Proof of The Lift of the Second Moment: a Quadratic Function of Class $C^2$ with Gradient 2X and Translation Laplacian 2d

Each result cited is universally quantified over the data in its own statement. Write $L^{2}$ for $L^{2}(\Omega;\mathbb{R}^{d})$, a real Hilbert space with the differential calculus of Real Hilbert Spaces: Series, Products, Orthonormal Bases and Differential Calculus §calculus in force by The Wasserstein Space and Its Lift to Square-Integrable Random Vectors: Standing Notation §space, and $0_{L^{2}}$ for its zero vector; the identities of Elementary Identities in a Real Inner Product Space are in force for it by Real Hilbert Spaces: Standing Notation and Background §background. The natural number $d$ is read in $\mathbb{R}$ through the canonical map $\iota$ of The Canonical Map from the Natural Numbers to a Field, as in the statement. Two identities are in play: $I$ is the identity form on $L^{2}$ of Real Hilbert Spaces: Standing Notation and Background §forms, while $I_{d}$ in Claim 3 is the identity matrix of size $d$.

Claim 1. For $X\in L^{2}$, $U(X)=u(\mathcal{L}(X))=M_{2}(\mathcal{L}(X))$ by The Lift of a Function on the Wasserstein Space to the Space of Square-Integrable Random Vectors §lift, and $M_{2}(\mathcal{L}(X))=\lVert X\rVert_{L^{2}}^{2}$ by The Space of Square-Integrable Random Vectors is a Real Hilbert Space; Its Laws Have Finite Second Moment; Constants and Translations §law.

Claim 2. Let $q:L^{2}\to\mathbb{R}$ be given by $q(X)=\tfrac{2}{2}\lVert X-0_{L^{2}}\rVert_{L^{2}}^{2}$. By Affine and Quadratic Functions on a Real Hilbert Space are of Class $C^2$ §quadratic, applied to the Hilbert space $L^{2}$ with $\alpha=2$ and $y_{0}=0_{L^{2}}$, the function $q$ belongs to $C^{2}(L^{2})$ with $Dq(X)=2(X-0_{L^{2}})$ and $D^{2}q(X)=2I$ for every $X\in L^{2}$. Since $\tfrac{2}{2}=1$ in the field $\mathbb{R}$ and $X-0_{L^{2}}=X$ by the axioms of the vector space $L^{2}$, we have $q(X)=\lVert X\rVert_{L^{2}}^{2}=U(X)$ for every $X$ by claim 1, that is, $q=U$. Hence $U\in C^{2}(L^{2})$, $DU(X)=2X$ and $D^{2}U(X)=2I$. A function of class $C^{2}$ is of class $C^{1}$ by The Classes $C^1$ and $C^2$ on an Open Subset of a Real Inner Product Space §c2, so $u$ is continuously $L$-differentiable by L-Differentiability of a Function on the Wasserstein Space via the Fréchet Derivative of Its Lift §c1; a function of class $C^{1}$ is differentiable on $L^{2}$ by The Classes $C^1$ and $C^2$ on an Open Subset of a Real Inner Product Space §c1, so $u$ is $L$-differentiable with $L$-gradient $DU(X)=2X$ at $X$ by L-Differentiability of a Function on the Wasserstein Space via the Fréchet Derivative of Its Lift §differentiable.

Claim 3. Fix $X\in L^{2}$ and let $\phi_{X}:\mathbb{R}^{d}\to\mathbb{R}$, $\phi_{X}(a)=U(X+c_{a})$, be the function of The Translation Laplacian of a Function on the Space of Square-Integrable Random Vectors. By claim 1 and Elementary Identities in a Real Inner Product Space §expansion,

where $\kappa=\lVert X\rVert_{L^{2}}^{2}$, $\ell(a)=\langle X,c_{a}\rangle_{L^{2}}$ and $q_{0}(a)=\lVert a\rVert^{2}$, the last equality using $\lVert c_{a}\rVert_{L^{2}}=\lVert a\rVert$ from The Wasserstein Space and Its Lift to Square-Integrable Random Vectors: Standing Notation §constants. We treat the three summands in turn; $\mathbb{R}^{d}$ is open by claim 1 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous, and partial derivatives, the classes $C^{k}$ and smoothness are those of Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation §derivatives.

The constant. The constant function with value $\kappa$ is smooth on $\mathbb{R}^{d}$ by claim 2 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set, hence of class $C^{2}$ by Smooth Map on a Euclidean Open Set; and every partial derivative of a constant function exists and vanishes at every point, since its difference quotients in Partial Derivative on a Euclidean Open Set are all $0$.

The linear term. For $a,b\in\mathbb{R}^{d}$ and $t\in\mathbb{R}$ the constant map with value $a+b$ is the pointwise sum of the constant maps with values $a$ and $b$, and the constant map with value $ta$ is $t$ times the constant map with value $a$; hence $c_{a+b}=c_{a}+c_{b}$ and $c_{ta}=t\,c_{a}$ by the operations on classes of The Space of Square-Integrable Random Vectors §classes, and therefore $\ell(a+b)=\ell(a)+\ell(b)$ and $\ell(ta)=t\,\ell(a)$ by Elementary Identities in a Real Inner Product Space §bilinear. Fix $i\in[d]$ and let $e_{i}\in\mathbb{R}^{d}$ be the point with $i$th component $1$ and every other component $0$ (claim 2 of Euclidean Points as Tuples of Real Numbers). For $a\in\mathbb{R}^{d}$ and $h\in\mathbb{R}$, the point with $i$th component $a_{i}+h$ and $k$th component $a_{k}$ for $k\ne i$ is $a+he_{i}$, by claim 1 of Euclidean Points as Tuples of Real Numbers together with Sum of Points of $\mathbb{R}^n$ and Scalar Multiple of a Point of $\mathbb{R}^n$. Hence for $h\ne0$

so the partial derivative of $\ell$ with respect to the $i$th variable exists at every $a$ with value $\ell(e_{i})$ (Partial Derivative on a Euclidean Open Set, with any $\delta$); that is, $\partial_{i}\ell$ is the constant function with value $\ell(e_{i})$, which is smooth by claim 2 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set, hence of class $C^{1}$ by Smooth Map on a Euclidean Open Set and continuous at every point by claim 3 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous. The function $\ell$ itself is continuous at every point: by The Cauchy-Schwarz Inequality in a Real Inner Product Space and The Wasserstein Space and Its Lift to Square-Integrable Random Vectors: Standing Notation §constants, $|\ell(a)-\ell(b)|=|\ell(a-b)|=|\langle X,c_{a-b}\rangle_{L^{2}}|\le\lVert X\rVert_{L^{2}}\lVert a-b\rVert$, so given $\varepsilon>0$ the choice $\delta=\varepsilon/(\lVert X\rVert_{L^{2}}+1)$ gives $|\ell(a)-\ell(b)|<\varepsilon$ whenever $\lVert a-b\rVert<\delta$ (claim 10 of Elementary Order Arithmetic in an Ordered Field, and $\lVert X\rVert_{L^{2}}<\lVert X\rVert_{L^{2}}+1$), which is continuity at $b$ relative to $\mathbb{R}^{d}$ in the sense of Continuous Map Between Metric Spaces for the metrics $d_{E}$ and $d_{\mathbb{R}}$ (Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ claim 2), hence continuity in the Euclidean sense by claim 1 of Euclidean Continuity Agrees with Metric Continuity for Real-Valued Functions. Therefore $\ell$ is of class $C^{1}$ by clause 1 of C^k Maps on a Euclidean Open Set, and since each $\partial_{i}\ell$ is of class $C^{1}$, $\ell$ is of class $C^{2}$ by clause 2 there, with $\partial_{j}\partial_{i}\ell=0$ for all $i,j\in[d]$ by the constant case above.

The quadratic term. By claim 2 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$, $q_{0}(a)=\lVert a-0_{\mathbb{R}^{d}}\rVert^{2}=d_{E}(a,0_{\mathbb{R}^{d}})^{2}$, where $a-0_{\mathbb{R}^{d}}=a$ by Euclidean Space $\mathbb{R}^n$ is a Real Vector Space. So A Scaled Squared Distance to a Point is of Class $C^2$, with Gradient and Hessian, applied with its open set taken to be $\mathbb{R}^{d}$, with $c=1$ and the point $0_{\mathbb{R}^{d}}$, shows that $q_{0}$ is of class $C^{2}$ on $\mathbb{R}^{d}$ (claim 2 there) with Hessian matrix $D^{2}q_{0}(a)=2I_{d}$ at every $a$ (claim 3 there). By Hessian Matrix of a C^2 Function, Scalar Multiple of a Real Matrix and Identity Matrix, the diagonal entries give $\partial_{i}\partial_{i}q_{0}(a)=(2I_{d})_{ii}=2\cdot1=2$ for every $i\in[d]$ and every $a$.

Assembly. By claims 2 and 3 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set, $\phi_{X}=\kappa+2\ell+q_{0}$ is of class $C^{2}$ on $\mathbb{R}^{d}$, so $U$ is twice continuously differentiable along translations at $X$ by The Translation Laplacian of a Function on the Space of Square-Integrable Random Vectors §translations, and, $X$ being arbitrary, on the whole space by The Translation Laplacian of a Function on the Space of Square-Integrable Random Vectors §on-space. By claim 1 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set applied twice, for every $i\in[d]$ and every $a$,

the partial derivatives of the constants $0$, $2\ell(e_{i})$ vanishing by the constant case. Hence, by The Translation Laplacian of a Function on the Space of Square-Integrable Random Vectors §laplacian and The Laplacian of a Twice Continuously Differentiable Function §laplacian,

the last two steps by claim 3 of Properties of Finite Sums and The Canonical Map from the Natural Numbers to a Field. This is the claim, $2d$ denoting $2\,\iota(d)$.