The Periodic Extension of a Function on the Unit Cell

We work in the setting of The Flat Torus: Standing Notation, used here with a natural number $n$ satisfying $1\le n$ and a real number $p$ with $1\le p$; the cells $Q$, $\mathring{Q}$, $\overline{Q}$, the lattice $\mathbb{Z}^{n}$, the wrapping map $\pi$, the measure space $(Q,\mathcal{B}_{Q},\lambda_{Q})$, the integral over $\mathbb{T}^{n}$, the classes $\mathcal{L}^{p}(\mathbb{T}^{n})$ and $L^{p}(\mathbb{T}^{n})$, the periodic class $C_{\mathrm{per}}$ and the restriction $u|_{Q}$ are the ones fixed there. In addition, let $\mathbb{Z}^{n}$-periodicity of a map on $\mathbb{R}^{n}$ be as defined there; let measurability of a map on $\mathbb{R}^{n}$ mean measurability with respect to $\mathcal{B}(\mathbb{R}^{n})$, integrals over $\mathbb{R}^{n}$ being taken with respect to $\lambda_{n}$ in the measure space $(\mathbb{R}^{n},\mathcal{B}(\mathbb{R}^{n}),\lambda_{n})$; let $\lVert\,\cdot\,\rVert_{r}$ denote the $L^{r}$ seminorm of the measure space $(Q,\mathcal{B}_{Q},\lambda_{Q})$, for a real number $r$ with $1\le r$; for a real number $r$ with $1\le r$ and a map $f$ into $\mathbb{R}$ defined on $Q$ or on $\mathbb{R}^{n}$, let $|f|^{r}$ denote the map whose value at a point is the $r$th power of the absolute value of $f$ at that point, the power being that of Real Powers; and for $A\subseteq\mathbb{R}^{n}$ and $h\in\mathbb{R}^{n}$ put $A+h=\{x+h:x\in A\}$.

Let $v:Q\to\mathbb{R}$, and let $\tilde{v}:\mathbb{R}^{n}\to\mathbb{R}$, the periodic extension of $v$, be given by $\tilde{v}(x)=v(\pi(x))$; this is defined because $\pi(x)\in Q$ for every $x\in\mathbb{R}^{n}$ by The Half-Open Unit Cell Tiles Euclidean Space (iii). Then the following hold.

1. (Extension)(i) $\tilde{v}$ is $\mathbb{Z}^{n}$-periodic and $\tilde{v}(x)=v(x)$ for every $x\in Q$. If $v$ is measurable with respect to $\mathcal{B}_{Q}$, then $\tilde{v}$ is measurable. If $u:\mathbb{R}^{n}\to\mathbb{R}$ is $\mathbb{Z}^{n}$-periodic, then the periodic extension of $u|_{Q}$ is $u$ itself.

2. (Linearity)(ii) Let $v':Q\to\mathbb{R}$ and $c\in\mathbb{R}$. Then the periodic extension of $v+c\,v'$ is $\tilde{v}+c\,\tilde{v}'$.

3. (Almost-everywhere equality)(iii) Let $v,v':Q\to\mathbb{R}$ be measurable with respect to $\mathcal{B}_{Q}$ and suppose $v=v'$ $\lambda_{Q}$-almost everywhere on $Q$. Then $\tilde{v}=\tilde{v}'$ $\lambda_{n}$-almost everywhere on $\mathbb{R}^{n}$.

4. (Power-integrable functions are integrable)(iv) Let $v\in\mathcal{L}^{p}(\mathbb{T}^{n})$. Then $v$ is integrable with respect to $\lambda_{Q}$, so that $v\in\mathcal{L}^{1}(\mathbb{T}^{n})$, and

$$\lVert v\rVert_{1}\le\lVert v\rVert_{p}.$$

5. (Local integrability of the extension)(v) Let $v:Q\to\mathbb{R}$ be measurable with respect to $\mathcal{B}_{Q}$ and integrable with respect to $\lambda_{Q}$. Then $\mathbf{1}_{B}\tilde{v}$ is integrable for every bounded $B\in\mathcal{B}(\mathbb{R}^{n})$, and for every $h\in\mathbb{R}^{n}$,

$$\int_{\mathbb{R}^{n}}\mathbf{1}_{Q+h}\,\tilde{v}\,d\lambda_{n}=\int_{\mathbb{T}^{n}}v\,dx .$$

6. (Powers of the extension)(vi) Let $v\in\mathcal{L}^{p}(\mathbb{T}^{n})$. Then $|v|^{p}$ is measurable with respect to $\mathcal{B}_{Q}$ and integrable with respect to $\lambda_{Q}$, its periodic extension is $|\tilde{v}|^{p}$, and for every $h\in\mathbb{R}^{n}$,

$$\int_{\mathbb{R}^{n}}\mathbf{1}_{Q+h}\,|\tilde{v}|^{p}\,d\lambda_{n}=\bigl(\lVert v\rVert_{p}\bigr)^{p}.$$

In particular $\mathbf{1}_{B}|\tilde{v}|^{p}$ is integrable for every bounded $B\in\mathcal{B}(\mathbb{R}^{n})$.