Proof of The Heat Gauge Form on Probability Measures: Algebra, Representation by Smoothed Densities, Positivity, Cauchy-Schwarz, Separation, the Triangle Inequality, the Wasserstein Bound and Derivative Bounds for the Potential

Each result cited is universally quantified over the data in its own statement. Throughout, for $\alpha\in\mathcal{P}(\mathbb{R}^{q})$ and $k\in\mathbb{N}$ we write $\alpha_{k}=g_{s_{k}}*\alpha$, a nonnegative Borel integrable function with $\int\alpha_{k}\,d\lambda_{q}=1$ whose square is integrable (Gaussian Smoothing of a Probability Measure on Euclidean Space: Regularity, Mass, Duality, Approximation of a Bounded Function with a Modulus of Continuity, and the Pairing Identity (iii) and Gaussian Smoothing of a Probability Measure on Euclidean Space: Regularity, Mass, Duality, Approximation of a Bounded Function with a Modulus of Continuity, and the Pairing Identity (v) with $\nu=\mu$, as $s_{k}\le\tfrac12$), and $D_{k}=\mu_{k}-\nu_{k}$, $D'_{k}=\mu'_{k}-\nu'_{k}$. By The Multiscale Gaussian Kernel: Definition by a Series, Regularity and Bounds, the Smoothed Potential of a Probability Measure and the Kernel Pairing of Two Measures (iii), for all $\alpha,\beta\in\mathcal{P}(\mathbb{R}^{q})$,

$$\mathcal{K}(\alpha,\beta)=\mathcal{K}(\beta,\alpha)=\int_{\mathbb{R}^{q}}K*\beta\,d\alpha=\sum_{k=1}^{\infty}w_{k}\int_{\mathbb{R}^{q}}\alpha_{k}\beta_{k}\,d\lambda_{q},\tag{$*$}$$

a convergent series with nonnegative terms, each product $\alpha_{k}\beta_{k}$ being integrable. Series of real numbers are added and multiplied by constants termwise by Elementary Properties of Series of Real Numbers (i), and integrals of integrable functions are linear by claim 2 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion. Elementary identities in the field $\mathbb{R}$ are those of Field; $2ab\le a^{2}+b^{2}$ for real $a,b$ follows from $0\le(a-b)^{2}=a^{2}-2ab+b^{2}$ (claim 5 of Zero Products and Elementary Identities in a Field, claim 2 of Nonnegativity of Squares in an Ordered Field, claim 3 of Elementary Arithmetic in an Ordered Field); $\sqrt{\cdot}$ is nondecreasing on $[0,\infty)$ and $\sqrt{a}\sqrt{b}=\sqrt{ab}$ (claim 5 of Real Powers Through the Exponential, and Elementary Asymptotic Tools: Monotonicity, Null Sequences of Negative Powers, Exponential Domination, Integer Rounding, and Square-Root and Exponential Inequalities), and $\sqrt{c^{2}}=|c|$ for real $c$ (claim 1 of Nonnegativity of Squares in an Ordered Field and Existence and Uniqueness of the Nonnegative Square Root).

Claim 1. Writing out the definition of $B$ and using the symmetry of $\mathcal{K}$ in $(*)$: $B(\mu',\nu';\mu,\nu)=\mathcal{K}(\mu',\mu)-\mathcal{K}(\mu',\nu)-\mathcal{K}(\nu',\mu)+\mathcal{K}(\nu',\nu)=B(\mu,\nu;\mu',\nu')$; $B(\nu,\mu;\mu',\nu')=\mathcal{K}(\nu,\mu')-\mathcal{K}(\nu,\nu')-\mathcal{K}(\mu,\mu')+\mathcal{K}(\mu,\nu')=-B(\mu,\nu;\mu',\nu')$; and $B(\mu,\nu;\mu',\nu')+B(\nu,\sigma;\mu',\nu')=\mathcal{K}(\mu,\mu')-\mathcal{K}(\mu,\nu')-\mathcal{K}(\sigma,\mu')+\mathcal{K}(\sigma,\nu')=B(\mu,\sigma;\mu',\nu')$, the terms in $\nu$ cancelling. Consequently $Q(\nu,\mu)=B(\nu,\mu;\nu,\mu)=-B(\mu,\nu;\nu,\mu)=-B(\nu,\mu;\mu,\nu)=B(\mu,\nu;\mu,\nu)=Q(\mu,\nu)$, using the second identity, then the first, then the second again; $Q(\mu,\mu)=\mathcal{K}(\mu,\mu)-\mathcal{K}(\mu,\mu)-\mathcal{K}(\mu,\mu)+\mathcal{K}(\mu,\mu)=0$; and, using additivity in the first pair, the symmetry of pairs, and additivity again,

$$Q(\mu,\sigma)=B(\mu,\nu;\mu,\sigma)+B(\nu,\sigma;\mu,\sigma)=B(\mu,\sigma;\mu,\nu)+B(\mu,\sigma;\nu,\sigma)=\bigl(Q(\mu,\nu)+B(\nu,\sigma;\mu,\nu)\bigr)+\bigl(B(\mu,\nu;\nu,\sigma)+Q(\nu,\sigma)\bigr),$$

and $B(\nu,\sigma;\mu,\nu)=B(\mu,\nu;\nu,\sigma)$ by the symmetry of pairs, which gives the displayed polarisation identity.

Claim 2. By $(*)$ applied to the four pairings in $B$ and termwise linearity of series,

$$B(\mu,\nu;\mu',\nu')=\sum_{k=1}^{\infty}w_{k}\int_{\mathbb{R}^{q}}\bigl(\mu_{k}\mu'_{k}-\mu_{k}\nu'_{k}-\nu_{k}\mu'_{k}+\nu_{k}\nu'_{k}\bigr)\,d\lambda_{q}=\sum_{k=1}^{\infty}w_{k}\int_{\mathbb{R}^{q}}D_{k}D'_{k}\,d\lambda_{q},$$

by linearity of the integral and the pointwise identity $(a-b)(c-d)=ac-ad-bc+bd$ in $\mathbb{R}$; each $D_{k}D'_{k}$ is integrable as a linear combination of the four integrable products (claim 2 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion). The $k$th term is bounded in absolute value by the $k$th term of the convergent series $\sum_{k}w_{k}\int(\mu_{k}\mu'_{k}+\mu_{k}\nu'_{k}+\nu_{k}\mu'_{k}+\nu_{k}\nu'_{k})\,d\lambda_{q}$ (claim 2 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion, the triangle inequality of claim 5 of Properties of the Absolute Value in an Ordered Field, and termwise linearity), so the series converges absolutely by An Absolutely Convergent Series of Real Numbers Converges (iii). With $(\mu',\nu')=(\mu,\nu)$ this gives $Q(\mu,\nu)=\sum_{k}w_{k}\int D_{k}^{2}\,d\lambda_{q}$, whose terms are nonnegative (claim 2 of Nonnegativity of Squares in an Ordered Field, and monotonicity in claim 2 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion against the zero function), so $Q(\mu,\nu)\ge0$ by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series (i).

Claim 3. Put $a=Q(\mu,\nu)$, $b=Q(\mu',\nu')$, $c=B(\mu,\nu;\mu',\nu')$, and let $t\in\mathbb{R}$. For each $k$, $\int(tD_{k}-D'_{k})^{2}\,d\lambda_{q}=t^{2}\int D_{k}^{2}-2t\int D_{k}D'_{k}+\int D'^{2}_{k}\ge0$ (expansion of the square, linearity, claim 1 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion); multiplying by $w_{k}$, summing (claim 2 and termwise linearity) and using Series of Nonnegative Real Numbers, Comparison, and the Geometric Series (i) gives $t^{2}a-2tc+b\ge0$ for every real $t$. If $a=0$, then $b\ge2tc$ for every $t$, which forces $c=0$ (if $c\ne0$, taking $t=(b+1)/(2c)$ gives $b\ge b+1$, a contradiction), so $|c|=0\le\sqrt{a}\sqrt{b}$. If $a>0$, taking $t=c/a$ gives $c^{2}/a-2c^{2}/a+b\ge0$, that is $-c^{2}/a+b\ge0$, and multiplying by $a>0$ (claim 5 of Elementary Arithmetic in an Ordered Field) gives $c^{2}\le ab$, whence $|c|=\sqrt{c^{2}}\le\sqrt{ab}=\sqrt{a}\sqrt{b}$.

Claim 4. If $\mu=\nu$ then $Q(\mu,\nu)=0$ by claim 1. Conversely let $Q(\mu,\nu)=0$. Each term $w_{k}\int D_{k}^{2}$ of the nonnegative series in claim 2 is at most a partial sum of it (claim 6 of Properties of Finite Sums), which is at most the sum $0$ (Series of Nonnegative Real Numbers, Comparison, and the Geometric Series (ii)); so $\int D_{k}^{2}\,d\lambda_{q}=0$ ($w_{k}>0$), hence $D_{k}^{2}=0$, that is $\mu_{k}=\nu_{k}$, $\lambda_{q}$-almost everywhere by The Lebesgue Integral and Null Sets (iv). Let $\phi:\mathbb{R}^{q}\to\mathbb{R}$ be Lipschitz, with a constant $L>0$ (a Lipschitz constant may be increased), with values in $[0,1]$; $\phi$ is Borel and bounded by $1$ (A Probability Measure on Euclidean Space Is Determined by the Integrals of Lipschitz Functions with Values in the Unit Interval, preamble). By The Lebesgue Integral and Null Sets (iii), $\int\phi\,\mu_{k}\,d\lambda_{q}=\int\phi\,\nu_{k}\,d\lambda_{q}$, so by Gaussian Smoothing of a Probability Measure on Euclidean Space: Regularity, Mass, Duality, Approximation of a Bounded Function with a Modulus of Continuity, and the Pairing Identity (iii)

$$\int_{\mathbb{R}^{q}}g_{s_{k}}*\phi\,d\mu=\int_{\mathbb{R}^{q}}g_{s_{k}}*\phi\,d\nu\qquad\text{for every }k\in\mathbb{N}.$$

Let $\epsilon>0$ and put $r=\epsilon/L$; then $|\phi(x)-\phi(x')|\le L\lVert x-x'\rVert\le\epsilon$ whenever $\lVert x-x'\rVert\le r$, so by Gaussian Smoothing of a Probability Measure on Euclidean Space: Regularity, Mass, Duality, Approximation of a Bounded Function with a Modulus of Continuity, and the Pairing Identity (iv) with $M=1$, $|(g_{s_{k}}*\phi)(x)-\phi(x)|\le\epsilon+2qs_{k}L^{2}/\epsilon^{2}$ for all $x$ and $k$. Choose $N\in\mathbb{N}$ with $N\epsilon^{3}>2qL^{2}$ (claim 2 of The Archimedean Property of the Real Numbers); since $N\le4^{N}$ (the set of $k$ with $k\le4^{k}$ contains $1$ and is closed under $k\mapsto k+1$, as $k+1\le4^{k}+3\cdot4^{k}=4^{k+1}$, so it is $\mathbb{N}$ by Principle of Induction for the Natural Numbers), $s_{N}=4^{-N}\le1/N$ and $2qs_{N}L^{2}/\epsilon^{2}\le2qL^{2}/(N\epsilon^{2})<\epsilon$. Hence $|(g_{s_{N}}*\phi)(x)-\phi(x)|\le2\epsilon$ for all $x$, and by claim 2 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion and claim 6(b) of Borel Measurability and Bounded Integration on a Metric Space,

$$\Bigl|\int\phi\,d\mu-\int\phi\,d\nu\Bigr|\le\Bigl|\int(\phi-g_{s_{N}}*\phi)\,d\mu\Bigr|+\Bigl|\int(g_{s_{N}}*\phi-\phi)\,d\nu\Bigr|\le4\epsilon,$$

using the displayed equality for $k=N$ and the triangle inequality. As $\epsilon>0$ was arbitrary, $\int\phi\,d\mu=\int\phi\,d\nu$ by Comparison of Real Numbers with Arbitrary Positive Slack. This holds for every Lipschitz $\phi$ with values in $[0,1]$, so $\mu=\nu$ by A Probability Measure on Euclidean Space Is Determined by the Integrals of Lipschitz Functions with Values in the Unit Interval.

Claim 5. By claim 1, claim 3 and $c\le|c|$ (claim 3 of Properties of the Absolute Value in an Ordered Field),

$$Q(\mu,\sigma)=Q(\mu,\nu)+2B(\mu,\nu;\nu,\sigma)+Q(\nu,\sigma)\le Q(\mu,\nu)+2\sqrt{Q(\mu,\nu)}\sqrt{Q(\nu,\sigma)}+Q(\nu,\sigma)=\bigl(\sqrt{Q(\mu,\nu)}+\sqrt{Q(\nu,\sigma)}\bigr)^{2},$$

and taking nonnegative square roots, which is order preserving, gives the triangle inequality.

Claim 6. Let $C_{Q}=q\,M_{2}$, with $M_{2}$ the bound on the second partial derivatives of $K$ in The Multiscale Gaussian Kernel: Definition by a Series, Regularity and Bounds, the Smoothed Potential of a Probability Measure and the Kernel Pairing of Two Measures (i) (here $M_{2}$ is that constant, not a second moment). Let $\mu,\nu\in\mathcal{P}_{2}(\mathbb{R}^{q})$ and $\pi\in\Pi(\mu,\nu)$, so that $(\mathrm{pr}_{1})_{\#}\pi=\mu$ and $(\mathrm{pr}_{2})_{\#}\pi=\nu$ (Couplings of Two Probability Measures on Euclidean Space and Their Quadratic Cost (i)). For $z,z'\in\mathbb{R}^{q+q}$ write $x=\mathrm{pr}_{1}(z)$, $y=\mathrm{pr}_{2}(z)$, $x'=\mathrm{pr}_{1}(z')$, $y'=\mathrm{pr}_{2}(z')$, and

$$\Delta(z,z')=K(x-x')-K(x-y')-K(y-x')+K(y-y').$$

By $(*)$ and The Multiscale Gaussian Kernel: Definition by a Series, Regularity and Bounds, the Smoothed Potential of a Probability Measure and the Kernel Pairing of Two Measures (ii), $\mathcal{K}(\mu,\mu)=\int\bigl(\int K(x-x')\,\mu(dx')\bigr)\mu(dx)$, and by the change-of-variables formula of Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iv), applied to the inner integral with the bounded Borel function $x'\mapsto K(x-x')$ and then to the outer one, this equals $\int\bigl(\int K(\mathrm{pr}_{1}(z)-\mathrm{pr}_{1}(z'))\,\pi(dz')\bigr)\pi(dz)$; the inner integral is a bounded Borel function of $z$ by The Convolution of a Bounded Continuous Function with a Probability Measure on Euclidean Space: Continuity, Boundedness and Differentiation Under the Integral (i) applied to the probability measure $(\mathrm{pr}_{1})_{\#}\pi=\mu$, read at the point $\mathrm{pr}_{1}(z)$. The same holds for the other three pairings with $\mathrm{pr}_{2}$ in place of $\mathrm{pr}_{1}$ in the appropriate slots, so, by linearity of the integral,

$$Q(\mu,\nu)=\int_{\mathbb{R}^{q+q}}\Bigl(\int_{\mathbb{R}^{q+q}}\Delta(z,z')\,\pi(dz')\Bigr)\pi(dz).$$

We bound $\Delta$. Fix $z,z'$ and let $h:\mathbb{R}^{q}\to\mathbb{R}$, $h(w)=K(w-x')-K(w-y')$, of class $C^{1}$ with $\partial_{i}h(w)=\partial_{i}K(w-x')-\partial_{i}K(w-y')$ (claims 1 and 3 of Constants, Affine Functions, Sums, Products and Quotients of Cᵏ Functions together with Partial Derivatives, Continuity and $C^k$ Regularity under a Scaling Substitution, claim 2, for the substitution $w\mapsto w-x'$). Since $\partial_{i}K$ is of class $C^{1}$ with partial derivatives bounded by $M_{2}$, part (i) of Multivariate Taylor Expansion with Uniform Second-Order Remainder (on $W=\mathbb{R}^{q}$, every segment lying in it) gives $|\partial_{i}h(w)|\le\sqrt{q}\,M_{2}\,\lVert x'-y'\rVert$ for all $w$; applying part (i) again to $h$, whose partial derivatives are bounded by $\sqrt{q}M_{2}\lVert x'-y'\rVert$, gives

$$|\Delta(z,z')|=|h(x)-h(y)|\le\sqrt{q}\cdot\sqrt{q}\,M_{2}\,\lVert x'-y'\rVert\,\lVert x-y\rVert\le\frac{q\,M_{2}}{2}\bigl(\lVert x-y\rVert^{2}+\lVert x'-y'\rVert^{2}\bigr),$$

by $2ab\le a^{2}+b^{2}$. Integrating in $z'$ against $\pi$ (monotonicity, claim 2 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion; the right side is integrable since $\int\lVert x'-y'\rVert^{2}\pi(dz')=I(\pi)<\infty$ by Couplings of Two Probability Measures on Euclidean Space and Their Quadratic Cost (ii) and Couplings on Euclidean Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, and the Lipschitz Bound (iii)), then in $z$, gives $Q(\mu,\nu)\le\frac{qM_{2}}{2}\bigl(I(\pi)+I(\pi)\bigr)=C_{Q}\,I(\pi)$, using $|\int\cdot|\le\int|\cdot|$ and $Q\ge0$. Finally, by The Quadratic Wasserstein Distance on Euclidean Space (i), $W_{2}(\mu,\nu)^{2}$ is the infimum of $\{I(\pi):\pi\in\Pi(\mu,\nu)\}$; if $C_{Q}=0$ then $Q(\mu,\nu)=0$ and there is nothing to prove, and otherwise $Q(\mu,\nu)/C_{Q}$ is a lower bound of that set, hence at most its infimum, so $Q(\mu,\nu)\le C_{Q}W_{2}(\mu,\nu)^{2}$ and $\sqrt{Q(\mu,\nu)}\le\sqrt{C_{Q}}\,W_{2}(\mu,\nu)$.

Claim 7. $K*\mu$ and $K*\nu$ are of class $C^{3}$ (The Multiscale Gaussian Kernel: Definition by a Series, Regularity and Bounds, the Smoothed Potential of a Probability Measure and the Kernel Pairing of Two Measures (ii)), so $\psi=K*\mu-K*\nu$ is of class $C^{3}$ by claim 3 of Constants, Affine Functions, Sums, Products and Quotients of Cᵏ Functions, with partial derivatives the differences of those of $K*\mu$ and $K*\nu$ (claim 1 there), and it is continuous and bounded by $2M_{0}$, hence Borel and bounded. Fix $k$ and let $\rho_{k}$ be the measure with density $\mu_{k}$ with respect to $\lambda_{q}$, a probability measure on $\mathcal{B}(\mathbb{R}^{q})$ since $\int\mu_{k}\,d\lambda_{q}=1$ (claim 3 of Image Measures and Measures with a Density: Measures, Integrals and Change of Variables), and similarly $\rho'_{k}$ with density $\nu_{k}$. For $x\in\mathbb{R}^{q}$, by The Gaussian Kernels on Euclidean Space: Scaling, Derivatives up to Order Three, the Convolution Identity, Moments and Tails (iv) with $s=s_{k}$, Tonelli (Tonelli's and Fubini's Theorems, the integrand $(u,y)\mapsto g_{s_{k}}(u-x)g_{s_{k}}(u-y)$ being nonnegative and measurable for the product $\sigma$-algebra as in Gaussian Smoothing of a Probability Measure on Euclidean Space: Regularity, Mass, Duality, Approximation of a Bounded Function with a Modulus of Continuity, and the Pairing Identity), the density identity of claim 3 of Image Measures and Measures with a Density: Measures, Integrals and Change of Variables and the evenness of $g_{s_{k}}$,

$$(g_{2s_{k}}*\mu)(x)=\int\Bigl(\int g_{s_{k}}(u-x)\,g_{s_{k}}(u-y)\,du\Bigr)\mu(dy)=\int g_{s_{k}}(x-u)\,\mu_{k}(u)\,du=(g_{s_{k}}*\rho_{k})(x).$$

Thus $g_{2s_{k}}*\mu=g_{s_{k}}*\rho_{k}$ and likewise $g_{2s_{k}}*\nu=g_{s_{k}}*\rho'_{k}$. By Gaussian Smoothing of a Probability Measure on Euclidean Space: Regularity, Mass, Duality, Approximation of a Bounded Function with a Modulus of Continuity, and the Pairing Identity (ii) applied to $\rho_{k}$ and $\rho'_{k}$, and the density identity again, for $\partial^{\beta}$ any of the identity, $\partial_{i}$, $\partial_{j}\partial_{i}$, $\partial_{l}\partial_{j}\partial_{i}$,

$$\partial^{\beta}\bigl(g_{2s_{k}}*\mu-g_{2s_{k}}*\nu\bigr)(x)=\int_{\mathbb{R}^{q}}\partial^{\beta}g_{s_{k}}(x-u)\,D_{k}(u)\,du .$$

By the Cauchy-Schwarz inequality, the case $p=2$ of Hölder's Inequality (i), applied to the square-integrable functions $u\mapsto\partial^{\beta}g_{s_{k}}(x-u)$ and $D_{k}$ (the former by The Gaussian Kernels on Euclidean Space: Scaling, Derivatives up to Order Three, the Convolution Identity, Moments and Tails (iii) together with the reflection and translation invariance of Translation and Reflection Invariance of Lebesgue Measure on ℝⁿ for the substitution $u\mapsto x-u$, the latter by claim 2 above), and $|\int fg|\le\int|fg|$ (claim 2 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion), $\bigl|\int\partial^{\beta}g_{s_{k}}(x-u)D_{k}(u)\,du\bigr|^{2}\le\int(\partial^{\beta}g_{s_{k}})^{2}\,d\lambda_{q}\int D_{k}^{2}\,d\lambda_{q}\le B_{|\beta|}\,s_{k}^{-(q+2|\beta|)/2}\int D_{k}^{2}\,d\lambda_{q}$, where $|\beta|\in\{0,1,2,3\}$ is the order and $B_{0},\dots,B_{3}$ are the constants of The Gaussian Kernels on Euclidean Space: Scaling, Derivatives up to Order Three, the Convolution Identity, Moments and Tails (iii); here $s_{k}^{-(q+2|\beta|)/2}=2^{k(q+2|\beta|)}$, since $\sqrt{s_{k}}=2^{-k}$ by $(2^{-k})^{2}=4^{-k}$ (exponent rules of Properties of Natural Number Powers in a Field) and the uniqueness of the nonnegative square root, so that the inverse of $(\sqrt{s_{k}})^{q+2|\beta|}=2^{-k(q+2|\beta|)}$ is $2^{k(q+2|\beta|)}$. Now, by The Multiscale Gaussian Kernel: Definition by a Series, Regularity and Bounds, the Smoothed Potential of a Probability Measure and the Kernel Pairing of Two Measures (i) and The Multiscale Gaussian Kernel: Definition by a Series, Regularity and Bounds, the Smoothed Potential of a Probability Measure and the Kernel Pairing of Two Measures (ii), $\partial^{\beta}(K*\mu)(x)=\int\partial^{\beta}K(x-y)\,\mu(dy)=\int\sum_{k}w_{k}\partial^{\beta}g_{2s_{k}}(x-y)\,\mu(dy)=\sum_{k}w_{k}\int\partial^{\beta}g_{2s_{k}}(x-y)\,\mu(dy)$, the interchange being justified by Dominated Convergence Theorem applied to the partial sums, which are dominated by a constant: by The Gaussian Kernels on Euclidean Space: Scaling, Derivatives up to Order Three, the Convolution Identity, Moments and Tails (iii) with $s=2s_{k}$, and $(2s_{k})^{-(q+|\beta|)/2}=(\sqrt{2})^{-(q+|\beta|)}2^{k(q+|\beta|)}\le2^{k(q+|\beta|)}$, one has $|w_{k}\partial^{\beta}g_{2s_{k}}(x-y)|\le A_{|\beta|}2^{-k(8-|\beta|)}$, so every partial sum is bounded in absolute value by $A_{|\beta|}\sum_{k}(2^{-(8-|\beta|)})^{k}$, a finite constant by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series (iv) (claims 2 and 1 of Comparison and Absolute Value Bounds for Finite Sums of Real Numbers and Series of Nonnegative Real Numbers, Comparison, and the Geometric Series (i)), integrable with respect to $\mu$; and $\int\partial^{\beta}g_{2s_{k}}(x-y)\mu(dy)=\partial^{\beta}(g_{2s_{k}}*\mu)(x)$ by Gaussian Smoothing of a Probability Measure on Euclidean Space: Regularity, Mass, Duality, Approximation of a Bounded Function with a Modulus of Continuity, and the Pairing Identity (ii). Subtracting the same for $\nu$ (termwise linearity),

$$\partial^{\beta}\psi(x)=\sum_{k=1}^{\infty}w_{k}\,\partial^{\beta}\bigl(g_{2s_{k}}*\mu-g_{2s_{k}}*\nu\bigr)(x),$$

and, for each $k$, the $k$th term is bounded in absolute value by $u_{k}v_{k}$, where $u_{k}=\sqrt{w_{k}}\sqrt{B_{|\beta|}}\,(\sqrt{2})^{k(q+2|\beta|)}$ and $v_{k}=\sqrt{w_{k}}\sqrt{\int D_{k}^{2}\,d\lambda_{q}}$ (here $\sqrt{s_{k}^{-(q+2|\beta|)/2}}=\sqrt{2^{k(q+2|\beta|)}}=(\sqrt{2})^{k(q+2|\beta|)}$ and $\sqrt{ab}=\sqrt{a}\sqrt{b}$). For real $t>0$ and real numbers $u,v$ one has $2uv\le tu^{2}+v^{2}/t$ (from $0\le(tu-v)^{2}=t^{2}u^{2}-2tuv+v^{2}$, divided by $t$). Applying this termwise and summing over $k\le m$ (claim 1 of Comparison and Absolute Value Bounds for Finite Sums of Real Numbers) gives, for every $m\in\mathbb{N}$ and every $t>0$,

$$2\sum_{k=1}^{m}u_{k}v_{k}\le t\sum_{k=1}^{m}w_{k}B_{|\beta|}2^{k(q+2|\beta|)}+\frac1t\sum_{k=1}^{m}w_{k}\int D_{k}^{2}\,d\lambda_{q}\le t\,N_{|\beta|}^{2}+\frac{Q(\mu,\nu)}{t},$$

where $N_{|\beta|}^{2}=B_{|\beta|}\sum_{k}w_{k}2^{k(q+2|\beta|)}=B_{|\beta|}\sum_{k}(2^{-(8-2|\beta|)})^{k}$ is finite by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series (iv), since $w_{k}2^{k(q+2|\beta|)}=2^{-k(8-2|\beta|)}$ and $8-2|\beta|\ge2$, and the two partial sums are bounded by the corresponding sums by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series (i) (claim 2 above for the second). Hence the partial sums of the nonnegative series $\sum_{k}u_{k}v_{k}$ are bounded (take $t=1$), so it converges by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series (i), and $|\partial^{\beta}\psi(x)|\le\sum_{k}u_{k}v_{k}$ by An Absolutely Convergent Series of Real Numbers Converges (iii); passing to the limit $m\to\infty$ in the display (Order Properties of Limits of Real Sequences) gives $2\sum_{k}u_{k}v_{k}\le tN_{|\beta|}^{2}+Q(\mu,\nu)/t$ for every $t>0$. If $Q(\mu,\nu)=0$ this gives $2\sum_{k}u_{k}v_{k}\le tN_{|\beta|}^{2}$ for every $t>0$, so $\sum_{k}u_{k}v_{k}=0\le N_{|\beta|}\sqrt{Q(\mu,\nu)}$ by Comparison of Real Numbers with Arbitrary Positive Slack. If $Q(\mu,\nu)>0$ and $N_{|\beta|}>0$, taking $t=\sqrt{Q(\mu,\nu)}/N_{|\beta|}$ gives $\sum_{k}u_{k}v_{k}\le N_{|\beta|}\sqrt{Q(\mu,\nu)}$; if $N_{|\beta|}=0$ then every $u_{k}=0$ and the sum is $0$. In all cases $|\partial^{\beta}\psi(x)|\le N_{|\beta|}\sqrt{Q(\mu,\nu)}$, with $N_{0},\dots,N_{3}$ depending only on $q$. Lastly, by $(*)$, $\mathcal{K}(\mu,\mu')=\mathcal{K}(\mu',\mu)=\int K*\mu\,d\mu'$ and similarly for the other three pairings, so by linearity of the integral

$$B(\mu,\nu;\mu',\nu')=\int K*\mu\,d\mu'-\int K*\nu\,d\mu'-\int K*\mu\,d\nu'+\int K*\nu\,d\nu'=\int\psi\,d\mu'-\int\psi\,d\nu' .$$

Claim 8. The function $(\mu,\nu)\mapsto\sqrt{Q(\mu,\nu)}$ takes nonnegative real values (claim 2); it is symmetric by claim 1; it vanishes at $(\mu,\nu)$ exactly when $Q(\mu,\nu)=0$ (Existence and Uniqueness of the Nonnegative Square Root), that is, by claim 4, exactly when $\mu=\nu$; and it satisfies the triangle inequality by claim 5. These are the defining properties of a metric on $\mathcal{P}(\mathbb{R}^{q})$.