Proof of Regular Functions: Mode Derivatives, the Free-Field Generator, the Gradient Energy, the Renormalised Operator at the Gaussian Penalty plus a Regular Function, and Linear Combinations

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

Conventions. Elementary arithmetic, order, absolute values, squares and finite sums of real numbers are used freely under The Real Numbers: Standing Notation and Background §background. By The Wick-Square Problem on the Torus: Standing Notation §modes, $1\le\mu_{k}$, so $0<\mu_{k}$ and $1/\mu_{k}^{l'}\le1/\mu_{k}^{l}$ whenever $l\le l'$. The vector operations of $H^{-1}$ and of $H^{-3}$ are the pointwise ones (The Negative-Order Sobolev Spaces of the Torus §space). By Lattice Sums of the Fourier Weights of the Torus, and the Negative Sobolev Norms as Lattice Sums §sobolev, a family $c$ lies in $H^{-l}(\mathbb{T}^{n})$ ($l\in\mathbb{N}$) exactly when $k\mapsto c(k)^{2}/\mu_{k}^{l}$ is cube-summable; in particular (The Wick-Square Problem on the Torus: Standing Notation §state-space) $k\mapsto x(k)^{2}/\mu_{k}$ is cube-summable for every $x\in H^{-1}$. 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 $s=2$. Let $Q(y)=\sum_{k\in\mathbb{Z}^{n}}\eta(k)\,y(k)^{2}$ for $y\in H^{-1}$, a lattice sum that exists by 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; by 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, for every mode $k$

$$|\eta(k)|\le\frac{C}{\mu_{k}^{2}},\qquad\mu_{k}\,|\eta(k)|\le\frac{C}{\mu_{k}}.\qquad(\mathrm{P}1)$$

For $x\in H^{-1}$ we write $z_{x}=D\phi(x)\in H^{-3}$ and $b_{x}=D^{2}\phi(x)\in\mathrm{Sym}(H^{-3})$ (Fréchet Differentiability and the Gradient on an Open Subset of a Real Inner Product Space §on-set, The Second Derivative and the Hessian on an Open Subset of a Real Inner Product Space §on-set). By The Sobolev Triple of Order Two for the Wick-Square Problem: the Domain Is the Sobolev Space of Order -1, Unit Families Are Square-Summable, the Gradient Form and the Riccati Drift §pairings, whose space $H$ is $H^{-3}$ with $\langle\cdot,\cdot\rangle_{H}=\langle\cdot,\cdot\rangle_{H^{-3}}$,

$$\langle z_{x},e_{k}\rangle_{H^{-3}}=\frac{z_{x}(k)}{\mu_{k}^{3}}\qquad(k\in\mathbb{Z}^{n}).\qquad(\mathrm{P}2)$$

Step 1 (The diagonal part along a mode). Let $y\in H^{-1}$, $k\in\mathbb{Z}^{n}$ and $t\in\mathbb{R}$; then $y+te_{k}\in H^{-1}$ (The Wick-Square Problem on the Torus: Standing Notation §units), with value $y(k')$ at $k'\ne k$ and $y(k)+t$ at $k$. The family $\delta(k')=\eta(k')(y+te_{k})(k')^{2}-\eta(k')y(k')^{2}$ is the difference of the two cube-summable families defining $Q(y+te_{k})$ and $Q(y)$; it vanishes off $\{k\}$ and $\delta(k)=2\eta(k)y(k)\,t+\eta(k)\,t^{2}$. By Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §finite-support its lattice sum is $\delta(k)$, and by Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §linear

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

Step 2 (Clause derivatives). The diagonal part. Fix $x\in H^{-1}$ and $k$. By Step 1 the section of $Q$ at $x$ along $k$ (Derivatives of a Function on the Sobolev Space of Order -1 along the Fourier Modes) is $p(t)=Q(x)+2\eta(k)x(k)\,t+\eta(k)\,t^{2}$. The maps $t\mapsto t$ and $t\mapsto t^{2}$ are differentiable at every point of $\mathbb{R}$ with derivatives $1$ and $2t$ (claim 1 of Derivative of a Polynomial Function on the Real Line), and constants have derivative $0$ (claim 1 of Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives); by claim 2 of Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives, $p$ is differentiable at every $t$ with $p'(t)=2\eta(k)x(k)+2\eta(k)\,t$, and, by the same claims, $p'$ is differentiable at $0$ with $(p')'(0)=2\eta(k)$. So $Q$ is twice differentiable along the modes (Derivatives of a Function on the Sobolev Space of Order -1 along the Fourier Modes §twice) with $\partial_{k}Q(x)=2\eta(k)x(k)$ and $\partial_{k}^{2}Q(x)=2\eta(k)$ (Derivatives of a Function on the Sobolev Space of Order -1 along the Fourier Modes §derivatives).

The weak part. By Mode Derivatives of Linear Combinations and of Restrictions of Twice Differentiable Functions on a Negative Sobolev Space §sobolev with $m=3$ and $\Phi=\phi$, the restriction $\phi|_{H^{-1}}$ is twice differentiable along the modes with $\partial_{k}(\phi|_{H^{-1}})(x)=\langle D\phi(x),e_{k}\rangle_{H^{-3}}$ and $\partial_{k}^{2}(\phi|_{H^{-1}})(x)=D^{2}\phi(x)(e_{k},e_{k})$.

The sum. By the identity of 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, $\psi=1\cdot Q+1\cdot\phi|_{H^{-1}}$ pointwise on $H^{-1}$. By Mode Derivatives of Linear Combinations and of Restrictions of Twice Differentiable Functions on a Negative Sobolev Space §linear with $a=b=1$, $\psi$ is twice differentiable along the modes and, for all $x$ and $k$,

$$\partial_{k}\psi(x)=2\eta(k)x(k)+\langle D\phi(x),e_{k}\rangle_{H^{-3}},\qquad\partial_{k}^{2}\psi(x)=2\eta(k)+D^{2}\phi(x)(e_{k},e_{k}).$$

Step 3 (Clause generator). Fix $x\in H^{-1}$ and let

$$h(k)=\nu\,\eta(k)-2\mu_{k}\eta(k)\,x(k)^{2},\qquad e(k)=\tfrac{\nu}{2}\,b_{x}(e_{k},e_{k})-\mu_{k}\,x(k)\,\langle z_{x},e_{k}\rangle_{H^{-3}} .$$

$h$ is cube-summable. By (P1), $|h(k)|\le\nu|\eta(k)|+2x(k)^{2}\mu_{k}|\eta(k)|\le\nu C/\mu_{k}^{2}+2C\,x(k)^{2}/\mu_{k}$. The right-hand family is cube-summable by the Conventions and Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §linear, so $h$ is cube-summable by Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §comparison.

$e$ is cube-summable. By Elementary Properties of Bounded Symmetric Bilinear Forms: Norm, Quadratic Form, Order and Continuity §bound (for the norm $\lVert\cdot\rVert$ of $\mathrm{Sym}(H^{-3})$ fixed in Real Hilbert Spaces: Standing Notation and Background §forms) and Lattice Sums of the Fourier Weights of the Torus, and the Negative Sobolev Norms as Lattice Sums §units with $m=3$, $|b_{x}(e_{k},e_{k})|\le\lVert b_{x}\rVert\,|e_{k}|_{H^{-3}}^{2}=\lVert b_{x}\rVert/\mu_{k}^{3}\le\lVert b_{x}\rVert/\mu_{k}^{2}$. By (P2), $\mu_{k}x(k)\langle z_{x},e_{k}\rangle_{H^{-3}}=x(k)z_{x}(k)/\mu_{k}^{2}$. Hence

$$|e(k)|\le\tfrac{\nu}{2}\lVert b_{x}\rVert\,\frac{1}{\mu_{k}^{2}}+\frac{|x(k)z_{x}(k)|}{\mu_{k}^{2}} .$$

The family $k\mapsto|x(k)z_{x}(k)|/\mu_{k}^{2}$ is cube-summable by The Sobolev Triple of Order Two for the Wick-Square Problem: the Domain Is the Sobolev Space of Order -1, Unit Families Are Square-Summable, the Gradient Form and the Riccati Drift §pairings (as $x\in H^{-1}$ and $z_{x}\in H^{-3}$), and $k\mapsto1/\mu_{k}^{2}$ by the Conventions; so the right-hand family is cube-summable (Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §linear) and $e$ is cube-summable by Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §comparison. These are the two families of clause generator.

The limit. By Step 2, for every mode $k$,

$$\tfrac{\nu}{2}\,\partial_{k}^{2}\psi(x)-\mu_{k}\,x(k)\,\partial_{k}\psi(x)=\tfrac{\nu}{2}\bigl(2\eta(k)+b_{x}(e_{k},e_{k})\bigr)-\mu_{k}x(k)\bigl(2\eta(k)x(k)+\langle z_{x},e_{k}\rangle_{H^{-3}}\bigr)=h(k)+e(k),$$

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 $h+e$. By Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §linear, $h+e$ is cube-summable with lattice sum $\sum_{k}h(k)+\sum_{k}e(k)$, i.e. (Cube Sums of Families on the Integer Lattice §lattice-sum) $(L_{N}\psi(x))_{N\in\mathbb{N}}$ converges to this number. By The Free-Field Generator with a Mode Cutoff §limit, $L\psi$ is defined at $x$ and $L\psi(x)=\sum_{k}h(k)+\sum_{k}e(k)$.

Step 4 (Clause gradient). Let $\theta\in\mathbb{R}$ and $\chi=\theta P+\psi$. 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 $\partial_{k}P(x)=\beta x(k)/\mu_{k}$; by Step 2 so is $\psi$. By Mode Derivatives of Linear Combinations and of Restrictions of Twice Differentiable Functions on a Negative Sobolev Space §linear with $a=\theta$, $b=1$, $\chi$ is twice differentiable along the modes, and with Step 2 and (P2), for $x\in H^{-1}$ and every $k$,

$$\partial_{k}\chi(x)=\lambda_{k}\,x(k)+\frac{z_{x}(k)}{\mu_{k}^{3}},\qquad\lambda_{k}=\frac{\theta\beta}{\mu_{k}}+2\eta(k).$$

Fix $x\in H^{-1}$ and let $w(k)=(\partial_{k}\chi(x))^{2}\ge0$. Put $K=|\theta|\beta+2C\ge0$. By (P1), $|\lambda_{k}|\le|\theta|\beta/\mu_{k}+2C/\mu_{k}^{2}\le K/\mu_{k}$, so $\lambda_{k}^{2}\le K^{2}/\mu_{k}^{2}$. Using $(u+v)^{2}\le2u^{2}+2v^{2}$,

$$0\le w(k)\le2\lambda_{k}^{2}x(k)^{2}+2\,\frac{z_{x}(k)^{2}}{\mu_{k}^{6}}\le2K^{2}\,\frac{x(k)^{2}}{\mu_{k}}+2\,\frac{z_{x}(k)^{2}}{\mu_{k}^{3}} .$$

The right-hand family is cube-summable: $k\mapsto x(k)^{2}/\mu_{k}$ by the Conventions, $k\mapsto z_{x}(k)^{2}/\mu_{k}^{3}$ by the Conventions with $l=3$ since $z_{x}\in H^{-3}$, and Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §linear. As $|w(k)|=w(k)$, Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §comparison shows that $w$ is cube-summable. By The Gradient Energy with a Mode Cutoff §gradient and Cube Sums of Families on the Integer Lattice §cube-sums, $|D_{N}\chi(x)|^{2}$ is the $N$th cube sum of $w$, so $(|D_{N}\chi(x)|^{2})_{N\in\mathbb{N}}$ converges to $\sum_{k}w(k)$ (Cube Sums of Families on the Integer Lattice §lattice-sum); by The Gradient Energy with a Mode Cutoff §limit, $|D\chi(x)|^{2}$ is defined and equals $\sum_{k\in\mathbb{Z}^{n}}w(k)$, limits being unique (Limit of a Sequence of Real Numbers, The Real Numbers: Standing Notation and Background §background).

Step 5 (Clause operator). Let $g$, $\mathcal{D}$ and $F$ be as in clause operator, and $\chi=P+\psi$, which equals $1\cdot P+\psi$ pointwise. By Step 4 with $\theta=1$, $\chi$ is twice differentiable along the modes and, for every $x\in H^{-1}$, $(|D_{N}\chi(x)|^{2})_{N}$ converges to $|D\chi(x)|^{2}$. Since $\chi(u)-P(u)=\psi(u)$ for every $u\in H^{-1}$, the function $\chi-P$ is $\psi$, and The Cutoff Wick Square is Minus the Free-Field Generator of the Gaussian Penalty, and the Penalised Form of the Cutoff Operators §penalised gives, for every $N\in\mathbb{N}$ and $x\in H^{-1}$,

$$F_{N}[\chi](x)=\gamma\,\chi(x)-L_{N}\psi(x)+\tfrac12\,|D_{N}\chi(x)|^{2}-g(x).$$

Fix $x\in H^{-1}$. By Step 3 and The Free-Field Generator with a Mode Cutoff §limit, $(L_{N}\psi(x))_{N}$ converges to $L\psi(x)$; constant sequences converge to their value (Limit of a Sequence of Real Numbers). By Arithmetic of Limits of Real Sequences §sums and Arithmetic of Limits of Real Sequences §scalar, $(F_{N}[\chi](x))_{N}$ converges to $\gamma\chi(x)-L\psi(x)+\tfrac12|D\chi(x)|^{2}-g(x)$. Hence $(\chi,x)\in\mathcal{D}$ by The Renormalised Hamilton-Jacobi-Bellman Operator of the Wick-Square Problem as the Limit of the Wick-Ordered Cutoff Operators §domain, and $F[\chi](x)$ is this limit by The Renormalised Hamilton-Jacobi-Bellman Operator of the Wick-Square Problem as the Limit of the Wick-Ordered Cutoff Operators §operator, which is the asserted formula.

Step 6 (Clause linear). Combinations. Let $\psi_{1},\psi_{2}$ be regular, with witnessing data $\eta_{i}$, $C_{i}$, $\phi_{i}$ ($i=1,2$) 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, and let $a,b\in\mathbb{R}$. Put $\eta=a\eta_{1}+b\eta_{2}$ (pointwise), $C=|a|C_{1}+|b|C_{2}\ge0$ and $\phi=a\phi_{1}+b\phi_{2}$. For every $k$, $\mu_{k}^{2}|\eta(k)|\le|a|\,\mu_{k}^{2}|\eta_{1}(k)|+|b|\,\mu_{k}^{2}|\eta_{2}(k)|\le C$, so $\eta$ is a curvature family with curvature bound $C$ (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). The space $H^{-3}$ is open in itself (Real Hilbert Spaces: Standing Notation and Background §topology), so by Constants, Sums, Scalar Multiples and Differences of Differentiable Functions on an Open Subset of a Real Inner Product Space §scalar (twice) and Constants, Sums, Scalar Multiples and Differences of Differentiable Functions on an Open Subset of a Real Inner Product Space §sum, $\phi$ is of class $C^{2}$ on $H^{-3}$, a weak part in the sense of 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 §weak-part. For $x\in H^{-1}$, Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §linear gives

$$a\psi_{1}(x)+b\psi_{2}(x)=\sum_{k\in\mathbb{Z}^{n}}\bigl(a\eta_{1}(k)+b\eta_{2}(k)\bigr)x(k)^{2}+a\phi_{1}(x)+b\phi_{2}(x)=\sum_{k\in\mathbb{Z}^{n}}\eta(k)\,x(k)^{2}+\phi(x),$$

so $a\psi_{1}+b\psi_{2}$ is regular.

Constants. Let $c\in\mathbb{R}$ and let $\psi$ have constant value $c$ on $H^{-1}$. Take $\eta$ the zero family with curvature bound $0$, and $\phi$ the constant function with value $c$ on $H^{-3}$, which is of class $C^{2}$ on $H^{-3}$ by Constants, Sums, Scalar Multiples and Differences of Differentiable Functions on an Open Subset of a Real Inner Product Space §constant. For $x\in H^{-1}$, the family $k\mapsto0\cdot x(k)^{2}$ is the zero family, which is $0$ times the cube-summable family $k\mapsto x(k)^{2}/\mu_{k}$; so its lattice sum is $0$ by Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §linear, and $\sum_{k}\eta(k)x(k)^{2}+\phi(x)=0+c=\psi(x)$. Hence $\psi$ is regular.