The Flat Torus: Standing Notation

This setting fixes the standing notation used by results of analysis on the flat torus of dimension $n$, realised as the half-open unit cell of $\mathbb{R}^{n}$ carrying Lebesgue measure. It introduces no new concept and asserts nothing beyond the identifications recorded below, each of which is justified by the reference attached to it.

1. (Background)¶ Throughout, $n$ denotes a natural number with $1\le n$ and $p$ a real number with $1\le p$. The notation of Euclidean Space and Lebesgue Measure: Standing Notation and of Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation is in force for the dimension $n$, as combined in Lattice-Periodic Functions and the Periodic Function Classes: in particular Euclidean space $\mathbb{R}^{n}$ with its norm, distance, balls, topology and the notions of open, closed, bounded and compact set, the constant $\sigma_{n}$, the Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R}^{n})$, Lebesgue measure $\lambda_{n}$ and the null sets, the real numbers, the integers, the natural numbers and the initial segments $[n]$, and the partial derivatives $\partial_{i}$, the iterated partial derivatives, the gradient and the classes $C^{k}$ on a Euclidean open set. The notation of Measure Spaces and the Lebesgue Integral: Standing Notation is in force with the measure space named in clause 3.

2. (Lattice, cells and wrapping)¶ $\mathbb{Z}^{n}$ is the integer lattice, and $Q$, $\mathring{Q}$ and $\overline{Q}$ are the half-open, open and closed unit cells, with $\pi$ the wrapping map. The tiling property The Half-Open Unit Cell Tiles Euclidean Space §tiling and the cell-integral identities The Half-Open Unit Cell Tiles Euclidean Space §translate and The Half-Open Unit Cell Tiles Euclidean Space §translate-integrable are in force by reference.

3. (Integration over the cell)¶ $\mathcal{B}_{Q}$ denotes $\{A\in\mathcal{B}(\mathbb{R}^{n}):A\subseteq Q\}$ and $\lambda_{Q}$ the map $A\mapsto\lambda_{n}(A)$ on it; by claim 1 of Assembly of Measure Spaces: Restriction, Transport, One-Point Spaces, and Countable Disjoint Unions, applied to the measure space $(\mathbb{R}^{n},\mathcal{B}(\mathbb{R}^{n}),\lambda_{n})$ and to $Q\in\mathcal{B}(\mathbb{R}^{n})$, the triple $(Q,\mathcal{B}_{Q},\lambda_{Q})$ is a measure space, and $\lambda_{Q}(Q)=1$ by The Half-Open Unit Cell Tiles Euclidean Space §cell. It is this measure space that instantiates the $(X,\mathcal{F},\mu)$ of Measure Spaces and the Lebesgue Integral: Standing Notation unless another is named explicitly. No new object is introduced here: $\mathcal{B}_{Q}$ and $\lambda_{Q}$ are the restriction furnished by the lemma just cited, given short names. For a measurable $v:Q\to[0,\infty]$, and for an integrable $v:Q\to\mathbb{R}$, we write

4. (The Lebesgue spaces of the torus)¶ $\mathcal{L}^{p}(\mathbb{T}^{n})$ denotes the set of $p$-integrable functions on $(Q,\mathcal{B}_{Q},\lambda_{Q})$, $L^{p}(\mathbb{T}^{n})$ the Lebesgue space of that measure space, $[v]$ the class of $v\in\mathcal{L}^{p}(\mathbb{T}^{n})$ under almost-everywhere equality, and $\lVert\,\cdot\,\rVert_{L^{p}(\mathbb{T}^{n})}$ the norm. By The Riesz-Fischer Theorem: the Lebesgue Space is a Real Banach Space §normed this is a real normed space and by The Riesz-Fischer Theorem: the Lebesgue Space is a Real Banach Space §complete a real Banach space; its metric and topology are the ones fixed there, and all metric and topological notions applied to $L^{p}(\mathbb{T}^{n})$ refer to them. By The Lebesgue Space of Square-Integrable Functions is a Real Hilbert Space §hilbert the space $L^{2}(\mathbb{T}^{n})$ is a real Hilbert space for the inner product recorded there.

5. (Periodic functions, gradient and Laplacian)¶ $C_{\mathrm{per}}$, $C^{k}_{\mathrm{per}}$ and $C^{\infty}_{\mathrm{per}}$ abbreviate the periodic function classes $C_{\mathrm{per}}(\mathbb{R}^{n})$, $C^{k}_{\mathrm{per}}(\mathbb{R}^{n})$ and $C^{\infty}_{\mathrm{per}}(\mathbb{R}^{n})$, and the properties recorded in Elementary Properties of Lattice-Periodic Functions are in force by reference. For $u$ of class $C^{1}$ on $\mathbb{R}^{n}$ we write $\nabla u$ as an alternative notation for the gradient $Du$, namely the $n$-tuple of the partial derivatives of $u$; this clause introduces no object beyond that abbreviation.

6. (Restriction convention)¶ For a map $u$ defined on $\mathbb{R}^{n}$, $u|_{Q}$ denotes its restriction to $Q$. This clause fixes notation only: nothing is asserted here about measurability of $u|_{Q}$ or about its membership in any $\mathcal{L}^{p}(\mathbb{T}^{n})$, which a result relying on it must establish for itself.