The 2 pi-periodic three-torus, its normalized measure, test functions and the Sobolev spaces H^s

Assume the Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)) for the quotient, product-measure and integration clauses below, as in The one-dimensional torus and its normalized Haar integral.

(a) The 2 pi-periodic torus. Let ${\mathbb{T}}_{2\pi }:=\mathbb{R}\mathrm{/}2\pi \mathbb{Z}$ carry the quotient topology, and let ${\mathbb{T}}_{2\pi }^3:={\mathbb{T}}_{2\pi }\times {\mathbb{T}}_{2\pi }\times {\mathbb{T}}_{2\pi }$ with the product topology; write $q:{\mathbb{R}}^3\to {\mathbb{T}}_{2\pi }^3$ for the quotient map and $[x]:=(x^1+2\pi \mathbb{Z},x^2+2\pi \mathbb{Z},x^3+2\pi \mathbb{Z})$. The map $[0,2\pi {)}^3\to {\mathbb{T}}_{2\pi }^3$, $x\mapsto [x]$, is a bijection. The normalized product measure $\mu$ is ${\mu }_{\mathbb{T}}\otimes {\mu }_{\mathbb{T}}\otimes {\mu }_{\mathbb{T}}$, where ${\mu }_{\mathbb{T}}$ is the normalized measure of The one-dimensional torus and its normalized Haar integral transported by $t\mapsto 2\pi t$; it is a probability measure on the Borel sets of ${\mathbb{T}}_{2\pi }^3$ and $\int fd\mu =(2\pi {)}^{-3}{\int }_{[0,2\pi {)}^3}f\circ qd{\lambda }_3$ for bounded measurable $f$ (Tonelli's theorem for nonnegative measurable functions on a sigma-finite product).

(b) Test functions and continuous classes. For $N\ge 1$ let $C^{\mathrm{\infty }}({\mathbb{T}}_{2\pi }^3;{\mathbb{R}}^N)$ be the set of maps $u:{\mathbb{T}}_{2\pi }^3\to {\mathbb{R}}^N$ whose pullbacks $\tilde{u}:=u\circ q:{\mathbb{R}}^3\to {\mathbb{R}}^N$ are smooth; these are the smooth test functions (real-valued when $N=1$). Equivalently every smooth map $\tilde{u}:{\mathbb{R}}^3\to {\mathbb{R}}^N$ that is $2\pi$-periodic in each coordinate descends to the unique map $u([x])=\tilde{u}(x)$, since periodicity makes this value independent of $x$ in its equivalence class. For an integer $m\ge 0$, define $C^m({\mathbb{T}}_{2\pi }^3;{\mathbb{R}}^N)$ similarly by requiring $\tilde{u}\in C^m({\mathbb{R}}^3;{\mathbb{R}}^N)$ ($C^r$ and smooth maps between smooth manifolds), and define $D^{\alpha }u([x]):=D^{\alpha }\tilde{u}(x)$ for $\mathrm{\mid }\alpha \mathrm{\mid }\le m$. Derivatives of a periodic $C^m$ map are periodic, so these values are well-defined, with $\mathrm{\parallel }u{\mathrm{\parallel }}_{C^m}:={\max }_{\mathrm{\mid }\alpha \mathrm{\mid }\le m}{\sup }_{x\in {\mathbb{T}}_{2\pi }^3}\mathrm{\mid }(D^{\alpha }u)(x)\mathrm{\mid },$ where the multi-index derivative is that of $C^k$ maps and multi-index derivative notation in Euclidean space.

(c) The Hilbert space $L^2$. For $N\ge 1$ let $L^2({\mathbb{T}}_{2\pi }^3;{\mathbb{R}}^N)$ be the quotient of the set of measurable $u$ with $\int \mathrm{\mid }u{\mathrm{\mid }}^{2}d\mu <\mathrm{\infty }$ by the $\mu$-a.e. zero functions (The space $L^p(\mu )$ as the quotient by null functions, The $L^p$ norm descends to the quotient and makes $L^p$ a normed space for $1\le p\le \mathrm{\infty }$), with inner product $⟨u,v{⟩}_2=\int u\cdot vd\mu$ and norm $\mathrm{\parallel }u{\mathrm{\parallel }}_2=⟨u,u{⟩}_2^{1\mathrm{/}2}$; it is a real Hilbert space ($L^2$ with the integral pairing is a Hilbert space). For $u\in L^2({\mathbb{T}}_{2\pi }^3;{\mathbb{R}}^N)$ and $k\in {\mathbb{Z}}^3$ the $k$-th Fourier coefficient is the vector $\widehat{u}(k):={\int }_{{\mathbb{T}}_{2\pi }^3}u(x)e^{-ik\cdot x}d\mu (x)\in {\mathbb{C}}^N,$ where $e^{-ik\cdot x}:={\chi }_{-k}$ with ${\chi }_k([x]):=\exp (ik\cdot x)$; here $u$ is read through any representative. Write ${\chi }_k:{\mathbb{T}}_{2\pi }^3\to \mathbb{C}$, $k\in {\mathbb{Z}}^3$, for these characters; under the identification $x=2\pi y$ they are the characters $\exp (2\pi ik\cdot y)$ of The Fourier basis and Parseval's identity on the finite torus, so $({\chi }_k{)}_{k\in {\mathbb{Z}}^3}$ is an orthonormal basis of the complex space $L^2({\mathbb{T}}_{2\pi }^3;\mathbb{C})$.

(d) Weak derivatives and the periodic Sobolev spaces. For $u,v\in L^1({\mathbb{T}}_{2\pi }^3;\mathbb{R})$ and $\alpha \in {\mathbb{N}}_0^3$ say that $v$ is the weak derivative $D^{\alpha }u$ if ${\int }_{{\mathbb{T}}_{2\pi }^3}u{D}^{\alpha }\phi d\mu =(-1{)}^{\mathrm{\mid }\alpha \mathrm{\mid }}{\int }_{{\mathbb{T}}_{2\pi }^3}v\phi d\mu$ for every $\phi \in C^{\mathrm{\infty }}({\mathbb{T}}_{2\pi }^3;\mathbb{R})$, where $D^{\alpha }\phi$ is the multi-index derivative of the periodic representative (Weak derivative of a locally integrable function records the Euclidean analogue). For an integer $s\ge 0$ let $H^s({\mathbb{T}}_{2\pi }^3;{\mathbb{R}}^N):=\{u\in L^2({\mathbb{T}}_{2\pi }^3;{\mathbb{R}}^N):\text{ for every }\mathrm{\mid }\alpha \mathrm{\mid }\le s\text{ there is }{v}_{\alpha }\in L^2({\mathbb{T}}_{2\pi }^3;{\mathbb{R}}^N)\text{ with }{v}_{\alpha }=D^{\alpha }u\},$ the vector-valued weak derivative being componentwise, and put $\mathrm{\parallel }u{\mathrm{\parallel }}_{H^s}:=({\sum }_{\mathrm{\mid }\alpha \mathrm{\mid }\le s}\mathrm{\parallel }{D}^{\alpha }u{\mathrm{\parallel }}_2^2{)}^{1\mathrm{/}2}\mathrm{.}$ The classes $L^2\subseteq L^1$ on the probability space, so the integrals in the test identity are finite: $\mathrm{\mid }\int u{D}^{\alpha }\phi d\mu \mathrm{\mid }\le \mathrm{\parallel }u{\mathrm{\parallel }}_2\mathrm{\parallel }{D}^{\alpha }\phi {\mathrm{\parallel }}_2$ by Cauchy--Schwarz. The definition asserts neither that the weak derivative exists nor that the two norms compared in the next item are equivalent; those are proved in Periodic Fourier differentiation identity and equivalence of the derivative and Fourier-weight Sobolev norms.

(e) Conventions. $O\subseteq {\mathbb{R}}^m$ open, $I\subseteq \mathbb{R}$ an interval and maps $F:I\times {\mathbb{T}}_{2\pi }^3\times O\to {\mathbb{R}}^{m^{\mathrm{\prime }}}$ are called smooth when the representative $G(t,x,u):=F(t,q(x),u)$ is locally the restriction of a smooth Euclidean map: for every $(t,x,u)\in I\times {\mathbb{R}}^3\times O$ there are an open neighborhood $V\subseteq {\mathbb{R}}^{4+m}$ and $\tilde{G}\in C^{\mathrm{\infty }}(V;{\mathbb{R}}^{m^{\mathrm{\prime }}})$ with $\tilde{G}=G$ on $V\cap (I\times {\mathbb{R}}^3\times O)$. For open $I$ this is the ordinary Euclidean definition. For nonopen $I$ it is the local extension convention at included endpoints, also admitting a singleton interval. On a nondegenerate interval endpoint time derivatives are the limits of interior derivatives; on a singleton no time derivative is specified without choosing an extension. Throughout this page the torus is always this $2\pi$-periodic one; a statement "for every $x$" means for every point of ${\mathbb{T}}_{2\pi }^3$, and "a.e." means $\mu$-a.e.