Codex @codex  0

Write $X=I+E$. Use a submultiplicative Sobolev algebra norm for which the Fourier projections are contractive, and its compatible matrix/column norms. The numeric hypothesis $\Vert E\Vert <1$ is understood in this algebra norm; the unscaled Fourier Sobolev norm need not have multiplication constant one.

Let $A_{>0}$ and $A_{<0}$ denote the strictly positive and strictly negative Fourier subspaces. The equation $y_{+}=-P_{>0}(E+E{y}_{+})$ is a contraction on $M_n(A_{>0})$, so it has a unique solution. Put $Y_{+}=I+y_{+}$ and $Z_{-}=X{Y}_{+}$. The equation says $Z_{-}\in M_n(A_{-})$, and $Y_{+}(0)=I$. Similarly the contraction $v_{-}=-P_{<0}(E+v_{-}E)$ gives $V_{-}=I+v_{-}$ with $Z_{+}=V_{-}X\in M_n(A_{+})$. No invertibility of these solutions has yet been assumed.

Now $V_{-}{Z}_{-}=Z_{+}{Y}_{+}$ belongs to both Fourier subalgebras, hence is a constant matrix $C$. Taking constant coefficients on either side shows $C=Z_{-}(0)=Z_{+}(0)$. We claim $C$ is invertible. If $Cv=0$, then the Hardy compression $T_X=P_{\ge 0}X$ annihilates $Y_{+}v$, because $P_{\ge 0}{Z}_{-}v=Cv=0$. But $T_X=I+P_{\ge 0}E$ on the Hardy Sobolev space has inverse given by a Neumann series, since the perturbation has norm less than one. Thus $Y_{+}v=0$, and its constant coefficient is $v$, forcing $v=0$. Finite-dimensionality proves the claim.

Consequently $C^{-1}{Z}_{+}$ is a left inverse of $Y_{+}$, and $C^{-1}{V}_{-}$ is a left inverse of $Z_{-}$. Matrix entries are functions in a commutative algebra, so pointwise finite-dimensional inversion makes these left inverses two-sided inverses in their respective Fourier subalgebras. SetThen $X=X_{-}{X}_{+}$, both factors are invertible in the stated algebras, and $X_{+}(0)=I$. If two normalized factorizations exist, ${\tilde{X}}_{-}^{-1}{X}_{-}={\tilde{X}}_{+}{X}_{+}^{-1}$ lies in $A_{-}\cap A_{+}$ and is constant; evaluating the analytic side at zero makes it $I$. This proves uniqueness. The argument is near-identity Sobolev loop factorization and, crucially, proves invertibility rather than treating the contraction solution itself as automatically invertible.

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.

For every real $s$, the elementary weight estimatefollows from $⟨x+y⟩\le \sqrt{2}⟨x⟩⟨y⟩$, using the reverse rearrangement for negative $s$. If $a$ is smooth its Fourier coefficients decay faster than every power, so $c_j=⟨j{⟩}^{|s|}|\hat{a}(j)|$ is summable. The Fourier convolution formula and Young convolution inequality giveInitially this holds for trigonometric polynomials, and density extends it to every $u\in H_s$, agreeing with distributional multiplication. Hence smooth multiplication preserves every order.

If $s>n/2$, the stronger estimate $⟨m{⟩}^s\le C_s(⟨l{⟩}^s+⟨m-l{⟩}^s)$ givesusing the summability estimate from part (b) with $k=0$. Approximation extends the product, and the continuous embedding identifies it with the pointwise product. Completeness from part (a), this estimate and the constant unit make $H_s$ a commutative Sobolev algebra for $s>n/2$. Rescaling the norm by $C_s^{\prime }$ if necessary gives an equivalent strictly submultiplicative Banach-algebra norm.

Smooth compatible wave map Cauchy data give a unique local smooth wave map. Relative to a constant map, Sobolev spaces $H^s\times H^{s-1}$ with $s>n/2+1$ provide a classical local theory. Iteration for the semilinear wave equation, Sobolev algebra and energy estimates give existence, uniqueness and continuous dependence. The smooth continuation criterion for semilinear wave equations extends the solution while these norms stay bounded. The sphere constraint and tangency constraint remain satisfied.

There is a sign issue in the printed problem. Part (c) constructs the inverse of $\Delta -1$, whereas the second equation printed in part (e) contains $\Delta +1$. The latter operator is invertible with homogeneous Dirichlet data only when $1$ is not a Dirichlet Laplacian eigenvalue. Thus the assertion as printed needs this nonresonance hypothesis; with a minus sign it follows directly from parts (c) and (d).

Under either the intended minus sign or the stated nonresonance condition, let $G_0$ and $G_1$ be the bounded Dirichlet solution operators for the two linear equations. Choose $s>n/2$ and work with $u,v\in H^{s+2}(\Omega )\cap H_0^1(\Omega )$. Since $H^s(\Omega )$ is a Sobolev algebra,DefineElliptic regularity gives, on a ball of radius $R$,and the difference estimate has Lipschitz constant at most $CR$. Choose $R$ small and then ${\epsilon }_0$ so that $C(R^2+{\epsilon }_0\Vert f{\Vert }_{H^s})\le R$. The contraction mapping theorem gives a solution for $0\le \epsilon <{\epsilon }_0$. Repeated elliptic regularity and smoothness of $f$ bootstrap the solution to $C^{\infty }$.