Periodic vector Sobolev spaces on the three-torus, the weight family, and the Leray projection

Assume the Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), which is used by the imported Fourier-basis theorem quoted below.

The three-torus and its measure. Let $\mathbb{T}:=\mathbb{R}\mathrm{/}\mathbb{Z}$ be the one-dimensional torus with its quotient topology and its normalized Haar integral $m_{\mathbb{T}}$ (The one-dimensional torus and its normalized Haar integral); $\mathbb{T}$ is compact Hausdorff (Finite tori are compact Hausdorff spaces separated by characters). Put ${\mathbb{T}}^3:=\mathbb{T}\times \mathbb{T}\times \mathbb{T}$ with the finite product topology. Let $dx:=m_{\mathbb{T}}\otimes m_{\mathbb{T}}\otimes m_{\mathbb{T}}$ be the product measure on the Borel $\sigma$-algebra of ${\mathbb{T}}^3$ (The product measure of two sigma-finite measure spaces, Lebesgue measurable sets, the family $\mathcal{L}({\mathbb{R}}^n)$, and the restricted set function ${\lambda }_n$); being a product of probability measures it is a probability measure. For a vector $k=(k_1,k_2,k_3)\in {\mathbb{Z}}^3$ and $x=(x_1,x_2,x_3)\in {\mathbb{T}}^3$ define the character $e_k(x):=\exp (2\pi i(k_1{x}_1+k_2{x}_2+k_3{x}_3))=e^{2\pi ik\cdot x},$ where each $x_j$ is inserted through any real representative and $\exp$ is the complex exponential (Fourier coefficients and trigonometric polynomials on the torus); the definition does not depend on the representatives. Then $\mathrm{\mid }{e}_k(x)\mathrm{\mid }=1$, $e_k{e}_l=e_{k+l}$, and $e_{-k}=\overline{e_k}$.

Function spaces. Write $L^2:=L^2({\mathbb{T}}^3;{\mathbb{R}}^3)$ for the real Hilbert space of (equivalence classes of) real Borel vector fields $u:{\mathbb{T}}^3\to {\mathbb{R}}^3$ with ${\int }_{{\mathbb{T}}^3}\mathrm{\mid }u{\mathrm{\mid }}_2^{2}dx<+\mathrm{\infty }$, inner product $⟨u,v⟩:={\int }_{{\mathbb{T}}^3}u\cdot vdx$ and norm $\mathrm{\mid }u{\mathrm{\mid }}_2:=⟨u,u{⟩}^{1\mathrm{/}2}$; here $u\cdot v$ is the Euclidean inner product of ${\mathbb{R}}^3$ (Complex Lp classes and Euclidean test-function conventions). Similarly $L^1({\mathbb{T}}^3;{\mathbb{C}}^3)$ denotes the integrable complex vector fields.

Fourier coefficients. For $u\in L^1({\mathbb{T}}^3;{\mathbb{C}}^3)$ and $k\in {\mathbb{Z}}^3$ define $\widehat{u}(k):={\int }_{{\mathbb{T}}^3}u(x)e_{-k}(x)dx\in {\mathbb{C}}^3\mathrm{.}$ The integral depends only on the class of $u$. A field $u\in L^1$ is real almost everywhere if and only if $\widehat{u}(-k)=\overline{\widehat{u}(k)}$ for every $k\in {\mathbb{Z}}^3$; the forward direction uses $\widehat{\bar{u}}(k)=\overline{\widehat{u}(-k)}$, and the reverse uses $L^1$ Fourier uniqueness, proved below. On the circle this uniqueness follows from Fejer means converge in L^p for 1 <= p < infinity with $p=1$: zero coefficients give zero Fejer means, whose $L^1$ limit is the function. To pass to ${\mathbb{T}}^d$ for $d=2,3$, use Fubini's theorem for L^1 functions on a sigma-finite product: for fixed $k^{\mathrm{\prime }}\in {\mathbb{Z}}^{d-1}$, $g_{k^{\mathrm{\prime }}}(x_1)={\int }_{{\mathbb{T}}^{d-1}}u(x_1,x^{\mathrm{\prime }})e_{-k^{\mathrm{\prime }}}(x^{\mathrm{\prime }})d{x}^{\mathrm{\prime }}$ is in $L^1(\mathbb{T})$ and all its coefficients vanish. Hence $g_{k^{\mathrm{\prime }}}=0$ almost everywhere. The countable union of the exceptional sets over $k^{\mathrm{\prime }}$ is null. For every remaining $x_1$, the $L^1$ section in $x^{\mathrm{\prime }}$ has all coefficients zero, so induction in $d$ makes it zero almost everywhere. Fubini then makes $u=0$ almost everywhere on ${\mathbb{T}}^d$. Apply this uniqueness to $u-\bar{u}$ to obtain the reverse real-valuedness implication. For $u\in L^2\subset L^1$ (finite measure) the coefficient map is the one of the imported theorem, which states: $(e_k{)}_{k\in {\mathbb{Z}}^3}$ is an orthonormal basis of the complex Hilbert space $L^2({\mathbb{T}}^3;\mathbb{C})$, the finite-subset-net expansion $u={\sum }_k\widehat{u}(k)e_k$ and the Parseval identities hold, and $u\mapsto \widehat{u}$ is a surjective linear isometry of $L^2({\mathbb{T}}^3;\mathbb{C})$ onto ${\mathrm{\ell }}^2({\mathbb{Z}}^3;\mathbb{C})$ (The Fourier basis and Parseval's identity on the finite torus); the last statement applied componentwise gives $⟨u,v⟩={\sum }_k\widehat{u}(k)\cdot \overline{\widehat{v}(k)}$ for real $u,v\in L^2$.

Weights. For $k\in {\mathbb{Z}}^3$ set $w_k:=(1+\mathrm{\mid }2\pi k{\mathrm{\mid }}_2^2{)}^{1\mathrm{/}2}=(1+4{\pi }^2\mathrm{\mid }k{\mathrm{\mid }}_2^2{)}^{1\mathrm{/}2}\ge 1,$ so that $w_k\ge \mathrm{\mid }2\pi k_j\mathrm{\mid }$ for $j=1,2,3$ and $w_{k+l}\le w_k+w_l$ for all $k,l\in {\mathbb{Z}}^3$.

The coefficient space and $H^s$. For $s\in \mathbb{R}$ let ${\mathrm{\ell }}_s^2$ be the complex Hilbert space of all $c:{\mathbb{Z}}^3\to {\mathbb{C}}^3$ with $\mathrm{\parallel }c{\mathrm{\parallel }}_{{\mathrm{\ell }}_s^2}^2:={\sum }_{k\in {\mathbb{Z}}^3}{w}_k^{2s}\mathrm{\mid }c(k){\mathrm{\mid }}_2^2<+\mathrm{\infty },⟨c,d{⟩}_{{\mathrm{\ell }}_s^2}:={\sum }_{k\in {\mathbb{Z}}^3}{w}_k^{2s}c(k)\cdot \overline{d(k)},$ and let ${\mathrm{\ell }}_{s,\mathbb{R}}^2:=\{c\in {\mathrm{\ell }}_s^2:c(-k)=\overline{c(k)}\mathrm{\forall }k\}$. The set ${\mathrm{\ell }}_{s,\mathbb{R}}^2$ is closed in ${\mathrm{\ell }}_s^2$ and is a real-linear subspace; on it the Hermitian pairing $⟨\cdot ,\cdot {⟩}_{{\mathrm{\ell }}_s^2}$ is real valued, because the substitution $k\mapsto -k$ shows that it equals its own conjugate. By the imported theorem the coefficient map restricts to a bijection of the set of real classes of $L^2$ onto ${\mathrm{\ell }}_{0,\mathbb{R}}^2$. For $s\ge 0$ define $H^s({\mathbb{T}}^3;{\mathbb{R}}^3):=\{u\in L^2:\widehat{u}\in {\mathrm{\ell }}_{s,\mathbb{R}}^2\},\mathrm{\parallel }u{\mathrm{\parallel }}_{H^s}:=\mathrm{\parallel }\widehat{u}{\mathrm{\parallel }}_{{\mathrm{\ell }}_s^2}\mathrm{.}$ For $s<0$ define $H^s$ to be the completion of the real trigonometric polynomials in the same weighted coefficient norm, equivalently the space of formal real coefficient families ${\mathrm{\ell }}_{s,\mathbb{R}}^2$. For an element of this completion $\widehat{u}(k)$ means its coefficient, not an integral of an assumed $L^2$ function. The embedding of $L^2$ into these negative-order spaces is its coefficient map. Truncation makes finite families dense, so this completion is all of ${\mathrm{\ell }}_{s,\mathbb{R}}^2$, not merely the coefficient families of $L^2$ fields. Consequently each $H^s$ is a real Hilbert space isometric to ${\mathrm{\ell }}_{s,\mathbb{R}}^2$ through $u\mapsto \widehat{u}$, and $\mathrm{\parallel }u{\mathrm{\parallel }}_{H^0}=\mathrm{\mid }u{\mathrm{\mid }}_2$. If $s\ge t$ then $H^s\subseteq H^t$ and $\mathrm{\parallel }u{\mathrm{\parallel }}_{H^t}\le \mathrm{\parallel }u{\mathrm{\parallel }}_{H^s}$. The trigonometric polynomials are the fields $p={\sum }_{k\in F}{c}_k{e}_k$ with $F\subseteq {\mathbb{Z}}^3$ finite and $c_{-k}=\overline{c_k}$ (equivalently, real coefficient families of finite support); they belong to every $H^s$, and the finitely supported families are dense in each ${\mathrm{\ell }}_{s,\mathbb{R}}^2$ (truncate a family to $\mathrm{\mid }k{\mathrm{\mid }}_{\mathrm{\infty }}\le M$ and let $M\to \mathrm{\infty }$), so trigonometric polynomials are dense in every $H^s$.

Scalar spaces. For $s\in \mathbb{R}$ define ${\mathrm{\ell }}_s^2({\mathbb{Z}}^3;\mathbb{C})$ to consist of families $c:{\mathbb{Z}}^3\to \mathbb{C}$ with ${\sum }_k{w}_k^{2s}\mathrm{\mid }c(k){\mathrm{\mid }}^2<\mathrm{\infty }$, with pairing ${\sum }_k{w}_k^{2s}c(k)\overline{d(k)}$; its real subspace is specified by $c(-k)=\overline{c(k)}$. For $s\ge 0$, $H^s({\mathbb{T}}^3;\mathbb{R})$ consists of real scalar $L^2$ classes with this weighted coefficient norm; for $s<0$ it is the completion of real scalar trigonometric polynomials, identified with the real coefficient subspace. Scalar Fourier coefficients are $\widehat{f}(k)=\int f{e}_{-k}dx$ when $f\in L^1$, and otherwise denote the completion coordinates. Parseval and truncation give the scalar Hilbert-space and density assertions exactly as above. A vector belongs to $H^s({\mathbb{T}}^3;{\mathbb{R}}^3)$ exactly when each of its three scalar components belongs to $H^s({\mathbb{T}}^3;\mathbb{R})$, with $\mathrm{\parallel }u{\mathrm{\parallel }}_{H^s}^2={\sum }_{j=1}^3\mathrm{\parallel }{u}_j{\mathrm{\parallel }}_{H^s}^2$. Below the value type distinguishes scalar and vector $H^s$; ${\mathrm{\Lambda }}^s$, ${\mathrm{\partial }}_j$ and $\mathrm{\Delta }$ have the same multipliers for scalars, $\mathrm{\nabla }$ maps scalar to vector spaces, and $\mathrm{div}$ maps vector to scalar spaces.

The operators ${\mathrm{\Lambda }}^s$, ${\mathrm{\partial }}_j$, $\mathrm{\Delta }$, $\mathrm{\nabla }$ and $\mathrm{div}$. For $s,t\in \mathbb{R}$ and $u\in H^t$ define ${\mathrm{\Lambda }}^{s}u\in H^{t-s}$ by the coefficient rule $\widehat{{\mathrm{\Lambda }}^{s}u}(k):=w_k^s\widehat{u}(k);$ then ${\mathrm{\Lambda }}^s$ is a linear isometric isomorphism of $H^t$ onto $H^{t-s}$, ${\mathrm{\Lambda }}^0=\mathrm{i}\mathrm{d}$, ${\mathrm{\Lambda }}^s{\mathrm{\Lambda }}^t={\mathrm{\Lambda }}^{s+t}$, and ${\mathrm{\Lambda }}^{-s}$ is the inverse of ${\mathrm{\Lambda }}^s$. For $j=1,2,3$ and $u\in H^{s+1}$ define ${\mathrm{\partial }}_{j}u\in H^s$ by $\widehat{{\mathrm{\partial }}_{j}u}(k):=2\pi i{k}_j\widehat{u}(k);$ for $u\in H^{s+2}$ define $\mathrm{\Delta }u\in H^s$ by $\widehat{\mathrm{\Delta }u}(k):=-4{\pi }^2\mathrm{\mid }k{\mathrm{\mid }}_2^2\widehat{u}(k)$; for a scalar field $f$ with coefficients $\widehat{f}(k)\in \mathbb{C}$ in the scalar spaces just defined, define $\mathrm{\nabla }f\in H^s$ for $f\in H^{s+1}$ by $\widehat{\mathrm{\nabla }f}(k):=2\pi ik\widehat{f}(k)$; and for $u\in H^{s+1}$ define $\mathrm{div}u:={\sum }_{j=1}^3{\mathrm{\partial }}_j{u}_j\in H^s$. These operators satisfy $\mathrm{\parallel }{\mathrm{\partial }}_{j}u{\mathrm{\parallel }}_{H^s}\le \mathrm{\parallel }u{\mathrm{\parallel }}_{H^{s+1}}$, $\mathrm{\parallel }\mathrm{\Delta }u{\mathrm{\parallel }}_{H^s}\le \mathrm{\parallel }u{\mathrm{\parallel }}_{H^{s+2}}$, and for every integer $m\ge 1$, ${\mathrm{\Lambda }}^{2m}=(I-\mathrm{\Delta }{)}^m$, the identity following coefficientwise from $w_k^2=1+4{\pi }^2\mathrm{\mid }k{\mathrm{\mid }}_2^2$. On trigonometric polynomials all four operators agree with the classical coordinatewise derivatives and divergence computed from the smooth functions $x\mapsto e^{2\pi ik\cdot x}$, by the classical rules ${\mathrm{\partial }}_j{e}_k=2\pi i{k}_j{e}_k$; this is the convention in which the notation is used below.

Divergence-free fields and the Leray projection. For $s\in \mathbb{R}$ put $H_{\sigma }^s({\mathbb{T}}^3;{\mathbb{R}}^3):=\{u\in H^s:k\cdot \widehat{u}(k)=0\text{ for every }k\in {\mathbb{Z}}^3\};$ this is a closed real-linear subspace of $H^s$, and $\mathrm{div}u=0$ in $H^s$ for $u\in H_{\sigma }^{s+1}$. For $k\mathrm{\ne }0$ let ${\pi }_k\in L({\mathbb{C}}^3)$ be the orthogonal projection onto the hyperplane $\{k{\}}^{\perp }$, ${\pi }_{k}v:=v-\frac{k\cdot v}{\mathrm{\mid }k{\mathrm{\mid }}_2^2}k,$ and set ${\pi }_0:={\mathrm{i}\mathrm{d}}_{{\mathbb{C}}^3}$. Define the Leray projection $P$ by $\widehat{Pu}(k):={\pi }_k\widehat{u}(k)(k\in {\mathbb{Z}}^3)\mathrm{.}$ Then: $P$ maps $H^s$ linearly into $H_{\sigma }^s$, and $\mathrm{\parallel }Pu{\mathrm{\parallel }}_{H^s}\le \mathrm{\parallel }u{\mathrm{\parallel }}_{H^s}$; $P^2=P$; $P$ is self-adjoint on $H^s$, i.e. $⟨Pu,v{⟩}_{H^s}=⟨u,Pv{⟩}_{H^s}$ for all $u,v\in H^s$; $P$ commutes with each ${\mathrm{\Lambda }}^t$, with each ${\mathrm{\partial }}_j$, and with $\mathrm{\Delta }$ wherever these are defined; and the identity $\mathrm{div}(Pu)=0$ holds in $H^{s-1}$ for $u\in H^s$. Moreover every $u\in H^s$ has the orthogonal decomposition $u=Pu+\mathrm{\nabla }f,f\in H^{s+1},\widehat{f}(0)=0,⟨Pu,\mathrm{\nabla }f{⟩}_{H^s}=0.$ Here zero mean means zero coefficient at $0$ (the usual integral when the field is integrable); orthogonality for all real orders is in the $H^s$ pairing. The scalar element $f$ is determined by $\widehat{f}(0):=0$ and $\widehat{f}(k):={\displaystyle \frac{k\cdot \widehat{u}(k)}{2\pi i\mathrm{\mid }k{\mathrm{\mid }}_2^2}}$ for $k\mathrm{\ne }0$; here $f\in H^{s+1}$ because $w_k^{2(s+1)}\mathrm{\mid }\widehat{f}(k){\mathrm{\mid }}^2\le C{w}_k^{2s}\mathrm{\mid }\widehat{u}(k){\mathrm{\mid }}^2$ for $k\mathrm{\ne }0$, and $\widehat{\mathrm{\nabla }f}(k)=(k\cdot \widehat{u}(k)\mathrm{/}\mathrm{\mid }k{\mathrm{\mid }}_2^2)k=\widehat{u}(k)-{\pi }_k\widehat{u}(k)$. In particular $P\mathrm{\nabla }f=0$ for every $f$ with $\mathrm{\nabla }f\in H^s$, so $P$ is the orthogonal projection of $H^s$ onto $H_{\sigma }^s$. On smooth fields the decomposition $u=Pu+\mathrm{\nabla }f$ derived above from the Fourier coefficients is called the Hodge decomposition of the torus; the operator $P$ is the Leray projection of the fluid literature.

Time-dependent fields. For an interval $[0,T]$ with $T>0$ and $s\in \mathbb{R}$, let $C([0,T],H^s)$ denote the set of continuous maps from $[0,T]$ to the metric space $(H^s,\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{H^s})$, with $\mathrm{\parallel }u{\mathrm{\parallel }}_{C([0,T],H^s)}:={\sup }_{0\le t\le T}\mathrm{\parallel }u(t){\mathrm{\parallel }}_{H^s}$. For a map $u:[0,T]\times {\mathbb{T}}^3\to {\mathbb{R}}^3$ of class $C^{\mathrm{\infty }}$ we mean that every mixed partial derivative ${\mathrm{\partial }}_t^r{\mathrm{\partial }}_x^{\alpha }u$ exists and is continuous on $[0,T]\times {\mathbb{T}}^3$, with one-sided time derivatives at $t=0,T$.

• The whole definition is built on the imported torus Fourier theory, so the countable-choice assumption and the identification of $L^2$ classes with weighted coefficient families are inherited from The Fourier basis and Parseval's identity on the finite torus; no other choice principle is used.
• The weights are normalized so that differentiation costs one weight: the symbol of ${\mathrm{\partial }}_j$ is $2\pi i{k}_j$ with $\mathrm{\mid }2\pi k_j\mathrm{\mid }\le w_k$, while $w_k$ itself is the symbol of ${\mathrm{\Lambda }}^1$ and $w_k^2$ is the symbol of $I-\mathrm{\Delta }$.
• The definition of $H^s$ is the Fourier-coefficient definition used in the fluid literature on the torus; the equivalent description by weak derivatives is not used anywhere below, so no extension theory is presupposed.