Mode Derivatives of Linear Combinations and of Restrictions of Twice Differentiable Functions on a Negative Sobolev Space

In the setting of The Wick-Square Problem on the Torus: Standing Notation, being twice differentiable along the modes and the mode derivatives $\partial_{k}$ and $\partial_{k}^{2}$ are those of Derivatives of a Function on the Sobolev Space of Order -1 along the Fourier Modes, and $e_{k}$ is the unit family of the mode $k$. Functions on $H^{-1}$ are added and multiplied by real numbers pointwise.

1. (Linear combinations and constants)¶ Let $\varphi,\psi:H^{-1}\to\mathbb{R}$ be twice differentiable along the modes and let $a,b\in\mathbb{R}$. Then $a\varphi+b\psi$ is twice differentiable along the modes, and for every $x\in H^{-1}$ and $k\in\mathbb{Z}^{n}$

$$\partial_{k}(a\varphi+b\psi)(x)=a\,\partial_{k}\varphi(x)+b\,\partial_{k}\psi(x),\qquad\partial_{k}^{2}(a\varphi+b\psi)(x)=a\,\partial_{k}^{2}\varphi(x)+b\,\partial_{k}^{2}\psi(x).$$

Every constant function on $H^{-1}$ is twice differentiable along the modes, with both mode derivatives zero at every point and every mode.

2. (Restrictions from a negative Sobolev space)¶ Let $m\in\mathbb{N}$, let $H^{-m}=H^{-m}(\mathbb{T}^{n})$ be the Sobolev space of order $-m$ with its inner product $\langle\cdot,\cdot\rangle_{H^{-m}}$, a real inner product space to which the calculus of Real Hilbert Spaces: Standing Notation and Background applies, and let $\Phi:H^{-m}\to\mathbb{R}$ be of class $C^{2}$ on $H^{-m}$, with gradient map $D\Phi$ and Hessian map $D^{2}\Phi$. One has $H^{-1}\subseteq H^{-m}$ by repeated use of 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, and $e_{k}\in H^{-m}$ for every mode $k$ by Lattice Sums of the Fourier Weights of the Torus, and the Negative Sobolev Norms as Lattice Sums §units. Then the restriction $\Phi|_{H^{-1}}$ of $\Phi$ to $H^{-1}$ is twice differentiable along the modes, and for every $x\in H^{-1}$ and $k\in\mathbb{Z}^{n}$

$$\partial_{k}(\Phi|_{H^{-1}})(x)=\langle D\Phi(x),e_{k}\rangle_{H^{-m}},\qquad\partial_{k}^{2}(\Phi|_{H^{-1}})(x)=D^{2}\Phi(x)(e_{k},e_{k}).$$