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

In the setting of The Wick-Square Problem on the Torus: Standing Notation, let $\psi:H^{-1}\to\mathbb{R}$ be regular, and fix one choice of data witnessing this: a curvature family $\eta$ with curvature bound $C$ and a weak part $\phi:H^{-3}\to\mathbb{R}$, for which the identity displayed 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 holds. Here $\langle\cdot,\cdot\rangle_{H^{-3}}$ is the inner product of the Sobolev space $H^{-3}=H^{-3}(\mathbb{T}^{n})$, and $D\phi$ and $D^{2}\phi$ are the gradient map and the Hessian map of $\phi$; $e_{k}$ is the unit family of the mode $k$, which lies in $H^{-3}$ by Lattice Sums of the Fourier Weights of the Torus, and the Negative Sobolev Norms as Lattice Sums §units, and $H^{-1}\subseteq H^{-3}$ 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 $m=2$. Let $P$ be the Gaussian penalty; mode derivatives, $L$ and $|D\cdot|^{2}$ are those of Derivatives of a Function on the Sobolev Space of Order -1 along the Fourier Modes §derivatives, The Free-Field Generator with a Mode Cutoff §limit and The Gradient Energy with a Mode Cutoff §limit. Then the following hold.

1. (Mode derivatives)¶ $\psi$ is twice differentiable along the modes, and for every $x\in H^{-1}$ and $k\in\mathbb{Z}^{n}$

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

2. (The generator converges)¶ For every $x\in H^{-1}$ the families

$$k\mapsto\nu\,\eta(k)-2\mu_{k}\eta(k)\,x(k)^{2}\qquad\text{and}\qquad k\mapsto\tfrac{\nu}{2}\,D^{2}\phi(x)(e_{k},e_{k})-\mu_{k}\,x(k)\,\langle D\phi(x),e_{k}\rangle_{H^{-3}}$$

are cube-summable, $L\psi$ is defined at $x$, and $L\psi(x)$ is the sum of their two lattice sums.

3. (The gradient energy converges)¶ Let $\theta\in\mathbb{R}$. Then $\theta P+\psi$ is twice differentiable along the modes, and for every $x\in H^{-1}$ the family $k\mapsto\bigl(\partial_{k}(\theta P+\psi)(x)\bigr)^{2}$ is cube-summable and $|D(\theta P+\psi)(x)|^{2}$ is defined and equals its lattice sum.

4. (The renormalised operator)¶ Let $g:H^{-1}\to\mathbb{R}$ be a running cost, and let $\mathcal{D}$ and $F$ be the domain and the renormalised operator for $g$. Then for every $x\in H^{-1}$ one has $(P+\psi,x)\in\mathcal{D}$ and

$$F[P+\psi](x)=\gamma\,(P+\psi)(x)-L\psi(x)+\tfrac12\,|D(P+\psi)(x)|^{2}-g(x).$$

5. (Linear combinations)¶ Let $\psi_{1},\psi_{2}:H^{-1}\to\mathbb{R}$ be regular and $a,b\in\mathbb{R}$. Then the pointwise combination $a\psi_{1}+b\psi_{2}$ is regular, and so is every constant function on $H^{-1}$.