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

In the setting of The Wick-Square Problem on the Torus: Standing Notation, a function $\psi:H^{-1}\to\mathbb{R}$ is regular if there are

¶(a) a family $\eta:\mathbb{Z}^{n}\to\mathbb{R}$ and a real number $C\ge0$ with $\mu_{k}^{2}\,|\eta(k)|\le C$ for every $k\in\mathbb{Z}^{n}$ (a curvature family of $\psi$, with curvature bound $C$), and

¶(b) a function $\phi:H^{-3}\to\mathbb{R}$ of class $C^{2}$ on $H^{-3}$, where $H^{-3}=H^{-3}(\mathbb{T}^{n})$ is the Sobolev space of order $-3$, a real inner product space with the inner product $\langle\cdot,\cdot\rangle_{H^{-3}}$, to which the calculus of Real Hilbert Spaces: Standing Notation and Background applies, and open in itself by Real Hilbert Spaces: Standing Notation and Background §topology (a weak part of $\psi$),

such that for every $x\in H^{-1}$

$$\psi(x)=\sum_{k\in\mathbb{Z}^{n}}\eta(k)\,x(k)^{2}+\phi(x),$$

where $\phi(x)$ is defined because $H^{-1}\subseteq H^{-2}\subseteq H^{-3}$ 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, applied with $m=1$ and $m=2$. The lattice sum exists: since $1\le\mu_{k}$ (The Wick-Square Problem on the Torus: Standing Notation §modes), one has $|\eta(k)\,x(k)^{2}|\le C\,x(k)^{2}/\mu_{k}^{2}\le C\,x(k)^{2}/\mu_{k}$, the family $k\mapsto C\,x(k)^{2}/\mu_{k}$ 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, and Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §comparison applies.