Proof of Regular Functions: Mode Derivatives, the Free-Field Generator and the Gradient Energy of a Regular Function plus a Multiple of the Gaussian Penalty

Each result cited below is universally quantified over the data in its own statement.

The data $\eta$, $m$, $k_{1},\dots,k_{m}$ and $f$ are those of one witness of the regularity of $\psi$ in Regular Functions for the Wick-Square Problem: Diagonal Quadratics of Curvature of Order the Inverse Squared Weight, plus Twice Differentiable Functions on the Sobolev Space of Order -3; we fix a curvature bound $C\ge0$ of $\eta$ as in Regular Functions for the Wick-Square Problem: Diagonal Quadratics of Curvature of Order the Inverse Squared Weight, plus Twice Differentiable Functions on the Sobolev Space of Order -3 §curvature, so that $\mu_{k}^{2}|\eta(k)|\le C$ for every mode $k$, and for every $y\in H^{-1}$

$$\psi(y)=Q(y)+f(\xi(y)),\qquad Q(y)=\sum_{k\in\mathbb{Z}^{n}}\eta(k)\,y(k)^{2},$$

the lattice sum $Q(y)$ existing as recorded in that definition. Let $E=\{k_{1},\dots,k_{m}\}$. Natural numbers start at $1$, so $[m]$ is nonempty and finite, and since the modes $k_{i}$ are pairwise distinct the map $i\mapsto k_{i}$ is a bijection from $[m]$ onto $E$; in particular $E$ is a nonempty finite subset of $\mathbb{Z}^{n}$.

Preliminary facts.

(P1) Real arithmetic. For a real $y$, $|y|$ equals $y$ or $-y$ and $0\le|y|$ (claim 1 of Properties of the Absolute Value in an Ordered Field); since $(-y)(-y)=yy$ (claim 2 of Zero Products and Elementary Identities in a Field), $y^{2}=|y|\,|y|$, and $0\le|y|\,|y|$ by claim 5 of Elementary Arithmetic in an Ordered Field (multiply $0\le|y|$ by $|y|$). Thus $0\le y^{2}$ and $|y^{2}|=y^{2}$. If $0\le y$, then $|y|=y$: otherwise $|y|=-y$ with $0\le -y$ and $0\le y$, forcing $y=0$ by claim 4 of Elementary Order Arithmetic in an Ordered Field and antisymmetry.

(P2) Weights. By The Wick-Square Problem on the Torus: Standing Notation §modes, $1\le\mu_{k}$, so $0<\mu_{k}$ (claims 6 and 2 of Elementary Order Arithmetic in an Ordered Field); $0<\mu_{k}^{2}=\mu_{k}\mu_{k}$ by claim 5 of that lemma, and the inverses $\mu_{k}^{-1}$, $\mu_{k}^{-2}$ exist and are positive by its claim 7. Multiplying $1\le\mu_{k}$ by $\mu_{k}^{-2}\ge0$ (claim 5 of Elementary Arithmetic in an Ordered Field) gives $\frac{1}{\mu_{k}^{2}}\le\frac{1}{\mu_{k}}$.

(P3) Curvature. Multiplying $\mu_{k}^{2}|\eta(k)|\le C$ by $\mu_{k}^{-2}\ge0$, respectively by $\mu_{k}^{-1}\ge0$ (claim 5 of Elementary Arithmetic in an Ordered Field), gives for every mode $k$

$$|\eta(k)|\le\frac{C}{\mu_{k}^{2}},\qquad\mu_{k}\,|\eta(k)|\le\frac{C}{\mu_{k}} .$$

(P4) The diagonal part along a mode. Let $y\in H^{-1}$, $k\in\mathbb{Z}^{n}$ and $t\in\mathbb{R}$; then $y+te_{k}\in H^{-1}$ by The Wick-Square Problem on the Torus: Standing Notation §units, and pointwise $(y+te_{k})(k')=y(k')$ for $k'\ne k$ and $(y+te_{k})(k)=y(k)+t$. Let $a(k')=\eta(k')y(k')^{2}$, $b(k')=\eta(k')(y+te_{k})(k')^{2}$ and $\delta=b-a$. Both $a$ and $b$ are cube-summable (the families whose lattice sums are $Q(y)$ and $Q(y+te_{k})$), $\delta(k')=0$ for $k'\ne k$, and by claim 5 of Zero Products and Elementary Identities in a Field

$$\delta(k)=\eta(k)\bigl((y(k)+t)^{2}-y(k)^{2}\bigr)=2\eta(k)y(k)\,t+\eta(k)\,t^{2}.$$

By Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §finite-support with the set $\{k\}$, $\delta$ is cube-summable with lattice sum $\sum_{k'\in\{k\}}\delta(k')=\delta(k)$ (Sum over a Finite Index Set, with the bijection from $[1]$ sending $1$ to $k$). Since $b=a+\delta$, Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §linear gives

$$Q(y+te_{k})=Q(y)+2\eta(k)y(k)\,t+\eta(k)\,t^{2}.$$

(P5) Coordinate lines. Let $x\in H^{-1}$, $i\in[m]$, and let $G:\mathbb{R}^{m}\to\mathbb{R}$ be a function whose partial derivative with respect to the $i$th variable (Partial Derivative on a Euclidean Open Set) exists at every point of $\mathbb{R}^{m}$. Put $g(t)=G(\xi(x+te_{k_{i}}))$ for $t\in\mathbb{R}$. Then $g$ is differentiable at every $t_{0}\in\mathbb{R}$, in the sense of Single-Variable Calculus on an Interval §derivative with $I=\mathbb{R}$, and $g'(t_{0})=\partial_{i}G(\xi(x+t_{0}e_{k_{i}}))$. Indeed, by (P4) and the distinctness of the modes, the $j$th component of $\xi(x+te_{k_{i}})$ is $x(k_{j})$ for $j\ne i$ and $x(k_{i})+t$ for $j=i$. Hence, for $a=\xi(x+t_{0}e_{k_{i}})$ and $h\in\mathbb{R}$, the point $(a_{1},\dots,a_{i-1},a_{i}+h,a_{i+1},\dots,a_{m})$ equals $\xi(x+(t_{0}+h)e_{k_{i}})$ by claim 1 of Euclidean Points as Tuples of Real Numbers, it lies in $\mathbb{R}^{m}$, and the difference quotient in Partial Derivative on a Euclidean Open Set at $a$ with increment $h$ is $(g(t_{0}+h)-g(t_{0}))/h$. So the defining condition of the partial derivative at $a$ with value $\partial_{i}G(a)$ is exactly the defining condition of differentiability of $g$ at $t_{0}$ with derivative $\partial_{i}G(a)$ in Single-Variable Calculus on an Interval §derivative (every point of $\mathbb{R}$ being an interior point of $\mathbb{R}$, as recorded in Derivatives of a Function on the Sobolev Space of Order -1 along the Fourier Modes), and the derivative is unique by that clause.

Clause 1. Fix $x\in H^{-1}$ and $k\in\mathbb{Z}^{n}$, and let $\sigma(t)=\psi(x+te_{k})$ be the section of Derivatives of a Function on the Sobolev Space of Order -1 along the Fourier Modes. By (P4), $\sigma=p+g$ with

$$p(t)=Q(x)+2\eta(k)x(k)\,t+\eta(k)\,t^{2},\qquad g(t)=f(\xi(x+te_{k})).$$

By the constant, power, sum and constant-multiple rules of Single-Variable Calculus on an Interval §derivative (the map $t\mapsto t$ has derivative $1$ and $t\mapsto t^{2}$ has derivative $2t$), $p$ is differentiable at every $t$ with $p'(t)=2\eta(k)x(k)+2\eta(k)\,t$, and $p'$ is differentiable at $0$ with $(p')'(0)=2\eta(k)$.

Suppose first that $k=k_{i}$ for some $i\in[m]$, unique by distinctness. Since $f$ is of class $C^{2}=C^{1+1}$ on $\mathbb{R}^{m}$, clause 2 of C^k Maps on a Euclidean Open Set (read with the scalar convention of its clause 3) shows that $f$ is of class $C^{1}$, so its partial derivative with respect to the $i$th variable exists at every point, and that $\partial_{i}f$ is of class $C^{1}$, so the partial derivative of $\partial_{i}f$ with respect to the $i$th variable exists at every point, its value being $\partial_{i}\partial_{i}f$ by clause 4 of that definition. By (P5) with $G=f$, $g$ is differentiable at every $t$ with $g'(t)=\partial_{i}f(\xi(x+te_{k_{i}}))$; by (P5) with $G=\partial_{i}f$, $g'$ is differentiable at $0$ with $(g')'(0)=\partial_{i}\partial_{i}f(\xi(x))$. Also $g'(0)=\partial_{i}f(\xi(x))$. These are $d_{k}(x)$ and $d_{k}^{2}(x)$.

Suppose next that $k\notin E$. Then $k_{j}\ne k$ for every $j$, so by (P4) $\xi(x+te_{k})=\xi(x)$ for every $t$ (claim 1 of Euclidean Points as Tuples of Real Numbers); $g$ is constant, hence differentiable everywhere with $g'$ the zero function, whose derivative at $0$ is $0$ (constant rule of Single-Variable Calculus on an Interval §derivative). Thus again $g'(0)=0=d_{k}(x)$ and $(g')'(0)=0=d_{k}^{2}(x)$.

In both cases the sum rule of Single-Variable Calculus on an Interval §derivative shows that $\sigma$ is differentiable at every point with $\sigma'=p'+g'$, and that $\sigma'$ is differentiable at $0$ with $(\sigma')'(0)=2\eta(k)+d_{k}^{2}(x)$. So $\psi$ is twice differentiable along the modes (Derivatives of a Function on the Sobolev Space of Order -1 along the Fourier Modes §twice), and by Derivatives of a Function on the Sobolev Space of Order -1 along the Fourier Modes §derivatives

$$\partial_{k}\psi(x)=\sigma'(0)=2\eta(k)x(k)+d_{k}(x),\qquad\partial_{k}^{2}\psi(x)=2\eta(k)+d_{k}^{2}(x).$$

Clause 2. Fix $x\in H^{-1}$ and define families on $\mathbb{Z}^{n}$ by

$$h(k)=\nu\,\eta(k)-2\mu_{k}\eta(k)\,x(k)^{2},\qquad e(k)=\frac{\nu}{2}\,d_{k}^{2}(x)-\mu_{k}\,x(k)\,d_{k}(x).$$

Cube-summability of $h$: by claims 5 and 2 of Properties of the Absolute Value in an Ordered Field, claim 4 of that lemma, (P1) (for the nonnegative numbers $\nu$, $2\mu_{k}$, $x(k)^{2}$) and (P3), with claim 5 of Elementary Arithmetic in an Ordered Field for scaling and claims 3 and 2 of that lemma for adding two inequalities,

$$|h(k)|\le\nu\,|\eta(k)|+2x(k)^{2}\,\mu_{k}|\eta(k)|\le\nu C\,\frac{1}{\mu_{k}^{2}}+2C\,\frac{x(k)^{2}}{\mu_{k}}=:B(k).$$

Since $n\le3<4=2\cdot2$ (The Wick-Square Problem on the Torus: Standing Notation §dimension), $k\mapsto1/\mu_{k}^{2}$ is cube-summable by Lattice Sums of the Fourier Weights of the Torus, and the Negative Sobolev Norms as Lattice Sums §convergent with $s=2$; $k\mapsto x(k)^{2}/\mu_{k}$ is cube-summable by The Wick-Square Problem on the Torus: Standing Notation §state-space; so $B$ is cube-summable by Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §linear, and $h$ is cube-summable by Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §comparison. This is the first assertion of clause 2.

The family $e$ vanishes off $E$: for $k\notin E$, $d_{k}(x)=d_{k}^{2}(x)=0$, and products with $0$ vanish (claim 1 of Zero Products and Elementary Identities in a Field). By Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §finite-support, $e$ is cube-summable with lattice sum $\sum_{k\in E}e(k)$, and by claims 2 and 1 of Properties of a Sum over a Finite Index Set along the bijection $i\mapsto k_{i}$ from $[m]$ onto $E$,

$$\sum_{k\in\mathbb{Z}^{n}}e(k)=\sum_{i=1}^{m}e(k_{i})=\sum_{i=1}^{m}\Bigl(\frac{\nu}{2}\,\partial_{i}\partial_{i}f(\xi(x))-\mu_{k_{i}}\,x(k_{i})\,\partial_{i}f(\xi(x))\Bigr),$$

because $d_{k_{i}}(x)=\partial_{i}f(\xi(x))$ and $d_{k_{i}}^{2}(x)=\partial_{i}\partial_{i}f(\xi(x))$.

By clause 1 and field arithmetic, for every mode $k$

$$\frac{\nu}{2}\,\partial_{k}^{2}\psi(x)-\mu_{k}\,x(k)\,\partial_{k}\psi(x)=\frac{\nu}{2}\bigl(2\eta(k)+d_{k}^{2}(x)\bigr)-\mu_{k}x(k)\bigl(2\eta(k)x(k)+d_{k}(x)\bigr)=h(k)+e(k),$$

so by The Free-Field Generator with a Mode Cutoff §generator and Cube Sums of Families on the Integer Lattice §cube-sums, $L_{N}\psi(x)$ is the $N$th cube sum of $h+e$ for every $N\in\mathbb{N}$. By Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §linear, $h+e$ is cube-summable, that is (Cube Sums of Families on the Integer Lattice §lattice-sum) the sequence $(L_{N}\psi(x))_{N\in\mathbb{N}}$ converges, to $\sum_{k\in\mathbb{Z}^{n}}h(k)+\sum_{k\in\mathbb{Z}^{n}}e(k)$. Hence $L\psi$ is defined at $x$ (The Free-Field Generator with a Mode Cutoff §limit) and $L\psi(x)$ is the displayed expression of clause 2.

Clause 3. By The Cutoff Wick Square is Minus the Free-Field Generator of the Gaussian Penalty, and the Penalised Form of the Cutoff Operators §derivatives, $P$ is twice differentiable along the modes with $\partial_{k}P(x)=\beta x(k)/\mu_{k}$. Fix $x\in H^{-1}$ and $k\in\mathbb{Z}^{n}$. The section of $\varphi$ at $x$ along $k$ is $t\mapsto\theta P(x+te_{k})+\psi(x+te_{k})$, that is, $\theta$ times the section of $P$ plus the section of $\psi$. Both sections are differentiable at every point and their derivatives are differentiable at $0$ (by The Cutoff Wick Square is Minus the Free-Field Generator of the Gaussian Penalty, and the Penalised Form of the Cutoff Operators §derivatives and clause 1), so by the sum and constant-multiple rules of Single-Variable Calculus on an Interval §derivative the section of $\varphi$ is differentiable everywhere, its derivative is $\theta$ times that of the section of $P$ plus that of the section of $\psi$, and this derivative is differentiable at $0$. So $\varphi$ is twice differentiable along the modes, and by clause 1

$$\partial_{k}\varphi(x)=\lambda_{k}\,x(k)+d_{k}(x),\qquad\lambda_{k}=\frac{\theta\beta}{\mu_{k}}+2\eta(k).$$

Let $w(k)=(\partial_{k}\varphi(x))^{2}$, $w_{1}(k)=(\lambda_{k}x(k))^{2}$ and $w_{2}=w-w_{1}$. For $k\notin E$, $d_{k}(x)=0$, so $w(k)=w_{1}(k)$ and $w_{2}(k)=0$; by Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §finite-support, $w_{2}$ is cube-summable.

Put $K=|\theta|\beta+2C$; $0\le K$ by (P1), claim 5 of Elementary Arithmetic in an Ordered Field and claim 2 of that lemma. By claims 5 and 4 of Properties of the Absolute Value in an Ordered Field, (P1), (P3) and (P2) (with claim 5 of Elementary Arithmetic in an Ordered Field),

$$|\lambda_{k}|\le\frac{|\theta|\beta}{\mu_{k}}+\frac{2C}{\mu_{k}^{2}}\le\frac{|\theta|\beta}{\mu_{k}}+\frac{2C}{\mu_{k}}=\frac{K}{\mu_{k}} .$$

Both $|\lambda_{k}|$ and $K/\mu_{k}$ are nonnegative, so claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field gives $|\lambda_{k}|^{2}\le K^{2}/\mu_{k}^{2}$; multiplying by $x(k)^{2}\ge0$ (claim 5 of Elementary Arithmetic in an Ordered Field) and using $|\lambda_{k}|^{2}=\lambda_{k}^{2}$ from (P1) and then (P2),

$$0\le w_{1}(k)=\lambda_{k}^{2}\,x(k)^{2}\le K^{2}\,\frac{x(k)^{2}}{\mu_{k}^{2}}\le K^{2}\,\frac{x(k)^{2}}{\mu_{k}} .$$

Here $w_{1}(k)\ge0$ by (P1), so $|w_{1}(k)|=w_{1}(k)$. The right-hand family is cube-summable by The Wick-Square Problem on the Torus: Standing Notation §state-space and Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §linear, so $w_{1}$ is cube-summable by Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §comparison. Then $w=w_{1}+w_{2}$ is cube-summable by Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §linear. Since $|D_{N}\varphi(x)|^{2}$ is the $N$th cube sum of $w$ (The Gradient Energy with a Mode Cutoff §gradient, Cube Sums of Families on the Integer Lattice §cube-sums), the sequence $(|D_{N}\varphi(x)|^{2})_{N\in\mathbb{N}}$ converges (Cube Sums of Families on the Integer Lattice §lattice-sum), and $|D\varphi(x)|^{2}$ is defined by The Gradient Energy with a Mode Cutoff §limit.