Proof 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

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. The triple is that of The Negative-Order Sobolev Triple of the Torus: a Diagonal Hilbert Triple whose Form Operator Is One Minus the Laplacian with $s=2$, so $H=H^{-3}(\mathbb{T}^{n})$ and $V=H^{-2}(\mathbb{T}^{n})$, with $\langle\cdot,\cdot\rangle_{H}=\langle\cdot,\cdot\rangle_{H^{-3}}$ and $\langle\cdot,\cdot\rangle_{V}=\langle\cdot,\cdot\rangle_{H^{-2}}$. Every space $H^{-l}(\mathbb{T}^{n})$ ($l\in\mathbb{N}$) is a linear subspace of $\mathrm{Map}(\mathbb{Z}^{n},\mathbb{R})$ under the pointwise operations (The Negative-Order Sobolev Spaces of the Torus §space), and $H^{-1}\subseteq V\subseteq H$ by The Negative-Order Sobolev Spaces of the Torus are Hilbert Spaces: the Embedding of the Square-Integrable Classes, the Rescaled Trigonometric Basis, the Series Form of the Inner Product and the Inclusion of the Scale §inclusion with $m=1$ and $m=2$; so sums, differences and multiples of states are the same families in each of these spaces. By The Wick-Square Problem on the Torus: Standing Notation §modes, $1\le\mu_{k}$, hence $0<\mu_{k}$ and $\mu_{k}^{l}\le\mu_{k}^{l'}$, i.e. $1/\mu_{k}^{l'}\le1/\mu_{k}^{l}$, whenever $l\le l'$. By Lattice Sums of the Fourier Weights of the Torus, and the Negative Sobolev Norms as Lattice Sums §sobolev, for $l\in\mathbb{N}$ a family $c$ lies in $H^{-l}(\mathbb{T}^{n})$ exactly when $k\mapsto c(k)^{2}/\mu_{k}^{l}$ is cube-summable, and then $|c|_{H^{-l}}^{2}=\sum_{k\in\mathbb{Z}^{n}}c(k)^{2}/\mu_{k}^{l}$. The enumeration $\kappa$ is a bijection from $\mathbb{N}$ onto $\mathbb{Z}^{n}$ (Properties of the Fourier Coefficients on the Torus, and the Realisation of Weighted Coefficient Families).

Step 1 (Series along $\kappa$). Let $a:\mathbb{Z}^{n}\to\mathbb{R}$ be such that $|a|$ is cube-summable. Then $a$ is cube-summable, and the series $\sum_{j=1}^{\infty}a(\kappa(j))$ converges with sum $\sum_{k\in\mathbb{Z}^{n}}a(k)$. Indeed, $a$ is cube-summable by Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §absolute, hence so is $a+|a|$ by Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §linear. Both $|a|$ and $a+|a|$ are nonnegative families, so by Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §enumeration the series $\sum_{j}|a|(\kappa(j))$ and $\sum_{j}(a+|a|)(\kappa(j))$ converge, with sums the lattice sums of $|a|$ and of $a+|a|$. Since $a(\kappa(j))=(a+|a|)(\kappa(j))+(-1)\,|a|(\kappa(j))$, Elementary Properties of Series of Real Numbers §linearity shows that $\sum_{j}a(\kappa(j))$ converges with sum $\sum_{k}(a(k)+|a(k)|)-\sum_{k}|a(k)|$, which is $\sum_{k\in\mathbb{Z}^{n}}a(k)$ by Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §linear.

Step 2 (Inner products as lattice sums). Let $l\in\mathbb{N}$ and $c,d\in H^{-l}(\mathbb{T}^{n})$. Then the families $k\mapsto|c(k)d(k)|/\mu_{k}^{l}$ and $k\mapsto c(k)d(k)/\mu_{k}^{l}$ are cube-summable, $|c(k)d(k)|/\mu_{k}^{l}\le\tfrac12\bigl(c(k)^{2}/\mu_{k}^{l}+d(k)^{2}/\mu_{k}^{l}\bigr)$ for every $k$, and $\langle c,d\rangle_{H^{-l}}=\sum_{k\in\mathbb{Z}^{n}}c(k)d(k)/\mu_{k}^{l}$. The inequality follows from $2|c(k)|\,|d(k)|\le c(k)^{2}+d(k)^{2}$ (as $0\le(|c(k)|-|d(k)|)^{2}$) on dividing by $2\mu_{k}^{l}>0$. The right-hand side is a cube-summable family by the Conventions and Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §linear, so both families are cube-summable by Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §comparison. By The Negative-Order Sobolev Spaces of the Torus are Hilbert Spaces: the Embedding of the Square-Integrable Classes, the Rescaled Trigonometric Basis, the Series Form of the Inner Product and the Inclusion of the Scale §series with $m=l$, the series $\sum_{j=1}^{\infty}(\rho_{\kappa(j)}^{l})^{2}c(\kappa(j))d(\kappa(j))$ converges with sum $\langle c,d\rangle_{H^{-l}}$, and $(\rho_{k}^{l})^{2}=1/\mu_{k}^{l}$ by The Negative-Order Sobolev Spaces of the Torus are Hilbert Spaces: the Embedding of the Square-Integrable Classes, the Rescaled Trigonometric Basis, the Series Form of the Inner Product and the Inclusion of the Scale §weights; so its terms are $a(\kappa(j))$ for the family $a(k)=c(k)d(k)/\mu_{k}^{l}$. By Step 1 the same series converges with sum $\sum_{k}a(k)$, and the two sums agree because the sum of a convergent series is the unique limit of its partial sums (Series of Real Numbers §convergent).

Step 3 (Pairings with unit families; first part of clause pairings). Let $l\in\mathbb{N}$, $c\in H^{-l}(\mathbb{T}^{n})$ and $k\in\mathbb{Z}^{n}$. Then $\langle c,e_{k}\rangle_{H^{-l}}=c(k)/\mu_{k}^{l}$. Indeed $e_{k}\in H^{-l}(\mathbb{T}^{n})$ by Lattice Sums of the Fourier Weights of the Torus, and the Negative Sobolev Norms as Lattice Sums §units, and the family $k'\mapsto c(k')e_{k}(k')/\mu_{k'}^{l}$ vanishes off $\{k\}$ and takes the value $c(k)/\mu_{k}^{l}$ at $k$; by Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §finite-support its lattice sum is $c(k)/\mu_{k}^{l}$, and Step 2 applies. With $l=3$ and $l=2$ this gives $\langle z,e_{k}\rangle_{H}=z(k)/\mu_{k}^{3}$ for $z\in H$ and $\langle z,e_{k}\rangle_{V}=z(k)/\mu_{k}^{2}$ for $z\in V$.

Step 4 (Clause domain). By Hilbert Triple: a Densely and Continuously Embedded Hilbert Space and Its Form Operator §operator, $D(A)$ is the set of $x\in V$ for which some $z\in H$ satisfies $\langle x,y\rangle_{V}=\langle z,y\rangle_{H}$ for every $y\in V$, and $Ax$ is that (unique) $z$.

$H^{-1}\subseteq D(A)$, with $Ax=(\mu_{k}x(k))_{k}$. Let $x\in H^{-1}$; then $x\in V$ (Conventions). Let $z$ be the family $k\mapsto\mu_{k}x(k)$. Then $z(k)^{2}/\mu_{k}^{3}=x(k)^{2}/\mu_{k}$, and $k\mapsto x(k)^{2}/\mu_{k}$ is cube-summable with lattice sum $|x|_{H^{-1}}^{2}$ (The Wick-Square Problem on the Torus: Standing Notation §state-space); so by the Conventions with $l=3$, $z\in H$ and $|z|_{H}^{2}=|x|_{H^{-1}}^{2}$. Let $y\in V$, so also $y\in H$. By Step 2 with $l=2$ and with $l=3$,

$$\langle x,y\rangle_{V}=\sum_{k\in\mathbb{Z}^{n}}\frac{x(k)y(k)}{\mu_{k}^{2}},\qquad\langle z,y\rangle_{H}=\sum_{k\in\mathbb{Z}^{n}}\frac{\mu_{k}x(k)y(k)}{\mu_{k}^{3}}=\sum_{k\in\mathbb{Z}^{n}}\frac{x(k)y(k)}{\mu_{k}^{2}} ,$$

the families summed being equal termwise. Hence $x\in D(A)$ and $Ax=z$. Since $|Ax|_{H}$ and $|x|_{H^{-1}}$ are nonnegative with equal squares, $|Ax|_{H}=|x|_{H^{-1}}$.

$D(A)\subseteq H^{-1}$. Let $x\in D(A)$ and $z=Ax\in H$. For each $k$, $e_{k}\in V$ (Lattice Sums of the Fourier Weights of the Torus, and the Negative Sobolev Norms as Lattice Sums §units with $m=2$), so $\langle x,e_{k}\rangle_{V}=\langle z,e_{k}\rangle_{H}$, i.e. $x(k)/\mu_{k}^{2}=z(k)/\mu_{k}^{3}$ by Step 3, whence $z(k)=\mu_{k}x(k)$. As $z\in H$, the family $k\mapsto z(k)^{2}/\mu_{k}^{3}=x(k)^{2}/\mu_{k}$ is cube-summable (Conventions, $l=3$), so $x\in H^{-1}$ (Conventions, $l=1$).

Thus $D(A)=H^{-1}$, and clause domain holds.

Step 5 (Second part of clause pairings). Let $x\in H^{-1}$ and $z\in H$. By Step 4, $Ax\in H$ is the family $k\mapsto\mu_{k}x(k)$. Step 2 with $l=3$, applied to $Ax$ and $z$, shows that $k\mapsto|\mu_{k}x(k)z(k)|/\mu_{k}^{3}$ and $k\mapsto\mu_{k}x(k)z(k)/\mu_{k}^{3}$ are cube-summable and that $\langle Ax,z\rangle_{H}$ is the lattice sum of the latter. Since $0<\mu_{k}$, these families are $k\mapsto|x(k)z(k)|/\mu_{k}^{2}$ and $k\mapsto x(k)z(k)/\mu_{k}^{2}$, which proves the assertion.

Step 6 (Clause infinite). By The Negative-Order Sobolev Spaces of the Torus are Hilbert Spaces: the Embedding of the Square-Integrable Classes, the Rescaled Trigonometric Basis, the Series Form of the Inner Product and the Inclusion of the Scale §hilbert with $m=3$, $H$ is a real Hilbert space, and by The Negative-Order Sobolev Spaces of the Torus are Hilbert Spaces: the Embedding of the Square-Integrable Classes, the Rescaled Trigonometric Basis, the Series Form of the Inner Product and the Inclusion of the Scale §basis with $m=3$ the sequence $(\zeta_{3,\kappa(j)})_{j\in\mathbb{N}}$ is an orthonormal basis of $H$, in particular an orthonormal sequence in $H$ (that definition is made for orthonormal sequences). By A Real Hilbert Space Containing an Orthonormal Sequence Is Not Finite-Dimensional §not-finite-dimensional, $H$ is not finite-dimensional.

Step 7 (Clause noise). For $j\in\mathbb{N}$, $f_{j}=e_{\kappa(j)}\in V$ and $|f_{j}|_{V}^{2}=1/\mu_{\kappa(j)}^{2}$ by Lattice Sums of the Fourier Weights of the Torus, and the Negative Sobolev Norms as Lattice Sums §units with $m=2$. Since $n\le3<4=2\cdot2$ (The Wick-Square Problem on the Torus: Standing Notation §dimension), the family $k\mapsto1/\mu_{k}^{2}$ is cube-summable by Lattice Sums of the Fourier Weights of the Torus, and the Negative Sobolev Norms as Lattice Sums §convergent with $s=2$; it is nonnegative, so by Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §enumeration the series $\sum_{j=1}^{\infty}1/\mu_{\kappa(j)}^{2}=\sum_{j=1}^{\infty}|f_{j}|_{V}^{2}$ converges with sum $\sum_{k\in\mathbb{Z}^{n}}(1/\mu_{k}^{2})$. By Square-Summable Sequences in the Form Space of a Hilbert Triple and the Trace of a Form along Them §square-summable, $f$ is square-summable in $V$ with $\sigma(f)=\sum_{k\in\mathbb{Z}^{n}}(1/\mu_{k}^{2})$.

Let $X\in\mathrm{Sym}(V)$ and $a(k)=X(e_{k},e_{k})$. By Elementary Properties of Bounded Symmetric Bilinear Forms: Norm, Quadratic Form, Order and Continuity §bound and Lattice Sums of the Fourier Weights of the Torus, and the Negative Sobolev Norms as Lattice Sums §units with $m=2$, $|a(k)|\le\lVert X\rVert\,|e_{k}|_{V}\,|e_{k}|_{V}=\lVert X\rVert/\mu_{k}^{2}$, and $k\mapsto\lVert X\rVert/\mu_{k}^{2}$ is cube-summable by Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §linear. By Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §comparison, $|a|$ and $a$ are cube-summable. By Square-Summable Sequences in the Form Space of a Hilbert Triple and the Trace of a Form along Them §trace, $\mathrm{Tr}_{f}X$ is the sum of the series $\sum_{j=1}^{\infty}X(f_{j},f_{j})=\sum_{j=1}^{\infty}a(\kappa(j))$, which by Step 1 is $\sum_{k\in\mathbb{Z}^{n}}X(e_{k},e_{k})$ (uniqueness of the sum of a series, Series of Real Numbers §convergent).

Step 8 (Clause gradient-form). Let $p,p'\in H$ and $g(k)=\langle p,e_{k}\rangle_{H}\langle p',e_{k}\rangle_{H}$. By Step 3, $g(k)=p(k)p'(k)/\mu_{k}^{6}$, so, as $1\le\mu_{k}^{3}$ and by Step 2 with $l=3$,

$$|g(k)|\le\frac{|p(k)p'(k)|}{\mu_{k}^{3}}\le\frac12\Bigl(\frac{p(k)^{2}}{\mu_{k}^{3}}+\frac{p'(k)^{2}}{\mu_{k}^{3}}\Bigr).$$

The right-hand family is cube-summable with lattice sum $\tfrac12(|p|_{H}^{2}+|p'|_{H}^{2})$ (Conventions, $l=3$, and Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §linear); by Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §comparison, $|g|$ and $g$ are cube-summable, and with Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §absolute,

$$|G(p,p')|\le\sum_{k\in\mathbb{Z}^{n}}|g(k)|\le\tfrac12\bigl(|p|_{H}^{2}+|p'|_{H}^{2}\bigr).\qquad(\ast)$$

$G\in\mathrm{Sym}(H)$. Symmetry holds because the families for $(p,p')$ and $(p',p)$ coincide. For $p,p'',p'\in H$ and $\lambda\in\mathbb{R}$, $\langle p+p'',e_{k}\rangle_{H}=\langle p,e_{k}\rangle_{H}+\langle p'',e_{k}\rangle_{H}$ and $\langle\lambda p,e_{k}\rangle_{H}=\lambda\langle p,e_{k}\rangle_{H}$ (the inner product is bilinear, Elementary Identities in a Real Inner Product Space §bilinear with symmetry); so the family for $(p+p'',p')$ is the sum of those for $(p,p')$ and $(p'',p')$, and that for $(\lambda p,p')$ is $\lambda$ times that for $(p,p')$, and Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §linear gives $G(p+p'',p')=G(p,p')+G(p'',p')$ and $G(\lambda p,p')=\lambda G(p,p')$. For the bound, let $p,p'\in H$. If $p=0_{H}$ or $p'=0_{H}$ then $G(p,p')=0$ by the homogeneity just proved (with $\lambda=0$) and symmetry. Otherwise $|p|_{H}>0$ and $|p'|_{H}>0$ (Elementary Identities in a Real Inner Product Space §vanishing); the vectors $u=|p|_{H}^{-1}p$ and $u'=|p'|_{H}^{-1}p'$ have norm $1$ (Elementary Identities in a Real Inner Product Space §homogeneity), so $(\ast)$ gives $|G(u,u')|\le1$, and by homogeneity in each argument $G(p,p')=|p|_{H}|p'|_{H}\,G(u,u')$, whence $|G(p,p')|\le|p|_{H}|p'|_{H}$. So $G$ satisfies the conditions of Bounded Symmetric Bilinear Forms on a Real Inner Product Space: Norm, Order, Identity Form and Restriction §form with constant $1$, and $G\in\mathrm{Sym}(H)$ (Real Hilbert Spaces: Standing Notation and Background §forms).

$0_{\mathrm{Sym}}\preceq G\preceq I_{H}$. Let $p\in H$. Then $G(p,p)=\sum_{k}p(k)^{2}/\mu_{k}^{6}$, a lattice sum of a nonnegative family, so $0=0_{\mathrm{Sym}}(p,p)\le G(p,p)$ by Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §nonnegative. Also $0\le p(k)^{2}/\mu_{k}^{6}\le p(k)^{2}/\mu_{k}^{3}$, so by Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §comparison and the Conventions, $G(p,p)\le\sum_{k}p(k)^{2}/\mu_{k}^{3}=|p|_{H}^{2}=\langle p,p\rangle_{H}=I_{H}(p,p)$, using Real Inner Product Space §norm and Bounded Symmetric Bilinear Forms on a Real Inner Product Space: Norm, Order, Identity Form and Restriction §identity. By Bounded Symmetric Bilinear Forms on a Real Inner Product Space: Norm, Order, Identity Form and Restriction §order, $0_{\mathrm{Sym}}\preceq G\preceq I_{H}$.

Step 9 (Clause drift). For each $k$, $0<q_{k}$ by The Riccati Coefficients of the Wick-Square Problem: Existence, the Gaussian Profile and Summable Constants §root and $q_{k}\le\beta/(2\mu_{k})$ by The Riccati Coefficients of the Wick-Square Problem: Existence, the Gaussian Profile and Summable Constants §profile; hence $0<2q_{k}\le\beta/\mu_{k}$ and $(2q_{k})^{2}\le\beta^{2}/\mu_{k}^{2}$.

$B_{q}x\in H$, with a bound. Let $x\in V$. For every $k$, $0\le(2q_{k}x(k))^{2}/\mu_{k}^{3}\le\beta^{2}x(k)^{2}/\mu_{k}^{5}\le\beta^{2}x(k)^{2}/\mu_{k}^{2}$, and $k\mapsto\beta^{2}x(k)^{2}/\mu_{k}^{2}$ is cube-summable with lattice sum $\beta^{2}|x|_{V}^{2}$ (Conventions with $l=2$, Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §linear). By Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §comparison the family $k\mapsto(B_{q}x)(k)^{2}/\mu_{k}^{3}$ is cube-summable with lattice sum at most $\beta^{2}|x|_{V}^{2}$; by the Conventions with $l=3$, $B_{q}x\in H$ and $|B_{q}x|_{H}^{2}\le\beta^{2}|x|_{V}^{2}$, hence $|B_{q}x|_{H}\le\beta|x|_{V}$, both sides being nonnegative. So $B_{q}:V\to H$ is defined; it is linear, since $2q_{k}(x(k)-y(k))=2q_{k}x(k)-2q_{k}y(k)$ gives $B_{q}x-B_{q}y=B_{q}(x-y)$ for $x,y\in V$.

Monotonicity. Let $x,y\in V$ and $w=x-y\in V\subseteq H$. By Step 2 with $l=3$, applied to $B_{q}w$ and $w$,

$$\langle B_{q}x-B_{q}y,x-y\rangle_{H}=\langle B_{q}w,w\rangle_{H}=\sum_{k\in\mathbb{Z}^{n}}\frac{2q_{k}\,w(k)^{2}}{\mu_{k}^{3}}\ge0,$$

the inequality by Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §nonnegative, each term being nonnegative. This is Monotone, $A$-Monotone and Locally Bounded Nonlinearities on a Hilbert Triple §monotone.

$A$-monotonicity. Let $x\in D(A)$; by Step 4, $x\in H^{-1}$ and $Ax$ is the family $k\mapsto\mu_{k}x(k)$. By Step 2 with $l=3$,

$$\langle B_{q}x,Ax\rangle_{H}=\sum_{k\in\mathbb{Z}^{n}}\frac{2q_{k}x(k)\,\mu_{k}x(k)}{\mu_{k}^{3}}=\sum_{k\in\mathbb{Z}^{n}}\frac{2q_{k}\,x(k)^{2}}{\mu_{k}^{2}}\ge0,$$

again by Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §nonnegative. This is Monotone, $A$-Monotone and Locally Bounded Nonlinearities on a Hilbert Triple §a-monotone.

Boundedness on $V$-bounded sets. Let $R>0$. Every $x\in V$ with $|x|_{V}\le R$ satisfies $|B_{q}x|_{H}\le\beta|x|_{V}\le\beta R$, so the real number $\beta R$ serves in Monotone, $A$-Monotone and Locally Bounded Nonlinearities on a Hilbert Triple §bounded.

By Monotone, $A$-Monotone and Locally Bounded Nonlinearities on a Hilbert Triple §nonlinearity, $B_{q}$ is a monotone nonlinearity for $(H,V,A)$.