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On the torus ${\mathbb{T}}^n=(\mathbb{R}/2\pi \mathbb{Z}{)}^n$, choose normalized Haar measure, so that $e_k(x)=e^{ik\cdot x}$, $k\in {\mathbb{Z}}^n$, is an orthonormal basis of $L^2$. A periodic Sobolev space is defined, for any real $s$, byFor negative $s$, the coefficients describe distributions rather than necessarily functions. For a nonnegative integer $s$, the weight is comparable to ${\sum }_{|\alpha |\le s}|k^{\alpha }{|}^2$, so this norm is equivalent to the sum of the squared $L^2$ norms of all weak derivatives through order $s$. Fourier truncation proves that smooth trigonometric polynomials are dense. The $L^2$ pairing identifies the continuous dual of $H^s$ with $H^{-s}$, giving Sobolev duality.

Two Fourier estimates explain the usefulness of these spaces. If $s>m+n/2$, the Cauchy-Schwarz inequality givesUniform convergence of the differentiated Fourier series proves the Sobolev embedding $H^s\hookrightarrow C^m$. If $\delta >0$, a bounded subset of $H^{s+\delta }$ has uniformly small high-frequency tails in $H^s$:Finite-dimensional Fourier truncation therefore proves the compact embedding $H^{s+\delta }\hookrightarrow H^s$, the periodic version of Rellich-Kondrachov compactness theorem. In addition, for $s>n/2$ multiplication is continuous $H^s\times H^s\to H^s$, making it a Sobolev algebra. Indeed the weighted coefficient of a product is bounded by the sum of the convolutions of $⟨k{⟩}^s|\hat{u}(k)|$ with $|\hat{v}(k)|$ and of $|\hat{u}(k)|$ with $⟨k{⟩}^s|\hat{v}(k)|$; Cauchy-Schwarz puts unweighted coefficients in ${\ell }^1$, and the ${\ell }^1\ast {\ell }^2\to {\ell }^2$ case of Young's convolution inequality finishes the estimate.

For eigenfunction expansions, distinguish ellipticity from self-adjointness. An arbitrary non-self-adjoint elliptic operator need not have an orthonormal basis of eigenfunctions. A useful definite setting iswith smooth real periodic coefficients, $a_{ij}=a_{ji}$, real $V$, and uniform ellipticityTake the self-adjoint realization in $L^2$ with domain $H^2$. Its construction and spectral properties can be seen from the associated form on $H^1$,For a sufficiently large constant $C$, $b_C=b+C(\cdot ,\cdot )$ is a positive coercive form: $b_C(u,u)\ge c_1\Vert u{\Vert }_{H^1}^2$, and $L+C\ge I$. The Lax-Milgram theorem supplies the unique weak solution of $(L+C)u=f$ for $f\in H^{-1}$; the form is Hermitian, which gives symmetry of its inverse on $L^2$.

The regularity input is the periodic elliptic estimateHere is why an elliptic operator gains two derivatives. With coefficients frozen at a point, its principal symbol $\sum a_{ij}{\xi }_i{\xi }_j$ dominates $c|\xi {|}^2$, and the estimate follows by multiplying Fourier coefficients by that symbol. On a sufficiently small coordinate patch, the difference of the leading coefficients from these constants has arbitrarily small supremum norm. In the integer-order differentiated estimate, this absorbs the highest derivatives into the left side; derivatives falling on coefficients contribute only lower-order terms. A partition of unity reduces the variable-coefficient problem to such patches. Its commutators with $L$ have order one, controlled bywhich follows by splitting low and high Fourier modes. Absorbing these terms gives the displayed estimate. For a weak $H^1$ solution, the same localization with difference quotients first bounds its second weak derivatives for $f\in L^2$; differentiating and repeating then gives $f\in H^s\Rightarrow u\in H^{s+2}$ for nonnegative integer $s$. Interpolation and duality extend the scale to real orders. In particular $(L+C{)}^{-1}:L^2\to H^2$ is bounded.

Let $R=(L+C{)}^{-1}$. Compactness of $H^2\hookrightarrow L^2$ makes $R$ a compact operator. Symmetry of the form makes $R$ self-adjoint, and positivity of $L+C$ makes $R$ positive and injective. Its range is dense because $\ker R^{\ast }=\ker R=0$. The spectral theorem for compact self-adjoint operators consequently gives an orthonormal basis ${\varphi }_j$ withElliptic regularity bootstraps every ${\varphi }_j$ to a smooth function. Each eigenvalue has finite multiplicity. Thus a self-adjoint uniformly elliptic second-order operator on the torus has a complete smooth orthonormal eigenfunction expansion with discrete eigenvalues tending to infinity.

Writing $u_j=(u,{\varphi }_j)$, every $u\in L^2$ has $u={\sum }_j{u}_j{\varphi }_j$ in $L^2$, with $\Vert u{\Vert }_2^2={\sum }_j|u_j{|}^2$. Iterating the elliptic estimate shows that the graph norm of $(L+C{)}^m$ is equivalent to the $H^{2m}$ norm. Interpolation and duality give the more general spectral descriptionThe comparison constants depend on the operator and $s$. In particular smooth functions have coefficients decaying faster than every power of ${\lambda }_j+C$, and their expansions converge in every Sobolev norm and hence in every $C^m$ norm.

This gives a practical way to solve elliptic equations. The equation $Lu=f$ is solvable if and only if $f_j=0$ whenever ${\lambda }_j=0$; then one may take $u_j=f_j/{\lambda }_j$ on the other eigenspaces and add an arbitrary vector in $\ker L$. The Sobolev description proves the two-derivative gain and the Fredholm alternative, rather than leaving them as formal series manipulations. Similarly the solution of the heat equation ${\partial }_{t}u+Lu=0$ isFor every $t>0$ the exponential decay at high eigenvalues places this solution in every Sobolev space, hence it is smooth.

For the Laplacian $-\Delta$ the eigenfunctions are precisely $e^{ik\cdot x}$ with eigenvalues $|k{|}^2$. The number with $|k{|}^2\le R$ is the number of lattice points in a radius-$\sqrt{R}$ ball, asymptotic to its volume by comparison with unit cubes. For the general self-adjoint operator above, bounds on its form above and below by positive multiples of $\Vert \nabla u{\Vert }_2^2$ plus constants, together with the Courant–Fischer min-max principle, imply ${\lambda }_j+C\asymp j^{2/n}$. Thus the growth reflects dimension and the two-derivative order. Compactness of the torus is essential: on ${\mathbb{R}}^n$, translated bumps prevent the compact embedding, and the Laplacian instead has continuous spectrum described by the Fourier transform.

Use ${\mathbb{T}}^n={\mathbb{R}}^n/(2\pi \mathbb{Z}{)}^n$ with normalized Haar measure and Fourier basis $e_k(x)=e^{ik\cdot x}$, $k\in {\mathbb{Z}}^n$. For any real $s$, the periodic Sobolev space isWeighted Fourier coefficients identify it isometrically with ${\ell }^2$, so it is a Hilbert space and complete. This definition includes negative orders and distributions. For nonnegative integer $s$ the norm is equivalent to the sum of squared L2 norms of weak derivatives through order $s$.

Start with $u\in H_s$ for some real $s$. The equation becomesSmooth $f$ belongs to every periodic Sobolev space, and multiplication by the smooth coefficient preserves $H_s$ by part (c), so the right side lies in $H_s$. The Fourier multiplier $(1+\Delta {)}^{-1}$ has coefficient $(1+|m{|}^2{)}^{-1}$ and maps $H_s$ isometrically to $H_{s+2}$. Thus $u\in H_{s+2}$. Repeat this argument to get $u\in H_{s+2j}$ for every nonnegative integer $j$. For any derivative order $k$, choose $j$ with $s+2j>k+n/2$ and apply part (b). Therefore $\boxed{u\in C^{\infty }({\mathbb{T}}^n)}$. This is elliptic bootstrapping directly from the Fourier multiplier, with no assumption that the solution initially has nonnegative Sobolev order.

Use ${\mathbb{T}}^n={\mathbb{R}}^n/(2\pi \mathbb{Z}{)}^n$ and the periodic Sobolev space conventionThe Cauchy-Schwarz inequality givesTo check the last sum, partition the lattice into $|k|<1$ and dyadic shells $2^j\le |k|<2^{j+1}$. A shell contains at most $C_n{2}^{jn}$ points and contributes at most $C_n{2}^{j(n-2s)}$. This is a convergent geometric series exactly when $s>n/2$.

Thus the Fourier series ${\sum }_k\hat{f}(k)e^{ik\cdot x}$ converges absolutely and uniformly on the torus. Its sum is a continuous function and has the original Fourier coefficients, so it represents the original periodic distribution. In particular the representative is unique andThis proves the requested inclusion, with its continuous Sobolev embedding estimate.

Interpret $H_{-\infty }({\mathbb{T}}^n)={\bigcup }_{s\in \mathbb{R}}{H}^s({\mathbb{T}}^n)$, the periodic distributions of finite negative Sobolev space order. Smooth multiplication is bounded on every periodic Sobolev space. One direct proof uses the rapid decay of the Fourier coefficients of $V$: the inequalityreduces the weighted convolution formula for $\widehat{Vf}$ to an ${\ell }^1\ast {\ell }^2$ estimate. Thus $\Vert Vf{\Vert }_{H^s}\le C_{V,s}\Vert f{\Vert }_{H^s}$ for every real $s$.

Suppose initially $f\in H^s$. The equation gives $\Delta f=u-Vf\in H^s$, since a smooth function belongs to every periodic Sobolev space. For $k\ne 0$, the Fourier multiplier of $\Delta$ is $|k{|}^2$, andThe zero coefficient is already finite. Hence $f\in H^{s+2}$. Iterating the argument gives $f\in H^{s+2j}$ for every $j\ge 0$. Applying the Sobolev embedding theorem also to each distributional partial derivative produces a smooth representative. Therefore $\boxed{f\in C^{\infty }({\mathbb{T}}^n)}$. This is an explicit elliptic regularity bootstrap, valid for complex or real $V$ and without a sign assumption on it.

Let $\mathbb{T}=\mathbb{R}/(2\pi \mathbb{Z})$ and define the positive self-adjoint operatoron $L^2(\mathbb{T})$. On the Fourier mode $e^{inx}$ it acts by multiplication by $\sqrt{1+n^2}$. ConsequentlyThe energy space is the periodic Sobolev spacewith inner productIf $u=\sum u_n{e}^{inx}$ and $v=\sum v_n{e}^{inx}$, thenwhich is precisely the stated energy norm.

With $v=u_t$, the periodic Klein-Gordon equation becomesFor $AZ$ to belong to $H^1\times L^2$, one needs $v\in H^1$ and $u\in H^2$. Thus