Proof of Renormalised Viscosity Solutions versus Viscosity Solutions of the Shifted Equation on the Sobolev Triple of Order Two

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

Conventions. By The Wick-Square Problem on the Sobolev Triple of Order Two: Standing Notation §triple and 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, $H=H^{-3}$ and $V=H^{-2}$ are spaces of families on $\mathbb{Z}^{n}$ with the inner products of orders $-3$ and $-2$, so $d_{H}=d_{H^{-3}}$, the vector operations are modewise, and $x\mapsto x(k)$ is linear for each mode $k$; $D(A)=H^{-1}\subseteq V\subseteq H$, and $Ax$ is the family $k\mapsto\mu_{k}x(k)$ for $x\in H^{-1}$ (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 §domain). The class $C^{2}(H)$ and the gradients and Hessians of its members are those of The Classes $C^1$ and $C^2$ on an Open Subset of a Real Inner Product Space §c2 for the inner product space $H^{-3}$ (Hilbert Triples: Standing Notation and Background §open-sets, Real Hilbert Spaces: Series, Products, Orthonormal Bases and Differential Calculus §calculus); so a function on $H^{-3}$ of class $C^{2}$ on $H^{-3}$ is a member of $C^{2}(H)$ and conversely. Elementary arithmetic in $\mathbb{R}$ (rearrangement, the triangle inequality, case distinctions, the limit laws and the order of limits) is used under The Real Numbers: Standing Notation and Background §background, and the Cauchy-Schwarz inequality The Cauchy-Schwarz Inequality in a Real Inner Product Space and the identities of inner products under Real Hilbert Spaces: Standing Notation and Background §background. Lattice sums are manipulated with Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §linear, Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §nonnegative, Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §comparison and Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §finite-support. By Lattice Sums of the Fourier Weights of the Torus, and the Negative Sobolev Norms as Lattice Sums §sobolev, for $m\in\{1,2,3\}$ a family $c$ lies in $H^{-m}$ exactly when $k\mapsto c(k)^{2}/\mu_{k}^{m}$ is cube-summable, and then $|c|_{H^{-m}}^{2}$ is its lattice sum; we call this (S). Since $1\le\mu_{k}$ (The Wick-Square Problem on the Torus: Standing Notation §modes), $\mu_{k}^{m}\le\mu_{k}^{m'}$ for natural numbers $m\le m'$. By Lattice Sums of the Fourier Weights of the Torus, and the Negative Sobolev Norms as Lattice Sums §units with $m=3$, $|e_{k}|_{H}^{2}=(1/\mu_{k}^{3})$, and $e_{k}(j)$ is $1$ for $j=k$ and $0$ otherwise (The Wick-Square Problem on the Torus: Standing Notation §units). Since $w$ is continuous on $H$, Basic Properties of the $\delta$-Envelopes: Semicontinuity, Duality, Closed Superlevel Sets, Bounds and Monotonicity §semicontinuous-case (with $U=H$, so $V\cap U=V$) shows that $w$ is bounded above and below near each point of $H$ and that, for every real $\delta>0$, $w^{-}_{\delta}(x)=w(x)-\delta h(x)$ and $w^{+}_{\delta}(x)=w(x)+\delta h(x)$ for $x\in V$.

The zero function on $H^{-1}$ is regular by 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 §linear; applying The Riccati Shift: the Renormalised Operator at the Free Solution plus a Regular Function is the Shifted Operator, Which Satisfies the Hilbert-Triple Comparison Hypotheses, and Its Touching Points Lie in the State Space §identity to it shows that $\psi_{0}=u_{0}-P$ is regular.

Step 1 (Generator and gradient energy through mode derivatives). Let $\tilde{\chi}$ be regular, with a curvature family $\tilde{\eta}$ and weak part $\tilde{\phi}$, and let $x\in H^{-1}$. By 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 §gradient with $\theta=0$, the family $k\mapsto(\partial_{k}\tilde{\chi}(x))^{2}$ is cube-summable and

$$|D\tilde{\chi}(x)|^{2}=\sum_{k\in\mathbb{Z}^{n}}\bigl(\partial_{k}\tilde{\chi}(x)\bigr)^{2}.\qquad\text{(G1)}$$

By 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 §generator, $L\tilde{\chi}(x)$ is the sum of the lattice sums of $k\mapsto\nu\tilde{\eta}(k)-2\mu_{k}\tilde{\eta}(k)x(k)^{2}$ and $k\mapsto\tfrac{\nu}{2}D^{2}\tilde{\phi}(x)(e_{k},e_{k})-\mu_{k}x(k)\langle D\tilde{\phi}(x),e_{k}\rangle_{H}$; by 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 §derivatives the sum of these two families at $k$ is $\tfrac{\nu}{2}\partial_{k}^{2}\tilde{\chi}(x)-\mu_{k}x(k)\partial_{k}\tilde{\chi}(x)$, so by linearity of lattice sums this family is cube-summable and

$$L\tilde{\chi}(x)=\sum_{k\in\mathbb{Z}^{n}}\Bigl(\tfrac{\nu}{2}\,\partial_{k}^{2}\tilde{\chi}(x)-\mu_{k}x(k)\,\partial_{k}\tilde{\chi}(x)\Bigr).\qquad\text{(G2)}$$

Step 2 (Clause 1: the reduction). Suppose $w$ is a viscosity subsolution (case $\tau=1$), respectively supersolution (case $\tau=-1$), of $F^{\sharp}$ on $H$. Let $\psi:H^{-1}\to\mathbb{R}$ be regular and let $\hat{x}\in H^{-1}$ be such that $u(\hat{x})=P(\hat{x})+\psi(\hat{x})$ and $u-P-\psi$ has a local maximum (case $\tau=1$), respectively minimum (case $\tau=-1$), at $\hat{x}$ relative to $H^{-1}$ in $(H^{-3},d_{H^{-3}})$. By Local Maximum of a Function Relative to a Subset of a Metric Space and Local Minimum of a Function Relative to a Subset of a Metric Space choose a real $r>0$ such that $\tau(u-P-\psi)(y)\le\tau(u-P-\psi)(\hat{x})$ for every $y\in H^{-1}$ with $|y-\hat{x}|_{H}<r$. Put $\chi=\psi-\psi_{0}$, regular by 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 §linear. On $H^{-1}$ one has $P+\psi=u_{0}+\chi$ and $u-P-\psi=u_{0}+w-P-\psi=w-\chi$; hence $w(\hat{x})=\chi(\hat{x})$ and

$$\tau\bigl(w(y)-\chi(y)\bigr)\le0\qquad\text{for all }y\in H^{-1}\text{ with }|y-\hat{x}|_{H}<r.\qquad\text{(2.1)}$$

Fix data witnessing that $\chi$ is regular: a curvature family $\eta$ with curvature bound $C_{\chi}$ and a weak part $\phi$, so $\phi\in C^{2}(H)$ and $\chi(x)=\sum_{k}\eta(k)x(k)^{2}+\phi(x)$ for $x\in H^{-1}$. Put $C=C_{\chi}+1$, so $0<C$ and $|\eta(k)|\le C/\mu_{k}^{2}$ for every $k$, and put $c=2C>0$. We must show $\tau F[P+\psi](\hat{x})\le0$, and $F[P+\psi](\hat{x})=F[u_{0}+\chi](\hat{x})$ since $P+\psi=u_{0}+\chi$.

Step 3 (The pieces of the test function). Fix $N\in\mathbb{N}$. Define $a_{N}(k)=0$ for $k\in\Gamma_{N}$ and $a_{N}(k)=C/\mu_{k}^{2}-\tau\eta(k)$ for $k\notin\Gamma_{N}$; since $|\tau\eta(k)|\le C/\mu_{k}^{2}$,

$$0\le a_{N}(k)\le\frac{2C}{\mu_{k}^{2}}\qquad\text{for every }k,\qquad\text{and }a_{N}(k)=0\text{ for }k\in\Gamma_{N}.\qquad\text{(3.1)}$$

(a) Let $y_{N}$ be the family $k\mapsto\mu_{k}^{2}a_{N}(k)\hat{x}(k)$. By (3.1), $y_{N}(k)^{2}/\mu_{k}=\mu_{k}^{3}a_{N}(k)^{2}\hat{x}(k)^{2}\le4C^{2}\hat{x}(k)^{2}/\mu_{k}$, and $k\mapsto\hat{x}(k)^{2}/\mu_{k}$ is cube-summable by The Wick-Square Problem on the Torus: Standing Notation §state-space; by comparison and (S), $y_{N}\in H^{-1}=D(A)$, and $Ay_{N}\in H$ is the family $k\mapsto\mu_{k}^{3}a_{N}(k)\hat{x}(k)$. Let $\ell_{N}:H\to\mathbb{R}$, $\ell_{N}(x)=\langle Ay_{N},x\rangle_{H}$. By the last statement of 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 (with $y_{N}\in H^{-1}$ and $x\in H$), for $x\in H$ the family $k\mapsto a_{N}(k)\hat{x}(k)x(k)=y_{N}(k)x(k)/\mu_{k}^{2}$ is cube-summable and $\ell_{N}(x)=\sum_{k}a_{N}(k)\hat{x}(k)x(k)$; by the first statement there, $\langle Ay_{N},e_{k}\rangle_{H}=a_{N}(k)\hat{x}(k)$. By Affine and Quadratic Functions on a Real Hilbert Space are of Class $C^2$ §affine, $\ell_{N}\in C^{2}(H)$ with $D\ell_{N}(x)=Ay_{N}$ and $D^{2}\ell_{N}(x)=0_{\mathrm{Sym}}$.

(b) Put $\beta_{N}(k)=2(\eta(k)-\tau C/\mu_{k}^{2})$ and $b_{N}(x,x')=\sum_{k\in\Gamma_{N}}\beta_{N}(k)x(k)x'(k)$ for $x,x'\in H$ (a finite sum, $\Gamma_{N}$ being finite by The Wick-Square Problem on the Torus: Standing Notation §cubes). It is symmetric, and additive and homogeneous in $x$ because $x\mapsto x(k)$ is linear. 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 and Cauchy-Schwarz, $|x(k)|=\mu_{k}^{3}|\langle x,e_{k}\rangle_{H}|\le\mu_{k}^{3}|e_{k}|_{H}|x|_{H}$, so $|x(k)x'(k)|\le\mu_{k}^{6}|e_{k}|_{H}^{2}|x|_{H}|x'|_{H}=\mu_{k}^{3}|x|_{H}|x'|_{H}$ and $|b_{N}(x,x')|\le\bigl(\sum_{k\in\Gamma_{N}}|\beta_{N}(k)|\mu_{k}^{3}\bigr)|x|_{H}|x'|_{H}$. Thus $b_{N}\in\mathrm{Sym}(H)$ (Bounded Symmetric Bilinear Forms on a Real Inner Product Space: Norm, Order, Identity Form and Restriction §form). Let $Q_{N}(x)=\tfrac12b_{N}(x,x)=\sum_{k\in\Gamma_{N}}(\eta(k)-\tau C/\mu_{k}^{2})x(k)^{2}$. By Affine and Quadratic Functions on a Real Hilbert Space are of Class $C^2$ §form, $Q_{N}\in C^{2}(H)$ with $DQ_{N}(x)=T_{b_{N}}x$ and $D^{2}Q_{N}(x)=b_{N}$, and by The Bounded Symmetric Operator Represented by a Bounded Symmetric Bilinear Form on a Real Hilbert Space §representation and the values of $e_{k}$, $\langle T_{b_{N}}x,e_{k}\rangle_{H}=b_{N}(x,e_{k})$ is $\beta_{N}(k)x(k)$ for $k\in\Gamma_{N}$ and $0$ for $k\notin\Gamma_{N}$; likewise $b_{N}(e_{k},e_{k})$ is $\beta_{N}(k)$ for $k\in\Gamma_{N}$ and $0$ otherwise.

(c) By (3.1), $|a_{N}(k)\hat{x}(k)^{2}|\le2C\hat{x}(k)^{2}/\mu_{k}$, so $k\mapsto a_{N}(k)\hat{x}(k)^{2}$ is cube-summable; let $K_{N}$ be its lattice sum.

(d) Define $\varphi_{N}:H\to\mathbb{R}$ by $\varphi_{N}(x)=\phi(x)+Q_{N}(x)-2\tau\ell_{N}(x)+\tau K_{N}$. By Constants, Sums, Scalar Multiples and Differences of Differentiable Functions on an Open Subset of a Real Inner Product Space §sum, Constants, Sums, Scalar Multiples and Differences of Differentiable Functions on an Open Subset of a Real Inner Product Space §scalar and Constants, Sums, Scalar Multiples and Differences of Differentiable Functions on an Open Subset of a Real Inner Product Space §constant, $\varphi_{N}\in C^{2}(H)$, and for $x\in H$

$$D\varphi_{N}(x)=D\phi(x)+T_{b_{N}}x-2\tau Ay_{N},\qquad D^{2}\varphi_{N}(x)=D^{2}\phi(x)+b_{N}.$$

Step 4 (The test function $\chi_{N}$). Let $\chi_{N}=(\varphi_{N}+\tau c\,h)|_{H^{-1}}$. For $x\in H^{-1}\subseteq V$, (S) with $m=2$ and linearity give $\tau c\,h(x)=\tau C|x|_{V}^{2}=\sum_{k}\tau Cx(k)^{2}/\mu_{k}^{2}$, so

$$\chi_{N}(x)=\sum_{k\in\mathbb{Z}^{n}}\frac{\tau C}{\mu_{k}^{2}}\,x(k)^{2}+\varphi_{N}(x);$$

as $\mu_{k}^{2}|\tau C/\mu_{k}^{2}|=C$ and $\varphi_{N}$ is of class $C^{2}$ on $H^{-3}$, $\chi_{N}$ is regular with curvature family $k\mapsto\tau C/\mu_{k}^{2}$ and weak part $\varphi_{N}$ (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).

(a) Values. Let $x\in H^{-1}$. By Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §finite-support, $Q_{N}(x)$ is the lattice sum of the family equal to $(\eta(k)-\tau C/\mu_{k}^{2})x(k)^{2}$ on $\Gamma_{N}$ and $0$ elsewhere. Using Step 3(a), 3(c) and linearity of lattice sums, $\chi_{N}(x)-\chi(x)$ is the lattice sum of the family whose value at $k$ is

$$\frac{\tau C}{\mu_{k}^{2}}x(k)^{2}+\mathbf{1}_{\Gamma_{N}}(k)\Bigl(\eta(k)-\frac{\tau C}{\mu_{k}^{2}}\Bigr)x(k)^{2}-\eta(k)x(k)^{2}-2\tau a_{N}(k)\hat{x}(k)x(k)+\tau a_{N}(k)\hat{x}(k)^{2},$$

where $\mathbf{1}_{\Gamma_{N}}(k)$ is $1$ on $\Gamma_{N}$ and $0$ elsewhere. For $k\in\Gamma_{N}$ this is $0=\tau a_{N}(k)(x(k)-\hat{x}(k))^{2}$; for $k\notin\Gamma_{N}$, since $\tau a_{N}(k)=\tau C/\mu_{k}^{2}-\eta(k)$ ($\tau^{2}=1$), it is $\tau a_{N}(k)(x(k)^{2}-2\hat{x}(k)x(k)+\hat{x}(k)^{2})=\tau a_{N}(k)(x(k)-\hat{x}(k))^{2}$. Hence, by (3.1) and Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §nonnegative,

$$\tau\bigl(\chi_{N}(x)-\chi(x)\bigr)=\sum_{k\in\mathbb{Z}^{n}}a_{N}(k)\bigl(x(k)-\hat{x}(k)\bigr)^{2}\ge0,\qquad\chi_{N}(\hat{x})=\chi(\hat{x}),\qquad\text{(4.1)}$$

the latter because at $x=\hat{x}$ every cube sum of the family is $0$.

(b) Mode derivatives. Let $x\in H^{-1}$ and $k\in\mathbb{Z}^{n}$. By 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 §derivatives for $\chi$ with $(\eta,\phi)$ and for $\chi_{N}$ with $(\tau C/\mu^{2},\varphi_{N})$, by Step 3(d), and by Step 3(a), 3(b),

$$\partial_{k}\chi(x)=2\eta(k)x(k)+\langle D\phi(x),e_{k}\rangle_{H},\qquad\partial_{k}\chi_{N}(x)=\frac{2\tau C}{\mu_{k}^{2}}x(k)+\langle D\phi(x),e_{k}\rangle_{H}+\mathbf{1}_{\Gamma_{N}}(k)\beta_{N}(k)x(k)-2\tau a_{N}(k)\hat{x}(k),$$ $$\partial_{k}^{2}\chi(x)=2\eta(k)+D^{2}\phi(x)(e_{k},e_{k}),\qquad\partial_{k}^{2}\chi_{N}(x)=\frac{2\tau C}{\mu_{k}^{2}}+D^{2}\phi(x)(e_{k},e_{k})+\mathbf{1}_{\Gamma_{N}}(k)\beta_{N}(k).$$

Distinguishing $k\in\Gamma_{N}$ (where $a_{N}(k)=0$ and $2\tau C/\mu_{k}^{2}+\beta_{N}(k)=2\eta(k)$) from $k\notin\Gamma_{N}$ (where $2\tau C/\mu_{k}^{2}-2\eta(k)=2\tau a_{N}(k)$), one obtains for every $k$

$$\partial_{k}\chi_{N}(x)=\partial_{k}\chi(x)+2\tau a_{N}(k)\bigl(x(k)-\hat{x}(k)\bigr),\qquad\partial_{k}^{2}\chi_{N}(x)=\partial_{k}^{2}\chi(x)+2\tau a_{N}(k).\qquad\text{(4.2)}$$

In particular $\partial_{k}\chi_{N}(\hat{x})=\partial_{k}\chi(\hat{x})$ for every $k$.

Step 5 (Touching from above or below). Define $\Theta:V\to\mathbb{R}$ by $\Theta(x)=\tau\bigl(w(x)-\tau c\,h(x)-\varphi_{N}(x)\bigr)$. On $H^{-1}$, $\Theta=\tau(w-\chi_{N})=\tau(w-\chi)-\tau(\chi_{N}-\chi)$, so by (2.1) and (4.1), $\Theta(y)\le0$ for every $y\in H^{-1}$ with $|y-\hat{x}|_{H}<r$, and $\Theta(\hat{x})=\tau(w(\hat{x})-\chi(\hat{x}))=0$. $\Theta$ is continuous on $(V,d_{V})$: if $x_{j}\to x$ in $V$ then $|x_{j}-x|_{H}\le|x_{j}-x|_{V}\to0$ (Hilbert Triples: Standing Notation and Background §triple), so $w(x_{j})\to w(x)$ by continuity of $w$ on $H$ and sequential continuity (Real Hilbert Spaces: Standing Notation and Background §topology), $\varphi_{N}(x_{j})\to\varphi_{N}(x)$ by Basic Properties of Differentiability on an Open Subset of a Real Inner Product Space §continuous, and $h(x_{j})\to h(x)$ because $||x_{j}|_{V}-|x|_{V}|\le|x_{j}-x|_{V}$. Now let $x\in V$ with $|x-\hat{x}|_{H}<r$. By The Resolvent of the Form Operator of a Hilbert Triple: Minimisation, Contraction, and Density of the Domain §dense-in-v (its hypothesis holds by Hilbert Triples: Standing Notation and Background §separable) the points $x_{j}=R_{j}x$, which lie in $D(A)=H^{-1}$ by The Resolvent of the Form Operator of a Hilbert Triple: Minimisation, Contraction, and Density of the Domain §euler-lagrange, converge to $x$ in $V$, hence in $H$; so $|x_{j}-\hat{x}|_{H}<r$ for all large $j$, $\Theta(x_{j})\le0$ for those $j$, and $\Theta(x)=\lim_{j}\Theta(x_{j})\le0=\Theta(\hat{x})$. Thus $\Theta$ has a local maximum at $\hat{x}$ relative to $V$ in $(H,d_{H})$: for $\tau=1$, $x\mapsto w(x)-c\,h(x)-\varphi_{N}(x)$ has a local maximum relative to $V$ at $\hat{x}$; for $\tau=-1$, $x\mapsto w(x)+c\,h(x)-\varphi_{N}(x)$ has a local minimum relative to $V$ at $\hat{x}$. Moreover $w(\hat{x})=\chi(\hat{x})=\chi_{N}(\hat{x})=\varphi_{N}(\hat{x})+\tau c\,h(\hat{x})$, i.e. $w(\hat{x})-\tau c\,h(\hat{x})=\varphi_{N}(\hat{x})$.

For $\tau=1$, The Riccati Shift: the Renormalised Operator at the Free Solution plus a Regular Function is the Shifted Operator, Which Satisfies the Hilbert-Triple Comparison Hypotheses, and Its Touching Points Lie in the State Space §touching-sub (with $\delta=c$, $\varphi=\varphi_{N}$, and $\hat{x}\in V$) gives $F^{\sharp-}_{c}(\hat{x},\varphi_{N}(\hat{x}),D\varphi_{N}(\hat{x}),D^{2}\varphi_{N}(\hat{x}))\le0$, and by The Riccati Shift: the Renormalised Operator at the Free Solution plus a Regular Function is the Shifted Operator, Which Satisfies the Hilbert-Triple Comparison Hypotheses, and Its Touching Points Lie in the State Space §shifts the left side is $F[u_{0}+(\varphi_{N}+c\,h)|_{H^{-1}}](\hat{x})=F[u_{0}+\chi_{N}](\hat{x})$. For $\tau=-1$, The Riccati Shift: the Renormalised Operator at the Free Solution plus a Regular Function is the Shifted Operator, Which Satisfies the Hilbert-Triple Comparison Hypotheses, and Its Touching Points Lie in the State Space §touching-super and the second formula of The Riccati Shift: the Renormalised Operator at the Free Solution plus a Regular Function is the Shifted Operator, Which Satisfies the Hilbert-Triple Comparison Hypotheses, and Its Touching Points Lie in the State Space §shifts give $0\le F^{\sharp+}_{c}(\hat{x},\varphi_{N}(\hat{x}),D\varphi_{N}(\hat{x}),D^{2}\varphi_{N}(\hat{x}))=F[u_{0}+\chi_{N}](\hat{x})$. In both cases

$$\tau\,F[u_{0}+\chi_{N}](\hat{x})\le0.\qquad\text{(5.1)}$$

Step 6 (Removing the tail). Apply The Riccati Shift: the Renormalised Operator at the Free Solution plus a Regular Function is the Shifted Operator, Which Satisfies the Hilbert-Triple Comparison Hypotheses, and Its Touching Points Lie in the State Space §identity to $\chi$ and to $\chi_{N}$ at $\hat{x}$, and subtract. By (4.1) the terms $\gamma\chi(\hat{x})$ and $\gamma\chi_{N}(\hat{x})$ agree; by (4.2) at $\hat{x}$ the cross families $k\mapsto2q_{k}\hat{x}(k)\partial_{k}\chi(\hat{x})$ and $k\mapsto2q_{k}\hat{x}(k)\partial_{k}\chi_{N}(\hat{x})$ coincide, and by (G1) so do $|D\chi(\hat{x})|^{2}$ and $|D\chi_{N}(\hat{x})|^{2}$; the terms $g(\hat{x})$ agree. By (G2) and (4.2) at $\hat{x}$, and linearity of lattice sums, $L\chi_{N}(\hat{x})-L\chi(\hat{x})$ is the lattice sum of $k\mapsto\tfrac{\nu}{2}\cdot2\tau a_{N}(k)-\mu_{k}\hat{x}(k)\cdot0$; so with $S_{N}=\sum_{k}a_{N}(k)$ (cube-summable by (3.1), comparison and Lattice Sums of the Fourier Weights of the Torus, and the Negative Sobolev Norms as Lattice Sums §convergent, below),

$$F[u_{0}+\chi](\hat{x})=F[u_{0}+\chi_{N}](\hat{x})+\bigl(L\chi_{N}(\hat{x})-L\chi(\hat{x})\bigr)=F[u_{0}+\chi_{N}](\hat{x})+\tau\nu S_{N},$$

and by (5.1), $\tau F[u_{0}+\chi](\hat{x})=\tau F[u_{0}+\chi_{N}](\hat{x})+\nu S_{N}\le\nu S_{N}$.

Tail bound. Let $m(k)=(1/\mu_{k}^{2})$; since $n\le3<4$, $m$ 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 $\sigma$ be its lattice sum and $S_{N}(m)=\sum_{k\in\Gamma_{N}}m(k)$ its $N$th cube sum (Cube Sums of Families on the Integer Lattice §cube-sums). The family $m_{N}$ equal to $m$ off $\Gamma_{N}$ and to $0$ on $\Gamma_{N}$ is $m$ minus a family supported in $\Gamma_{N}$ whose lattice sum is $S_{N}(m)$ (Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §finite-support); so $m_{N}$ is cube-summable with lattice sum $\sigma-S_{N}(m)$. By (3.1), $0\le a_{N}(k)\le2C\,m_{N}(k)$ for every $k$, so by comparison $S_{N}\le2C(\sigma-S_{N}(m))$. Hence

$$\tau F[P+\psi](\hat{x})=\tau F[u_{0}+\chi](\hat{x})\le2\nu C\bigl(\sigma-S_{N}(m)\bigr)\qquad\text{for every }N\in\mathbb{N}.$$

The left side does not depend on $N$, and $S_{N}(m)\to\sigma$ as $N\to\infty$ (Cube Sums of Families on the Integer Lattice §lattice-sum); by the order of limits, $\tau F[P+\psi](\hat{x})\le0$. For $\tau=1$ this is $F[P+\psi](\hat{x})\le0$, so $u$ is a renormalised viscosity subsolution (Renormalised Viscosity Subsolutions, Supersolutions and Solutions of the Wick-Square Problem, Tested by the Gaussian Penalty plus a Regular Function §subsolution); for $\tau=-1$ it is $0\le F[P+\psi](\hat{x})$, so $u$ is a renormalised viscosity supersolution (Renormalised Viscosity Subsolutions, Supersolutions and Solutions of the Wick-Square Problem, Tested by the Gaussian Penalty plus a Regular Function §supersolution). This proves clause 1.

Step 7 (Clause 2: the data). Let $w$ be Lipschitz from $(H,d_{H})$ to $\mathbb{R}$ with a constant $\Lambda\ge0$ (Lipschitz Map Between Metric Spaces). Suppose $u$ is a renormalised viscosity subsolution (case $\tau=1$), respectively supersolution (case $\tau=-1$). Recall that $w$ is bounded above and below near each point of $H$ (Conventions). Let $\delta>0$, $\varphi\in C^{2}(H)$, $\hat{x}\in V$ and $\varepsilon>0$ be as in Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on a Hilbert Triple §subsolution (case $\tau=1$), respectively Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on a Hilbert Triple §supersolution (case $\tau=-1$), with $U=H$, $V\cap U=V$ and $W=D(A)=H^{-1}$. Since $w^{\mp}_{\delta}=w\mp\delta h$ on $V$, the function $\Xi:V\to\mathbb{R}$, $\Xi(x)=\tau\bigl(w(x)-\tau\delta h(x)-\varphi(x)\bigr)$, has a local maximum at $\hat{x}$ relative to $V$ in $(H,d_{H})$; choose a real $r>0$ with $\Xi(x)\le\Xi(\hat{x})$ for all $x\in V$ with $|x-\hat{x}|_{H}<r$.

Step 8 ($\hat{x}\in H^{-1}$). By Basic Properties of Differentiability on an Open Subset of a Real Inner Product Space §bound with $\varepsilon=1$ at $\hat{x}$, choose a real $\rho>0$ with $|\varphi(\hat{x}+z)-\varphi(\hat{x})|\le(|D\varphi(\hat{x})|_{H}+1)|z|_{H}$ for $z\in H$, $|z|_{H}<\rho$, and put $K=\Lambda+|D\varphi(\hat{x})|_{H}+1$. Let $v\in V$ with $v\ne0_{H}$ and let $t$ be real with $0<t$ and $t|v|_{H}<\min\{r,\rho\}$; put $x=\hat{x}+tv\in V$. From $\Xi(x)\le\Xi(\hat{x})$, multiplying by $\tau$ and rearranging,

$$\delta\bigl(h(x)-h(\hat{x})\bigr)\ge\tau\bigl(w(x)-w(\hat{x})\bigr)-\tau\bigl(\varphi(x)-\varphi(\hat{x})\bigr)\ge-\Lambda t|v|_{H}-(|D\varphi(\hat{x})|_{H}+1)t|v|_{H}=-Kt|v|_{H}.$$

By The Penalty Function $h=\tfrac12|\cdot|_V^2$ of a Hilbert Triple: Expansion Identities, Closed Sublevel Sets, Density and Local Bounds §expansion-v, $h(x)-h(\hat{x})=\langle\hat{x},tv\rangle_{V}+h(tv)=t\langle\hat{x},v\rangle_{V}+t^{2}h(v)$. Dividing by $t$, $\delta\langle\hat{x},v\rangle_{V}\ge-K|v|_{H}-\delta t\,h(v)$, and letting $t\to0$ gives $\delta\langle\hat{x},v\rangle_{V}\ge-K|v|_{H}$. Applying this to $-v$ as well,

$$|\langle\hat{x},v\rangle_{V}|\le\frac{K}{\delta}\,|v|_{H}\qquad\text{for every }v\in V\qquad\text{(8.1)}$$

(trivially for $v=0_{H}$). Fix $N\in\mathbb{N}$, let $v_{N}$ be the family equal to $\mu_{k}\hat{x}(k)$ for $k\in\Gamma_{N}$ and to $0$ elsewhere, and let $s_{N}=\sum_{k\in\Gamma_{N}}\hat{x}(k)^{2}/\mu_{k}$. By (S) and Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §finite-support, $v_{N}\in V$ and $|v_{N}|_{H}^{2}=\sum_{k\in\Gamma_{N}}\mu_{k}^{2}\hat{x}(k)^{2}/\mu_{k}^{3}=s_{N}$. Modewise, $v_{N}=\sum_{k\in\Gamma_{N}}\mu_{k}\hat{x}(k)e_{k}$, so by linearity of the inner product and the second statement of 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 (with $\hat{x}\in V$), $\langle\hat{x},v_{N}\rangle_{V}=\sum_{k\in\Gamma_{N}}\mu_{k}\hat{x}(k)\hat{x}(k)/\mu_{k}^{2}=s_{N}$. By (8.1), $s_{N}^{2}\le(K/\delta)^{2}s_{N}$, so $s_{N}\le(K/\delta)^{2}$ (trivially if $s_{N}=0$). By Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §nonnegative the family $k\mapsto\hat{x}(k)^{2}/\mu_{k}$ is cube-summable, so $\hat{x}\in H^{-1}=D(A)$ by The Wick-Square Problem on the Torus: Standing Notation §state-space.

Step 9 (Testing the renormalised notion). Put $a=w(\hat{x})-\tau\delta h(\hat{x})-\varphi(\hat{x})$ and $\varphi'=\varphi+a$; by Constants, Sums, Scalar Multiples and Differences of Differentiable Functions on an Open Subset of a Real Inner Product Space §sum and Constants, Sums, Scalar Multiples and Differences of Differentiable Functions on an Open Subset of a Real Inner Product Space §constant, $\varphi'\in C^{2}(H)$ with $D\varphi'=D\varphi$ and $D^{2}\varphi'=D^{2}\varphi$. Let $\chi'=(\varphi'+\tau\delta h)|_{H^{-1}}$, regular by The Riccati Shift: the Renormalised Operator at the Free Solution plus a Regular Function is the Shifted Operator, Which Satisfies the Hilbert-Triple Comparison Hypotheses, and Its Touching Points Lie in the State Space §triple-form (with $c=\tau\delta$), and $\psi'=\psi_{0}+\chi'$, regular by 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 §linear. On $H^{-1}$, $P+\psi'=u_{0}+\chi'$ and $u-P-\psi'=w-\chi'=w-\varphi-a-\tau\delta h$, so $\tau(u-P-\psi')(y)=\Xi(y)-\tau a$ for $y\in H^{-1}$. Since $H^{-1}\subseteq V$ and $d_{H}=d_{H^{-3}}$, $u-P-\psi'$ has at $\hat{x}\in H^{-1}$ a local maximum ($\tau=1$), respectively minimum ($\tau=-1$), relative to $H^{-1}$ in $(H^{-3},d_{H^{-3}})$, and $(u-P-\psi')(\hat{x})=w(\hat{x})-\varphi(\hat{x})-a-\tau\delta h(\hat{x})=0$, i.e. $u(\hat{x})=P(\hat{x})+\psi'(\hat{x})$. By Renormalised Viscosity Subsolutions, Supersolutions and Solutions of the Wick-Square Problem, Tested by the Gaussian Penalty plus a Regular Function §subsolution ($\tau=1$), respectively Renormalised Viscosity Subsolutions, Supersolutions and Solutions of the Wick-Square Problem, Tested by the Gaussian Penalty plus a Regular Function §supersolution ($\tau=-1$), $\tau F[P+\psi'](\hat{x})\le0$. Now $\varphi'(\hat{x})=w(\hat{x})-\tau\delta h(\hat{x})$, which is $w^{-}_{\delta}(\hat{x})$ for $\tau=1$ and $w^{+}_{\delta}(\hat{x})$ for $\tau=-1$. By The Riccati Shift: the Renormalised Operator at the Free Solution plus a Regular Function is the Shifted Operator, Which Satisfies the Hilbert-Triple Comparison Hypotheses, and Its Touching Points Lie in the State Space §shifts applied to $\varphi'$,

$$F^{\sharp-}_{\delta}\bigl(\hat{x},w^{-}_{\delta}(\hat{x}),D\varphi(\hat{x}),D^{2}\varphi(\hat{x})\bigr)=F[u_{0}+\chi'](\hat{x})\le0\qquad(\tau=1),$$ $$0\le F[u_{0}+\chi'](\hat{x})=F^{\sharp+}_{\delta}\bigl(\hat{x},w^{+}_{\delta}(\hat{x}),D\varphi(\hat{x}),D^{2}\varphi(\hat{x})\bigr)\qquad(\tau=-1).$$

Step 10 (Conclusion of clause 2). Take $y=\hat{x}\in W$, $q=D\varphi(\hat{x})\in H$, $Y=D^{2}\varphi(\hat{x})\in\mathrm{Sym}(H)$, and $s=w^{-}_{\delta}(\hat{x})$ for $\tau=1$, $s=w^{+}_{\delta}(\hat{x})$ for $\tau=-1$. Then $|y-\hat{x}|_{H}=0$, the differences of envelope values and $|s-w^{\mp}_{\delta}(\hat{x})|$ are $0$, $|q-D\varphi(\hat{x})|_{H}=0$ and $\lVert Y-D^{2}\varphi(\hat{x})\rVert=\lVert0_{\mathrm{Sym}}\rVert=0$ (Elementary Properties of Bounded Symmetric Bilinear Forms: Norm, Quadratic Form, Order and Continuity §norm-axioms), all less than $\varepsilon$; and by Step 9, $F^{\sharp-}_{\delta}(y,s,q,Y)\le0<\varepsilon$ for $\tau=1$, respectively $-\varepsilon<0\le F^{\sharp+}_{\delta}(y,s,q,Y)$ for $\tau=-1$. As $\delta,\varphi,\hat{x},\varepsilon$ were arbitrary, $w$ is a viscosity subsolution, respectively supersolution, of $F^{\sharp}$ on $H$. This proves clause 2.