The Negative-Order Sobolev Triple of the Torus: a Diagonal Hilbert Triple whose Form Operator Is One Minus the Laplacian

We work in the setting of The Flat Torus: Standing Notation, used here with a natural number $n$ satisfying $1\le n$, and in the settings of Real Hilbert Spaces: Standing Notation and Background and Real Hilbert Spaces: Series, Products, Orthonormal Bases and Differential Calculus, whose standing space $H$ is not used here; the integer lattice $\mathbb{Z}^{n}$, Euclidean space $\mathbb{R}^{n}$ with its norm $\lVert\,\cdot\,\rVert$, the cell $Q$, the set $\mathcal{L}^{2}(\mathbb{T}^{n})$, the real Hilbert space $L^{2}(\mathbb{T}^{n})$ with the class map $[\,\cdot\,]$ and inner product $\langle\,\cdot\,,\cdot\,\rangle_{L^{2}}$, the periodic class $C^{2}_{\mathrm{per}}$, the restriction $u|_{Q}$ and the Laplacian $\Delta$ of a map of class $C^{2}$ on $\mathbb{R}^{n}$ are the ones fixed there. Let $\pi$ be the real number of The Number Pi (i) (the wrapping map of The Flat Torus: Standing Notation (ii) is not used here), let $2=1+1$ and $4=2+2$, and let $\mu_{k}=1+4\pi^{2}\lVert k\rVert^{2}$ for $k\in\mathbb{Z}^{n}$ be the Fourier weights (that lemma used with $s=n$ there; its $s$ is unrelated to the $s$ below), the positive numbers $\rho_{k}$ with $\rho_{k}^{2}=\tfrac{1}{\mu_{k}}$, the natural powers $\rho_{k}^{l}$, the classes $E_{k}$ of the trigonometric system introduced in The Trigonometric System on the Torus is Orthonormal (i), the Fourier coefficient family $\hat{U}$ of a class $U$, and the enumerations $\kappa$ of the lattice, as in Properties of the Fourier Coefficients on the Torus, and the Realisation of Weighted Coefficient Families. For $l\in\mathbb{N}$, $H^{-l}(\mathbb{T}^{n})$ is the Sobolev space of order $-l$ with its inner product, norm and distance, a real Hilbert space 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 (ii), and $\zeta_{l,k}$ for $k\in\mathbb{Z}^{n}$ are the elements of its rescaled trigonometric basis. Let $s\in\mathbb{N}$, and write

$$H=H^{-(s+1)}(\mathbb{T}^{n}),\qquad V=H^{-s}(\mathbb{T}^{n}),$$

$V$ carrying the inner product $\langle\,\cdot\,,\cdot\,\rangle_{H^{-s}}$, so that $\langle\,\cdot\,,\cdot\,\rangle_{H}$ and $|\cdot|_{H}$ are $\langle\,\cdot\,,\cdot\,\rangle_{H^{-(s+1)}}$ and $|\cdot|_{H^{-(s+1)}}$, and $\langle\,\cdot\,,\cdot\,\rangle_{V}$ and $|\cdot|_{V}$ are $\langle\,\cdot\,,\cdot\,\rangle_{H^{-s}}$ and $|\cdot|_{H^{-s}}$. Series of real numbers and series in a real inner product space are as defined there, and for a real number $t$ we write $t^{2}=tt$. Then the following hold.

1. (The Sobolev triple)(i) $V$ is a linear subspace of $H$ satisfying $|x|_{H}\le|x|_{V}$ for every $x\in V$, and $V$ is dense in $H$; hence $H$, $V$ with $\langle\,\cdot\,,\cdot\,\rangle_{V}$, and the form operator $A$ they determine form a Hilbert triple $(H,V,A)$, called the Sobolev triple of order $s$, and the metric space $(V,d_{V})$ is separable, so that the standing hypothesis Hilbert Triples: Standing Notation and Background (ii) holds for it. Moreover, for every enumeration $\kappa$, the triple $(H,V,A)$ is the diagonal Hilbert triple determined by the orthonormal basis $(\zeta_{s+1,\kappa(j)})_{j\in\mathbb{N}}$ of $H$ and the weights $(\mu_{\kappa(j)})_{j\in\mathbb{N}}$.

2. (The domain and the operator in coefficients)(ii) Let $\kappa$ be an enumeration, and for $x\in H$ write $x_{j}=\langle x,\zeta_{s+1,\kappa(j)}\rangle_{H}$, which equals $\rho_{\kappa(j)}^{s+1}x(\kappa(j))$. Then $D(A)$ is the set of those $x\in H$ for which the series $\sum_{j=1}^{\infty}\mu_{\kappa(j)}^{2}x_{j}^{2}$ converges, and for such $x$ the series $\sum_{j=1}^{\infty}\mu_{\kappa(j)}x_{j}\zeta_{s+1,\kappa(j)}$ converges in $H$ with sum $Ax$.

3. (Eigenvectors)(iii) Let $k\in\mathbb{Z}^{n}$. Then $\hat{E}_{k}=\rho_{k}^{s+1}\zeta_{s+1,k}$, both $\zeta_{s+1,k}$ and $\hat{E}_{k}$ lie in $D(A)$, and

$$A\zeta_{s+1,k}=\mu_{k}\,\zeta_{s+1,k},\qquad A\hat{E}_{k}=\mu_{k}\,\hat{E}_{k}.$$

4. (The form operator is one minus the Laplacian)(iv) Let $u\in C^{2}_{\mathrm{per}}$. Then $u|_{Q}$ and $(\Delta u)|_{Q}$ lie in $\mathcal{L}^{2}(\mathbb{T}^{n})$, the coefficient family $\widehat{[\,u|_{Q}\,]}$ lies in $D(A)$, and

$$A\,\widehat{[\,u|_{Q}\,]}=\widehat{[\,u|_{Q}\,]}-\widehat{[\,(\Delta u)|_{Q}\,]} .$$