Proof of Heat Averages of Vector Fields on the Torus: Regularity, Contraction, Tangency, the Divergence Identity and Synchronous Couplings

Each result cited below is universally quantified over the data in its own statement. The lemmas The Convolution of a Bounded Continuous Function with a Probability Measure on Euclidean Space: Continuity, Boundedness and Differentiation Under the Integral and The Gaussian Kernels on Euclidean Space: Scaling, Derivatives up to Order Three, the Convolution Identity, Moments and Tails are read with $q=d$, and Translation and Reflection Invariance of Lebesgue Measure on $\mathbb{R}^n$ with $n=d$.

Step 0 (Conventions and tools). Fix $s$, $\mu$ and $F$ as in the statement, write $\Theta=\Theta_{s}$ and $\sigma=S_{s}\mu$, and let $F_{1},\dots,F_{d}$ be the components of $F$. Integrals $\int f(y)\,dy$ are over $\mathbb{R}^{d}$ with respect to $\lambda_{d}$. Let $c_{s}$ and $C_{s}$ be the constants of The Torus Heat Kernel: Regularity and Bounds, the Density of a Heat-Smoothed Measure, and Its Lipschitz Dependence on the Measure §regularity: $\Theta\in C^{2}_{\mathrm{per}}$, $\Theta(-v)=\Theta(v)$, $c_{s}\le\Theta(v)\le C_{s}$ and $|\partial_{i}\Theta(v)|\le C_{s}$ for all $v$ and $i$. Let $p_{\mu}$ be the function of The Torus Heat Kernel: Regularity and Bounds, the Density of a Heat-Smoothed Measure, and Its Lipschitz Dependence on the Measure §density, $p_{\mu}(y)=\int\Theta(y-x)\,\mu(dx)$; that clause gives $p_{\mu}\in C^{2}_{\mathrm{per}}$, $c_{s}\le p_{\mu}\le C_{s}$, $\partial_{i}p_{\mu}(y)=\int\partial_{i}\Theta(y-x)\,\mu(dx)$, and that $\mathbf{1}_{Q}p_{\mu}$ is a density of $\sigma$ with respect to $\lambda_{d}$. By the meaning of a density in The Radon-Nikodym Theorem for a Finite Measure and a Sigma-Finite Measure, and Uniqueness of Densities, $\sigma$ is the measure with density $\mathbf{1}_{Q}p_{\mu}$ of claim 3 of Image Measures, Measures with Densities, and Change of Variables; similarly $N_{s}$ is the measure with density $g_{s}$. So claim 3 of Image Measures, Measures with Densities, and Change of Variables gives, for every nonnegative Borel $f$, and for every Borel $f:\mathbb{R}^{d}\to\mathbb{R}$ that is $\sigma$-integrable (respectively $N_{s}$-integrable), which holds exactly when the product on the right is integrable,

$$\int f\,d\sigma=\int f(y)\,\mathbf{1}_{Q}(y)\,p_{\mu}(y)\,dy,\qquad \int f\,dN_{s}=\int f(z)\,g_{s}(z)\,dz. \qquad(\mathrm{D})$$

Integrability of a real function means that of Integrable Function and the Lebesgue Integral: both $\int f^{+}$ and $\int f^{-}$ are finite, and $\int f=\int f^{+}-\int f^{-}$; equivalently $\int|f|<\infty$.

(0a) Lattice facts. For $u\in\mathbb{R}^{d}$ and $k\in\mathbb{Z}^{d}$, The Half-Open Unit Cell Tiles Euclidean Space §wrap gives $\pi(u)\in Q$, $u-\pi(u)\in\mathbb{Z}^{d}$ and $\pi(u+k)=\pi(u)$. As $\mathbb{Z}^{d}$ is closed under sums and negatives (Lattice-Periodic Functions and the Periodic Function Classes §lattice), for all $u,v,z\in\mathbb{R}^{d}$ the point $(\pi(v)-\pi(u))-(v-u)$ lies in $\mathbb{Z}^{d}$ and $\pi(\pi(u)+z)=\pi(u+z)$, since $\pi(u)+z=(u+z)-(u-\pi(u))$. Hence, by The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §range ($\varpi(w+k)=\varpi(w)$),

$$\varpi\bigl(\pi(v)-\pi(u)\bigr)=\varpi(v-u)\qquad(u,v\in\mathbb{R}^{d}). \qquad(\mathrm{L})$$

The functions $\Theta$, $\partial_{i}\Theta$ (Elementary Properties of Lattice-Periodic Functions §derivative) and $p_{\mu}$ are $\mathbb{Z}^{d}$-periodic. Finally $\partial_{i}\Theta(-v)=-\partial_{i}\Theta(v)$ for all $v$ and $i$: since $\Theta(-u)=\Theta(u)$ for all $u$, the difference quotient of $\Theta$ at $-v$ in the $i$th variable with increment $t\ne0$, namely $t^{-1}(\Theta(-v+te_{i})-\Theta(-v))$ with $e_{i}$ the $i$th unit vector, equals $-(-t)^{-1}(\Theta(v+(-t)e_{i})-\Theta(v))$, minus the difference quotient of $\Theta$ at $v$ with increment $-t$; as $t\mapsto-t$ preserves the condition $0<|t|<\delta$, the limit condition of Partial Derivative on a Euclidean Open Set for $\partial_{i}\Theta(v)$ yields that for $\partial_{i}\Theta(-v)$ with value $-\partial_{i}\Theta(v)$.

(0b) Iterated integrals. The measures $\lambda_{d}$, $\mu$, $N_{s}$ and $\gamma$ are $\sigma$-finite: $\lambda_{d}$ by Lebesgue Measure on Euclidean Space is Sigma-Finite §sigma-finite, the others being probability measures. Let $\alpha$ on $\mathbb{R}^{q}$ and $\beta$ on $\mathbb{R}^{p}$ be two of them ($q,p\in\{d,d+d\}$), let $\hat G:\mathbb{R}^{q+p}\to\mathbb{R}$ be Borel and put $f(w,z)=\hat G(\iota^{q,p}(w,z))$. Then $f$ is measurable for $\mathcal{B}(\mathbb{R}^{q})\otimes\mathcal{B}(\mathbb{R}^{p})$, because $\iota^{q,p}$ is measurable for that $\sigma$-algebra (Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §product); and $f(w,z)$ is obtained from any formula for $\hat G(\zeta)$ by substituting $w$ and $z$ for $\mathrm{pr}^{q,p}_{1}(\zeta)$ and $\mathrm{pr}^{q,p}_{2}(\zeta)$ (Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §projections). Every two-variable integrand below has this form, its $\hat G$ being built from the Borel coordinate projections, the Borel maps $\pi$ (The Half-Open Unit Cell Tiles Euclidean Space §wrap), $\varpi$ (The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §lipschitz) and $F$, the continuous functions $\Theta$, $\partial_{i}\Theta$ and $g_{s}$, indicators of Borel sets, vector sums, and the Borel norm and dot-product functions of Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions, by composition, sums, products, positive parts and absolute values; it is therefore Borel by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps, claims 1 to 4 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions and Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions. The same references show that all one-variable functions below built in this way are Borel. We use two consequences of Tonelli and Fubini Theorems.

(T) If $f\ge0$, then $w\mapsto\int f(w,z)\,\beta(dz)$ is measurable and both iterated integrals equal $\int f\,d(\alpha\otimes\beta)$ in $[0,\infty]$ (Tonelli clause).

(F) If $\int|f|\,d(\alpha\otimes\beta)<\infty$ and every section $z\mapsto f(w,z)$ and $w\mapsto f(w,z)$ is integrable, then

$$\int\Bigl(\int f(w,z)\,\beta(dz)\Bigr)\alpha(dw)=\int\Bigl(\int f(w,z)\,\alpha(dw)\Bigr)\beta(dz).$$

Indeed $f$ is integrable for $\alpha\otimes\beta$; by (T) applied to $f^{\pm}=\hat G^{\pm}\circ\iota^{q,p}$ the section integrals of $f$ are differences of measurable everywhere finite functions, hence measurable (claim 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions); the functions produced by the Fubini clause agree with them off null sets, so The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §comparison transfers the identity of that clause to the section integrals.

(P) For probability measures $\alpha,\beta$ and Borel $\hat G\ge0$, Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §product gives $\int\hat G\,d(\alpha\boxtimes\beta)=\int\hat G\circ\iota^{q,p}\,d(\alpha\otimes\beta)$. If $\hat G$ is real and $(\alpha\boxtimes\beta)$-integrable, applying this to $\hat G^{+}$ and $\hat G^{-}$ shows that $\hat G\circ\iota^{q,p}$ is $(\alpha\otimes\beta)$-integrable with the same integral.

Step 1 (The key identity). We show: for every $x\in\mathbb{R}^{d}$ and every Borel $h:\mathbb{R}^{d}\to[0,\infty)$,

$$\int h\bigl(\pi(x+z)\bigr)\,N_{s}(dz)=\int\mathbf{1}_{Q}(y)\,h(y)\,\Theta(y-x)\,dy\qquad\text{in }[0,\infty]; \qquad(\mathrm{K})$$

and if $h:\mathbb{R}^{d}\to\mathbb{R}$ is Borel with $\mathbf{1}_{Q}h$ integrable, then $\mathbf{1}_{Q}h\,\Theta(\cdot-x)$ is integrable, $z\mapsto h(\pi(x+z))$ is $N_{s}$-integrable, and (K) holds in $\mathbb{R}$.

Put $x_{0}=\pi(x)\in Q$ and $m=x-x_{0}\in\mathbb{Z}^{d}$ (0a). The Dirac measure $\delta_{x_{0}}$ is a probability measure (Dirac Measures on Euclidean Space: Probability Measure, Integrals, Push-Forwards and the Coupling of Two Dirac Measures §measure) with $\delta_{x_{0}}(Q)=1$, so $\delta_{x_{0}}\in\mathcal{P}(\mathbb{T}^{d})$ (Probability Measures on the Flat Torus, the Torus Cost of a Coupling, the Torus Wasserstein Distance and Optimal Couplings §measures). By Dirac Measures on Euclidean Space: Probability Measure, Integrals, Push-Forwards and the Coupling of Two Dirac Measures §integral and periodicity of $\Theta$, the function of The Torus Heat Kernel: Regularity and Bounds, the Density of a Heat-Smoothed Measure, and Its Lipschitz Dependence on the Measure §density for $\delta_{x_{0}}$ is $y\mapsto\Theta(y-x_{0})=\Theta((y-x)+m)=\Theta(y-x)$; so that clause, with the argument that gave (D) run for $\delta_{x_{0}}$ in place of $\mu$, gives $\int h\,d(S_{s}\delta_{x_{0}})=\int\mathbf{1}_{Q}(y)h(y)\Theta(y-x)\,dy$. On the other hand, by The Heat Semigroup on the Probability Measures on the Torus §heat, $S_{s}\delta_{x_{0}}$ is the push-forward under $\pi$ of the measure with density $g_{s}*\delta_{x_{0}}$, and $(g_{s}*\delta_{x_{0}})(y)=g_{s}(y-x_{0})$ by the definition of the smoothing and Dirac Measures on Euclidean Space: Probability Measure, Integrals, Push-Forwards and the Coupling of Two Dirac Measures §integral. By the change of variables of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward, claim 3 of Image Measures, Measures with Densities, and Change of Variables, claim 2 of Translation and Reflection Invariance of Lebesgue Measure on $\mathbb{R}^n$ with $a=x_{0}$ applied to $y\mapsto h(\pi(y))g_{s}(y-x_{0})$, the identity $\pi(z+x_{0})=\pi(x+z-m)=\pi(x+z)$ of (0a), and (D),

$$\int h\,d(S_{s}\delta_{x_{0}})=\int h\bigl(\pi(y)\bigr)\,g_{s}(y-x_{0})\,dy=\int h\bigl(\pi(z+x_{0})\bigr)\,g_{s}(z)\,dz=\int h\bigl(\pi(x+z)\bigr)\,N_{s}(dz).$$

Comparing the two expressions proves (K) for $h\ge0$. For real $h$ with $\mathbf{1}_{Q}h$ integrable, $|\mathbf{1}_{Q}h\,\Theta(\cdot-x)|\le C_{s}\mathbf{1}_{Q}|h|$ gives integrability (monotonicity, claim 1 of Linearity and Monotonicity of the Lebesgue Integral); applying (K) to $h^{+}$ and $h^{-}$, whose composites with $z\mapsto\pi(x+z)$ are the positive and negative parts of $h(\pi(x+\cdot))$ and for which the right side of (K) is finite (as $\Theta>0$, $(\mathbf{1}_{Q}h\Theta(\cdot-x))^{\pm}=\mathbf{1}_{Q}h^{\pm}\Theta(\cdot-x)$), and subtracting, gives the real case.

Step 2 (Integrability). By (D) and $p_{\mu}\ge c_{s}$, monotonicity gives $\int\mathbf{1}_{Q}\lVert F\rVert^{2}\,dy\le c_{s}^{-1}\int\mathbf{1}_{Q}\lVert F\rVert^{2}p_{\mu}\,dy=c_{s}^{-1}\int\lVert F\rVert^{2}\,d\sigma<\infty$. As $t\le1+t^{2}$ for real $t\ge0$ and $\lambda_{d}(Q)=1$ (The Half-Open Unit Cell Tiles Euclidean Space §cell), $\int\mathbf{1}_{Q}\lVert F\rVert\,dy\le1+c_{s}^{-1}\int\lVert F\rVert^{2}\,d\sigma<\infty$. Since $|F_{j}(y)|\le\lVert F(y)\rVert$ (claim 4 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$), each $\mathbf{1}_{Q}F_{j}$ is integrable, and for each $x$ the Borel function $y\mapsto\mathbf{1}_{Q}(y)\Theta(y-x)F_{j}(y)$, bounded in absolute value by $C_{s}\mathbf{1}_{Q}\lVert F\rVert$, is integrable. This is the first assertion of clause 1, and $a$ is well defined.

Step 3 (Clause 1: the Gaussian formula). By Step 1 with $h=F_{j}$ (legitimate by Step 2), $a_{j}(x)=\int F_{j}(\pi(x+z))\,N_{s}(dz)$ for every $x$ and $j$, that is, $a(x)=\int F(\pi(x+z))\,N_{s}(dz)$.

Step 4 (Clause 1: regularity). Fix $j$ and a sign $\pm$, and put $u=\mathbf{1}_{Q}F_{j}^{\pm}$, a nonnegative Borel integrable function (Step 2), and $m_{u}=\int u\,dy$. For $x\in\mathbb{R}^{d}$ and $i\in[d]$ put $A(x)=\int u(y)\Theta(y-x)\,dy$ and $B_{i}(x)=\int u(y)\partial_{i}\Theta(y-x)\,dy$, both integrands being dominated by $C_{s}u$. We claim that $A$ is of class $C^{1}$ with $\partial_{i}A=-B_{i}$.

If $m_{u}=0$, then $u=0$ $\lambda_{d}$-almost everywhere (The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §vanishing), so both integrands vanish almost everywhere and $A=B_{i}=0$ identically (The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §comparison); the zero function is smooth (claim 2 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set) with zero partial derivatives.

If $m_{u}>0$, let $\kappa$ be the measure with density $m_{u}^{-1}u$ with respect to $\lambda_{d}$ (claim 3 of Image Measures, Measures with Densities, and Change of Variables); it has total mass $1$, so $\kappa\in\mathcal{P}(\mathbb{R}^{d})$, and for bounded Borel $\phi$ the same claim gives $\int\phi\,d\kappa=m_{u}^{-1}\int\phi\,u\,dy$. Using $\Theta(y-x)=\Theta(x-y)$ and $\partial_{i}\Theta(y-x)=-\partial_{i}\Theta(x-y)$ (0a), and the notation $H*\kappa$ of The Convolution of a Bounded Continuous Function with a Probability Measure on Euclidean Space: Continuity, Boundedness and Differentiation Under the Integral §continuous,

$$A(x)=m_{u}\int\Theta(x-y)\,\kappa(dy)=m_{u}\,(\Theta*\kappa)(x),\qquad B_{i}(x)=-m_{u}\bigl((\partial_{i}\Theta)*\kappa\bigr)(x).$$

$\Theta$ is of class $C^{1}$ (claim 2 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous) with $|\Theta|\le C_{s}$ and $|\partial_{i}\Theta|\le C_{s}$, so The Convolution of a Bounded Continuous Function with a Probability Measure on Euclidean Space: Continuity, Boundedness and Differentiation Under the Integral §derivative with $G=\Theta$ and $C=C_{s}$ shows that $\Theta*\kappa$ is of class $C^{1}$ with $\partial_{i}(\Theta*\kappa)=(\partial_{i}\Theta)*\kappa$. By claims 1 and 3 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set, $A$ is of class $C^{1}$ with $\partial_{i}A=m_{u}(\partial_{i}\Theta)*\kappa=-B_{i}$.

Now $\mathbf{1}_{Q}F_{j}=\mathbf{1}_{Q}F_{j}^{+}-\mathbf{1}_{Q}F_{j}^{-}$, so by linearity of the integral (claim 2 of Linearity and Monotonicity of the Lebesgue Integral) $a_{j}=A^{+}-A^{-}$, with $A^{\pm}$, $B^{\pm}_{i}$ the functions above for the two signs. By the claim and Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set (claims 1 and 3) $a_{j}$ is of class $C^{1}$ and, by linearity again,

$$\partial_{i}a_{j}(x)=-B^{+}_{i}(x)+B^{-}_{i}(x)=-\int\mathbf{1}_{Q}(y)F_{j}(y)\,\partial_{i}\Theta(y-x)\,dy .$$

For $k\in\mathbb{Z}^{d}$, periodicity of $\Theta$ gives $\Theta(y-x-k)=\Theta(y-x)$, so $a_{j}(x+k)=a_{j}(x)$. Hence $a_{j}\in C^{1}_{\mathrm{per}}$ (Lattice-Periodic Functions and the Periodic Function Classes §classes). Consequently $a_{j}$ and $\partial_{i}a_{j}$ are continuous (claim 3 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous) and periodic (Elementary Properties of Lattice-Periodic Functions §derivative), hence bounded (Elementary Properties of Lattice-Periodic Functions §bounded) and Borel, and integrable against every probability measure (Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures); in particular $a$ is Borel and its class lies in $L^{2}(\mu;\mathbb{R}^{d})$.

Step 5 (Clause 1: independence of the representative). Let $F'$ be Borel and agree with $F$ $\sigma$-almost everywhere. The set $N=\{y:F'(y)\ne F(y)\}$ is Borel (a finite union of sets $\{F'_{j}\ne F_{j}\}$, with claim 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions) and contained in a $\sigma$-null set, so $\sigma(N)=0$ by monotonicity of measures. By (D), $\int\mathbf{1}_{N}\mathbf{1}_{Q}p_{\mu}\,dy=\sigma(N)=0$, so by The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §vanishing the function $\mathbf{1}_{N}\mathbf{1}_{Q}p_{\mu}$ vanishes off a $\lambda_{d}$-null set; since $p_{\mu}>0$, it is nonzero exactly on $N\cap Q$, which is therefore $\lambda_{d}$-null. Also $\int\lVert F'\rVert^{2}\,d\sigma=\int\lVert F\rVert^{2}\,d\sigma<\infty$ by The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §comparison, so the average $a'$ of $F'$ is defined, and for each $x$ and $j$ the integrands $\mathbf{1}_{Q}\Theta(\cdot-x)F'_{j}$ and $\mathbf{1}_{Q}\Theta(\cdot-x)F_{j}$ agree off $N\cap Q$, so they have the same integral by the same clause: $a'=a$.

Step 6 (Clause 1: contraction). Fix $x$ and put $G(z)=F(\pi(x+z))$, a Borel map. By (K) with $h=\lVert F\rVert^{2}$, Step 2 and $\Theta\le C_{s}$,

$$\int\lVert G\rVert^{2}\,dN_{s}=\int\mathbf{1}_{Q}(y)\lVert F(y)\rVert^{2}\,\Theta(y-x)\,dy\le C_{s}c_{s}^{-1}\int\lVert F\rVert^{2}\,d\sigma<\infty .$$

As $G_{j}^{2}\le\lVert G\rVert^{2}$ (claim 1 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$), each $G_{j}$ is a square-integrable random variable on the probability space $(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}),N_{s})$ (Square-Integrable Random Variables and the Mean-Square Inner Product), and so is the constant $1$, with $\lVert1\rVert_{2}=1$. Claim 1 of Cauchy-Schwarz and Triangle Inequalities for the Mean-Square Norm with $X=G_{j}$, $Y=1$ and Step 3 give $a_{j}(x)^{2}=(\int G_{j}\,dN_{s})^{2}\le\int G_{j}^{2}\,dN_{s}$. Summing over $j$ (claim 1 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ and claim 1 of Linearity and Monotonicity of the Lebesgue Integral),

$$\lVert a(x)\rVert^{2}\le\int\lVert G\rVert^{2}\,dN_{s}=\int\mathbf{1}_{Q}(y)\lVert F(y)\rVert^{2}\,\Theta(y-x)\,dy .$$

Integrate in $\mu$ (monotonicity; the right side is a measurable function of $x$ by (T)), apply (T) with $\alpha=\mu$, $\beta=\lambda_{d}$, take the factor $\mathbf{1}_{Q}(y)\lVert F(y)\rVert^{2}$ out of the inner integral (claim 1 of Linearity and Monotonicity of the Lebesgue Integral), and use the definition of $p_{\mu}$ and (D):

$$\int\lVert a\rVert^{2}\,d\mu\le\int\mathbf{1}_{Q}(y)\lVert F(y)\rVert^{2}\Bigl(\int\Theta(y-x)\,\mu(dx)\Bigr)dy=\int\mathbf{1}_{Q}\lVert F\rVert^{2}p_{\mu}\,dy=\int\lVert F\rVert^{2}\,d\sigma .$$

This completes clause 1. We also record that the average is linear: if $F$ and $\tilde F$ both satisfy the hypotheses, with averages $a$ and $\tilde a$, then $F-\tilde F$ is Borel (claim 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, applied to each component) with $\int\lVert F-\tilde F\rVert^{2}\,d\sigma\le2\int\lVert F\rVert^{2}\,d\sigma+2\int\lVert\tilde F\rVert^{2}\,d\sigma<\infty$ (Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions), and its average is $a-\tilde a$ by claim 2 of Linearity and Monotonicity of the Lebesgue Integral; so $\int\lVert a-\tilde a\rVert^{2}\,d\mu\le\int\lVert F-\tilde F\rVert^{2}\,d\sigma$. $\qquad(\mathrm{C})$

Step 7 (Clause 2). Suppose the class of $F$ lies in $T_{\sigma}$, the closure of $G_{\sigma}$ (The Tangent Space of the Torus Wasserstein Space at a Probability Measure §tangent) in $L^{2}(\sigma;\mathbb{R}^{d})$ with the distance of its norm (Optimal Transport on the Flat Torus: Standing Notation §fields). By Sequential Characterization of the Closure in a Metric Space and The Tangent Space of the Torus Wasserstein Space at a Probability Measure §gradients there are $f_{k}\in C^{\infty}_{\mathrm{per}}$, $k\in\mathbb{N}$, with $\lVert\nabla f_{k}-F\rVert_{\sigma}\to0$.

Fix $k$. Let $\rho:\mathbb{R}^{d}\to\mathbb{R}^{d}$, $\rho(z)=-z$, which is continuous, hence Borel, and let $\tilde N=\rho_{\#}N_{s}\in\mathcal{P}(\mathbb{R}^{d})$ (Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward). For $G\in C^{\infty}_{\mathrm{per}}$, $G$ is of class $C^{1}$ (Smooth Map on a Euclidean Open Set), and $G$ and each $\partial_{i}G$ belong to $C_{\mathrm{per}}$ (Elementary Properties of Lattice-Periodic Functions §derivative), hence are bounded (Elementary Properties of Lattice-Periodic Functions §bounded); so The Convolution of a Bounded Continuous Function with a Probability Measure on Euclidean Space: Continuity, Boundedness and Differentiation Under the Integral §derivative applies to $G$ and $\tilde N$, with $C$ the largest of these finitely many bounds: $G*\tilde N$ is of class $C^{1}$ with $\partial_{i}(G*\tilde N)=(\partial_{i}G)*\tilde N$. As $\partial_{i}G\in C^{\infty}_{\mathrm{per}}$ again (Elementary Properties of Lattice-Periodic Functions §derivative), induction on $r$, reapplying The Convolution of a Bounded Continuous Function with a Probability Measure on Euclidean Space: Continuity, Boundedness and Differentiation Under the Integral §derivative to each iterated partial derivative of $G$ (again in $C^{\infty}_{\mathrm{per}}$, hence bounded with bounded derivatives, by Elementary Properties of Lattice-Periodic Functions §derivative and Elementary Properties of Lattice-Periodic Functions §bounded) and using clause 2 of C^k Maps on a Euclidean Open Set, shows that $G*\tilde N$ is of class $C^{r}$ for every $r$, hence smooth. It is periodic, since $(G*\tilde N)(x+l)=\int G(x+l-w)\,\tilde N(dw)=(G*\tilde N)(x)$ for $l\in\mathbb{Z}^{d}$. So $h_{k}=f_{k}*\tilde N\in C^{\infty}_{\mathrm{per}}$, and the class of $\nabla h_{k}$ belongs to $G_{\mu}$ (The Tangent Space of the Torus Wasserstein Space at a Probability Measure §gradients). By the change of variables for $\tilde N=\rho_{\#}N_{s}$, periodicity of $\partial_{i}f_{k}$ with $x+z-\pi(x+z)\in\mathbb{Z}^{d}$ (0a), and Step 1 applied to the bounded Borel $h=\partial_{i}f_{k}$ (so $\mathbf{1}_{Q}h$ is integrable),

$$\partial_{i}h_{k}(x)=\int\partial_{i}f_{k}(x-w)\,\tilde N(dw)=\int\partial_{i}f_{k}(x+z)\,N_{s}(dz)=\int\partial_{i}f_{k}\bigl(\pi(x+z)\bigr)\,N_{s}(dz)=\int\mathbf{1}_{Q}(y)\,\partial_{i}f_{k}(y)\,\Theta(y-x)\,dy .$$

Thus $\nabla h_{k}$ is the average of $\nabla f_{k}$, which is Borel with $\int\lVert\nabla f_{k}\rVert^{2}\,d\sigma<\infty$ by Optimal Transport on the Flat Torus: Standing Notation §calculus. By (C) with $\tilde F=\nabla f_{k}$, $\lVert a-\nabla h_{k}\rVert_{\mu}\le\lVert F-\nabla f_{k}\rVert_{\sigma}$, which tends to $0$. So the classes of $\nabla h_{k}$, members of $G_{\mu}$, converge to the class of $a$ in $L^{2}(\mu;\mathbb{R}^{d})$ (Convergent Sequence in a Metric Space), and by Sequential Characterization of the Closure in a Metric Space the class of $a$ lies in the closure of $G_{\mu}$, which is $T_{\mu}$ (The Tangent Space of the Torus Wasserstein Space at a Probability Measure §tangent).

Step 8 (Clause 3). Fix $i\in[d]$ and let $f(x,y)=\mathbf{1}_{Q}(y)F_{i}(y)\,\partial_{i}\Theta(y-x)$ on $\mathbb{R}^{d}\times\mathbb{R}^{d}$, with $\alpha=\mu$ and $\beta=\lambda_{d}$ in (0b). As $|f(x,y)|\le C_{s}\mathbf{1}_{Q}(y)|F_{i}(y)|$, (T) and Step 2 give $\int|f|\,d(\mu\otimes\lambda_{d})\le C_{s}\int\mathbf{1}_{Q}|F_{i}|\,dy<\infty$; for each $x$ the function $y\mapsto f(x,y)$ is integrable (Step 2), and for each $y$ the function $x\mapsto f(x,y)$ is bounded by the constant $C_{s}\mathbf{1}_{Q}(y)|F_{i}(y)|$, hence $\mu$-integrable. By Step 4, (F), claim 2 of Linearity and Monotonicity of the Lebesgue Integral (to take the constant $\mathbf{1}_{Q}(y)F_{i}(y)$ out of the inner integral) and the derivative formula for $p_{\mu}$ (Step 0),

$$\int\partial_{i}a_{i}\,d\mu=-\int\Bigl(\int f(x,y)\,dy\Bigr)\mu(dx)=-\int\mathbf{1}_{Q}(y)F_{i}(y)\Bigl(\int\partial_{i}\Theta(y-x)\,\mu(dx)\Bigr)dy=-\int\mathbf{1}_{Q}F_{i}\,\partial_{i}p_{\mu}\,dy .$$

By The Torus Heat Kernel: Regularity and Bounds, the Density of a Heat-Smoothed Measure, and Its Lipschitz Dependence on the Measure §density, $\sigma\in\mathcal{P}^{\mathcal{I}}(\mathbb{T}^{d})$ and $\xi_{\sigma}$ is the class of $r=p_{\mu}^{-1}\nabla p_{\mu}$, a Borel map whose class is in $L^{2}(\sigma;\mathbb{R}^{d})$. The function $F\cdot r$ is Borel with $|F\cdot r|\le\tfrac12(\lVert F\rVert^{2}+\lVert r\rVert^{2})$ (Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions), hence $\sigma$-integrable, and $(F\cdot r)\mathbf{1}_{Q}p_{\mu}=\sum_{i}\mathbf{1}_{Q}F_{i}\partial_{i}p_{\mu}$. Summing over $i$ (claim 2 of Linearity and Monotonicity of the Lebesgue Integral) and using (D) and Optimal Transport on the Flat Torus: Standing Notation §fields,

$$\int\sum_{i=1}^{d}\partial_{i}a_{i}\,d\mu=-\int(F\cdot r)\,\mathbf{1}_{Q}\,p_{\mu}\,dy=-\int F\cdot r\,d\sigma=-\langle F,\xi_{\sigma}\rangle_{\sigma}.$$

Step 9 (Clause 4). Write $\Gamma=\gamma\boxtimes N_{s}$ and $\iota'=\iota^{d+d,d}$. By Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §projections and the definition of $\tau$, $\mathrm{pr}_{1}(\tau(\iota'(w,z)))=\pi(\mathrm{pr}_{1}(w)+z)$ and $\mathrm{pr}_{2}(\tau(\iota'(w,z)))=\pi(\mathrm{pr}_{2}(w)+z)$. For a Borel $\Psi\ge0$ on $\mathbb{R}^{d+d}$, or a $\gamma^{s}$-integrable Borel $\Psi$, the change of variables of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward and (P) (with $\alpha=\gamma$, $\beta=N_{s}$) give

$$\int\Psi\,d\gamma^{s}=\int\Psi\circ\tau\,d\Gamma=\int\Psi\bigl(\tau(\iota'(w,z))\bigr)\,(\gamma\otimes N_{s})(d(w,z)). \qquad(\mathrm{I})$$

(9a) Marginals. Let $A\in\mathcal{B}(\mathbb{R}^{d})$. By (T) with $\alpha=\mu$, $\beta=N_{s}$, the function $\phi_{A}(x)=\int\mathbf{1}_{A}(\pi(x+z))\,N_{s}(dz)$ is Borel. By (I) with $\Psi=\mathbf{1}_{A}\circ\mathrm{pr}_{1}$, (T) with $\alpha=\gamma$, $\beta=N_{s}$, the change of variables for $(\mathrm{pr}_{1})_{\#}\gamma=\mu$ (Couplings of Two Probability Measures on Euclidean Space and Their Quadratic Cost §coupling), (K) with $h=\mathbf{1}_{A}$, (T) with $\alpha=\mu$, $\beta=\lambda_{d}$, and (D),

$$\gamma^{s}\bigl(\mathrm{pr}_{1}^{-1}(A)\bigr)=\int\phi_{A}\bigl(\mathrm{pr}_{1}(w)\bigr)\,\gamma(dw)=\int\phi_{A}\,d\mu=\int\Bigl(\int\mathbf{1}_{Q}(y)\mathbf{1}_{A}(y)\,\Theta(y-x)\,dy\Bigr)\mu(dx)=\int\mathbf{1}_{A}\mathbf{1}_{Q}\,p_{\mu}\,dy=S_{s}\mu(A).$$

The same computation with $\mathrm{pr}_{2}$, $(\mathrm{pr}_{2})_{\#}\gamma=\nu$ and $p_{\nu}$ gives $\gamma^{s}(\mathrm{pr}_{2}^{-1}(A))=S_{s}\nu(A)$. As $\gamma^{s}\in\mathcal{P}(\mathbb{R}^{d+d})$ (claim 1 of Image Measures, Measures with Densities, and Change of Variables), $\gamma^{s}\in\Pi(S_{s}\mu,S_{s}\nu)$ by Couplings of Two Probability Measures on Euclidean Space and Their Quadratic Cost §coupling, and $S_{s}\mu,S_{s}\nu\in\mathcal{P}(\mathbb{T}^{d})$ by The Heat Semigroup on the Probability Measures on the Torus §heat.

(9b) Cost. For $u,v,z\in\mathbb{R}^{d}$, The Wrapped Displacement and the Flat Torus Distance §distance and (L) give $d_{\mathbb{T}}(\pi(u+z),\pi(v+z))=\lVert\varpi(\pi(v+z)-\pi(u+z))\rVert=\lVert\varpi(v-u)\rVert=d_{\mathbb{T}}(u,v)$. By Probability Measures on the Flat Torus, the Torus Cost of a Coupling, the Torus Wasserstein Distance and Optimal Couplings §cost, (I) with $\Psi(w')=d_{\mathbb{T}}(\mathrm{pr}_{1}(w'),\mathrm{pr}_{2}(w'))^{2}$, (T), and $\int c\,dN_{s}=c$ for a constant $c\ge0$ (claim 1 of Linearity and Monotonicity of the Lebesgue Integral),

$$I_{\mathbb{T}}(\gamma^{s})=\int\Bigl(\int d_{\mathbb{T}}\bigl(\mathrm{pr}_{1}(w),\mathrm{pr}_{2}(w)\bigr)^{2}\,N_{s}(dz)\Bigr)\gamma(dw)=\int d_{\mathbb{T}}\bigl(\mathrm{pr}_{1}(w),\mathrm{pr}_{2}(w)\bigr)^{2}\,\gamma(dw)=I_{\mathbb{T}}(\gamma).$$

(9c) Pairing. Let $\Psi(w')=F(\mathrm{pr}_{1}(w'))\cdot\varpi(\mathrm{pr}_{2}(w')-\mathrm{pr}_{1}(w'))$, which is Borel and $\gamma^{s}$-integrable with $\int\Psi\,d\gamma^{s}=\mathcal{J}_{\mathbb{T}}(F,\gamma^{s})$ (The Torus Displacement Pairing of a Vector Field Along a Coupling §pairing, with the representative $F$ of the class of $F$ in $L^{2}(\sigma;\mathbb{R}^{d})$). By (L), with $u=\mathrm{pr}_{1}(w)+z$ and $v=\mathrm{pr}_{2}(w)+z$,

$$f(w,z)=\Psi\bigl(\tau(\iota'(w,z))\bigr)=F\bigl(\pi(\mathrm{pr}_{1}(w)+z)\bigr)\cdot\varpi\bigl(\mathrm{pr}_{2}(w)-\mathrm{pr}_{1}(w)\bigr),$$

and by (I) this $f$ is $(\gamma\otimes N_{s})$-integrable with $\int f\,d(\gamma\otimes N_{s})=\mathcal{J}_{\mathbb{T}}(F,\gamma^{s})$. For fixed $w$, with $x=\mathrm{pr}_{1}(w)$ and $e=\varpi(\mathrm{pr}_{2}(w)-\mathrm{pr}_{1}(w))$, the section $z\mapsto f(w,z)=\sum_{j}e_{j}F_{j}(\pi(x+z))$ is $N_{s}$-integrable with integral $a(x)\cdot e$, by Step 3 and claim 2 of Linearity and Monotonicity of the Lebesgue Integral. The Fubini clause of Tonelli and Fubini Theorems, applied to $f$ (measurable for the product $\sigma$-algebra by (0b)), provides a $\gamma$-null Borel set $M$ off which the $z$-sections are integrable, and an integrable function equal to $\int f(w,z)\,N_{s}(dz)=a(\mathrm{pr}_{1}(w))\cdot\varpi(\mathrm{pr}_{2}(w)-\mathrm{pr}_{1}(w))$ off $M$ whose $\gamma$-integral is $\int f\,d(\gamma\otimes N_{s})$. The Borel function $w\mapsto a(\mathrm{pr}_{1}(w))\cdot\varpi(\mathrm{pr}_{2}(w)-\mathrm{pr}_{1}(w))$ agrees with it off $M$, so by The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §comparison and The Torus Displacement Pairing of a Vector Field Along a Coupling §pairing, with the representative $a$ of the class of $a$ in $L^{2}(\mu;\mathbb{R}^{d})$ (Step 4),

$$\mathcal{J}_{\mathbb{T}}(F,\gamma^{s})=\int a\bigl(\mathrm{pr}_{1}(w)\bigr)\cdot\varpi\bigl(\mathrm{pr}_{2}(w)-\mathrm{pr}_{1}(w)\bigr)\,\gamma(dw)=\mathcal{J}_{\mathbb{T}}(a,\gamma). \qquad\blacksquare$$