R}^{d})$ with a representative,

by Hoelder's Inequality, for Two and for Finitely Many Factors §holder with $p=q=2$ (conjugate exponents in the sense of Conjugate Exponents and Young's Inequality §conjugate, 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 §inner-product).

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

Indeed, let $F^{+}=\max(F,0)$ and $F^{-}=\max(-F,0)$, nonnegative Borel functions (claim 4 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions) 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 §product, $F^{\pm}\circ\iota$ 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 and Fubini 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, $\int F^{\pm}(\iota(x,y))\beta(dy)\le g(x)+\int h\,d\beta<\infty$ 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, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions) and integrable, $F$ is $\alpha\boxtimes\beta$-integrable (Integrable Function and the 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 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 §lipschitz, 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 §c1).

(b) By The Lift of a Linear Functional of the Measure: Integrability, L-Gradient, Lipschitz Gradient Map and Translation Laplacian §derivative, $DU(X)=Df\circ X$ 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 §composition 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 §tangent, 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 §gradient).

(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 §translation.

(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 §integrable, $U(X)-U(Y)=\mathbb{E}[f\circ X]-\mathbb{E}[f\circ Y]=\mathbb{E}[f\circ X-f\circ Y]$, 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 (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 §space), 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 §lipschitz-converse, the space being rich, and continuous by A Lipschitz Map is Uniformly 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 §onto), and $\phi_{X}(a)=U(X+c_{a})$. By The Wasserstein Space and Its Lift to Square-Integrable Random Vectors: Standing Notation §constants, The Lift of a Linear Functional of the Measure: Integrability, L-Gradient, Lipschitz Gradient Map and Translation Laplacian §integrable, Basic Properties of Random Vectors: Coordinates, Borel Images and Arithmetic, Change of Variables, Almost Sure Equality and Pairs §expectation 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 §integrable 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 §translation, 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 A Real-Valued C^1 Function is Differentiable at Every Point; 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 on a Euclidean Open Set 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, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions; 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 §derivative 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, so $\phi_{X}$ is of class $C^{2}$ (C^k Maps on a Euclidean Open Set), and by Hessian Matrix of a C^2 Function and Test Functions on the Wasserstein Space: the Intrinsic Gradient and the Translation Hessian §hessian, $H_{u_{f}}(\mu)=D^{2}\phi_{X}(0_{\mathbb{R}^{d}})$ 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 §hessian, 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 on a Euclidean Open Set 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 §vanishing.

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 §projections, claim 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions), 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 §derivative 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 C^k Maps on a Euclidean Open Set, 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 §continuous. 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 §projections),

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 §translation. 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 Basic Properties of Random Vectors: Coordinates, Borel Images and Arithmetic, Change of Variables, Almost Sure Equality and Pairs §pair, 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, Basic Properties of Random Vectors: Coordinates, Borel Images and Arithmetic, Change of Variables, Almost Sure Equality and Pairs §composition 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 $\mathbb{R}^{d+(d+d)}$),

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, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions). $F_{1},F_{2}$ 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 Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$), 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 Tonelli for the dominating function,

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 change of variables with $(\mathrm{pr}_{1})_{\#}\pi=\mu$), and then $\int\Gamma_{\mu}(\mathrm{pr}_{1}(z))\cdot\mathrm{pr}_{2}(z)\,\pi(dz)=\mathbb{E}[\Gamma_{\mu}(X)\cdot H]=\langle\Gamma_{\mu}\circ X,H\rangle_{L^{2}}$ (Basic Properties of Random Vectors: Coordinates, Borel Images and Arithmetic, Change of Variables, Almost Sure Equality and Pairs §expectation for the pairing $(X,H)$, The Space of Square-Integrable Random Vectors §inner-product). 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

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 §differentiable, Fréchet Differentiability and the Gradient on an Open Subset of a Real Inner Product Space §gradient), 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 §triangle 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 §composition), $\lVert D\Phi(X)-D\Phi(Y)\rVert_{L^{2}}\le2\lVert\Gamma_{\mu}\circ X-\Gamma_{\mu}\circ Y\rVert_{L^{2}}+2\lVert(\Gamma_{\mu}-\Gamma_{\nu})\circ Y\rVert_{L^{2}}$. 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 Basic Properties of Random Vectors: Coordinates, Borel Images and Arithmetic, Change of Variables, Almost Sure Equality and Pairs §expectation 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 (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 $\Phi\in C^{1}(L^{2}(\Omega;\mathbb{R}^{d}))$.

(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 §tangent, 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 §closed). 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 §composition). 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 §gradient).

(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 §constants and translation invariance; a constant function is smooth (claim 2 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set) and its partial derivatives, and theirs, vanish identically (the difference quotients of Partial Derivative on a Euclidean Open Set are $0$). Hence $\Phi$ is twice continuously differentiable along translations at every $X$, and $H_{\mathcal{K}_{K}}(\mu)=D^{2}\phi_{X}(0_{\mathbb{R}^{d}})=0_{d}$ (Hessian Matrix of a C^2 Function, Test Functions on the Wasserstein Space: the Intrinsic Gradient and the Translation Hessian §hessian).

(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 $\nu$ (Basic Properties of Random Vectors: Coordinates, Borel Images and Arithmetic, Change of Variables, Almost Sure Equality and Pairs §pair). By (R) with $S=\mathrm{pr}_{1}$ and with $S=\mathrm{pr}_{2}$ (the same argument, $(\mathrm{pr}_{2})_{\#}\pi=\nu$), and (FB),

By (Lip) and claims 5 and 6 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$, 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, $|\Phi(X)-\Phi(Y)|\le2\sqrt{d}\,M\lVert X-Y\rVert_{L^{2}}$. 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 §lipschitz-converse and continuous by A Lipschitz Map is Uniformly Continuous. Hence $\mathcal{K}_{K}$ is a test function with the stated gradient and translation Hessian.