A normal gauge that keeps the boundary condition

The weak-source lift removes the natural-dual forcing without discarding its boundary action. The next operation removes the first normal derivative from the interior equation. With matrix coefficients these two operations leave a complex Robin condition. We construct the matrix gauge, regularize that condition exactly, and prove the full normal-commutator identity, including its boundary form and its matrix lower-order terms.

The identity first holds on actual twice weakly differentiable approximants. We then prove its weak graph extension using complementary Sobolev dual pairings, including the normal trace. The weak graph identity alone does not supply higher microlocal estimates, incoming-error bounds or strict-diffractive propagation.

1. The problem and the exact earlier proofs

N0. Conventions and inputs. Work in a flat collar $X=[0,\infty)_x\times\mathbb R^d_z$, with values in $\mathbb C^N$, finite $N$, and Lebesgue density. We use $D_j=-i\partial_j$; all inner products are linear in their first entry. Tangential adjoints are actual adjoints in $L^2_z$, with $x$ as a parameter. Coefficients are smooth up to the boundary with all derivatives bounded on the localized collar. After extension they may be taken bounded with all derivatives on the half-space. Every function to which the identity is applied has bounded normal support. Tangential pseudodifferential symbols are classical $S^m_{1,0}$ with uniform bounds for the required normal derivatives.

The normalized form retained from WSL:L0 is

For zero-trace tests the interior equation is

Exact dependencies are collected in the proof map. WSL:L6 supplies graph density and the signed Green formula. SY:H1–H2 supplies Fourier inversion, Plancherel and all real Sobolev spaces. SY:H7–H8 supplies ordered composition and actual adjoints, with finite-seminorm remainders; SY:H10 supplies their Sobolev bounds. BF:A0 supplies the two energy domains and their smooth density. These are complete earlier programme proofs, not external theorem citations.

2. An ordered matrix gauge

N1. Construction and inverse. On the chosen collar solve

The right equation $H_x=-HF$, $H(0)=I$, has the same construction, with right multiplication and its corresponding order. Differentiation gives $(HS)_x=0$, hence $HS=I$. A square finite-dimensional matrix with a left inverse is injective and therefore surjective, by elementary elimination; thus $H=S^{-1}$. This proves invertibility and $\|S\|,\|S^{-1}\|\le e^{Ma}$. Uniqueness also follows directly: the difference of two solutions is bounded by $M\int_0^x\|W(t)\|dt$; iterating this inequality $n$ times gives $\sup\|W\|(Ma)^n/n!$, which tends to zero. Extend $K$ to vanish past the collar if a half-space operator is wanted; $S$ is then constant in $x$ there. No self-adjointness or simultaneous diagonalization is used.

N2. Transform both slots of the weak form. Set $v=Su$, $\phi=S^{-*}\psi$, and $q'(u,\psi)=q(Su,S^{-*}\psi)$. Multiplication by either smooth bounded invertible matrix is an isomorphism on $H^1$ and on $H^1_0$, by the product rule, the inverse bounds and smooth density. For the transformed antidual use exactly $g'(\psi)=g(S^{-*}\psi)$. For an interior distribution this is multiplication $g'=S^{-1}g$.

Write $g_{xx}=1$, $g_{xa}=g_{ax}=0$, $g_{ab}=h^{ab}$. Expanding both slots, without commuting matrices, gives the same scalar principal form and

3. Boundary traces through the gauge

N3. The retained Robin condition. We first record the positive trace as well as the earlier negative trace. Put $\Lambda=(1+|D_z|^2)^{1/2}$. For smooth compactly supported $u$, integrate the derivative of $\|\Lambda^{1/2}u(x)\|^2$, using tangential Plancherel:

Smooth multiplication commutes with both traces. For the graph trace, the product rule bounds $\partial_x(MN)=(\partial_xM)N+M\partial_xN$ in $L^2_xH^{-1}_z$; SY:H10 bounds multiplication on $H^{-1}$ and $H^{-1/2}$. Apply the smooth identity to graph-dense approximants. For the positive trace use $H^1$ density. Therefore the exact flux factorization

For later use, the same density argument in $H^2$ applies to each first derivative. Its traces lie in $H^{1/2}$ by NC8. Tangential differentiation commutes with trace on smooth approximants; Fourier weights then show $u|_0\in H^{3/2}$, because the sum of its $H^{1/2}$ norm and those of its tangential first derivatives is equivalent to its $H^{3/2}$ norm. Also $D_xu|_0\in H^{1/2}$. Density in $H^2$ follows by inward translation first, then ordinary convolution at a smaller radius and cutoffs, with convergence of every derivative of order at most two. No boundary condition is required for this density statement.

4. Smoothing with an exact boundary correction

N4. Localization and the Gaussian commutators. Choose $\omega(x,z)=\chi_x(x)\chi_z(z)$ of compact support, equal to one on a smaller collar, with $\partial_x\omega(0,z)=0$. Let $w=\omega u$ and $H=P'u$. Its equation is exactly

For $0<\epsilon\le1$ let $J_\epsilon=e^{-\epsilon^2|D_z|^2}$, a scalar multiplier independent of $x$, and $j_\epsilon(\eta)=e^{-\epsilon^2|\eta|^2}$. Plancherel and dominated convergence give strong convergence to identity in every fixed $H^t_z$, and also in the corresponding $L^2_xH^t_z$ spaces. Every derivative of the Gaussian has the form $\epsilon^{|\alpha|}p_\alpha(\epsilon\eta)e^{-\epsilon^2|\eta|^2}$. Splitting at $\epsilon\langle\eta\rangle=1$ proves

In the ordered composition formula SY:H7, the zero-order product cancels in a commutator with a scalar multiplier. The remainder with one derivative, including its full finite-seminorm estimate and actual operator interpretation SY:H8, therefore gives

N5. Actual $H^2$ approximants and their full equations. In the Robin case put $\beta_\epsilon=[\kappa_0,J_\epsilon]\gamma w$, where $\gamma w=w|_0$. Choose a fixed smooth compactly supported normal function $\theta$, equal to one near zero, and define

The graph trace commutes with $J_\epsilon$ by its bounded action and graph-density approximation. Hence the Robin defect of $J_\epsilon w$ is precisely $\beta_\epsilon$. Since $C_\epsilon|_0=0$ and $D_xC_\epsilon|_0=-\beta_\epsilon$,

5. The full normal commutator

N6. An exact identity including the matrix lower terms. In this section write $P=D_x^2-R$ for the transformed operator. Decompose $R=R_h+iR_a$, with $R_h=(R+R^*)/2$ and $R_a=(R-R^*)/(2i)$. Both are tangentially self-adjoint; $R_h$ has order two and scalar real principal symbol $rI$, and $R_a$ has order at most one.

Choose scalar, componentwise operators $A=A^*\in\Psi^{-1}$, $C=C^*\in\Psi^0$, smoothly depending on $x$. A dot means $\partial_x$. Set

Here is the full calculation. Prove it first for smooth inputs compactly supported in the closed half-space. The signed Green identity is $(f,D_xh)_X=(D_xf,h)_X-i(f(0),h(0))_\partial$. Consequently $2\operatorname{Im}(h,Qh)_X=-(Ah(0),h(0))_\partial$. Apply Green once to $(D_x^2u,Qu)$, use $[D_x,Q]=-i(\dot A D_x+\dot B)$, and then apply this last formula to $h=D_xu$. The resulting boundary terms are $(Au_1,u_1)+2\operatorname{Re}(Bu_0,u_1)$; the volume terms are $2(\dot A u_1,u_1)+2\operatorname{Re}(\dot B u,u_1)$.

For the real part of $R$, move $A$ tangentially and split $AR_h=S_0+J_0$. Direct differentiation gives

Finally the part $iR_a$ of $R$ contributes

N7. Principal polynomial and the surviving boundary form. Let $a=\sigma_{-1}(A)$, $c_0=\sigma_0(C)$, $k=\sigma_1(R_a)$, so $k=k^*$ is a matrix. For the tangential Poisson bracket use $\{f,g\}_z=f_\eta g_z-f_zg_\eta$, with contraction of indices. SY:H7 applied to NC19 gives

In the Dirichlet case the boundary form is exactly $(Au_1,u_1)_\partial$. Choosing $A=-T^*T$ with $T\in\Psi^{-1/2}$ gives $-\|Tu_1\|^2$, including all lower terms. In the Robin case $u_1=-\kappa_0u_0$, so instead

N8. The identity on the actual weak graph. Suppose $u\in H^1(X)$ has bounded normal support and $Pu\in L^2_xH^{-1}_z$. Coefficients retain all the bounds above. Put $b_0=\gamma u\in H^{1/2}_z$ and $b_1=\gamma D_xu\in H^{-1/2}_z$; the latter is defined because $D_x^2u=Ru+Pu\in L^2_xH^{-1}_z$. For an operator of order $m$, SY:H10 gives every required tangential mapping. In particular,

To prove NC27, take the approximants of N5 with $w=u$. Its construction and estimates use precisely these global graph norms; the localization NC11 is needed only when starting from a local solution. If no boundary condition is specified, use $J_\epsilon u$ alone. If one is specified, retain the exact Dirichlet or Robin correction. In all cases,

For a Dirichlet solution and $A=-T^*T$, the remaining boundary term is exactly $-\|Tb_1\|_{L^2_z}^2$: $T:H^{-1/2}\to L^2$, and adjoint duality follows by Sobolev density from SY:H2 and SY:H8. For a Robin solution, $b_1=-\kappa_0b_0$ in $H^{-1/2}$, and NC25 is valid as the $H^{-1/2},H^{1/2}$ pairing of its order-one operator. This proves the identity in both weak domains with the actual complex boundary coefficient. It does not bound that Robin form by a favorable sign, nor supply the higher microlocal estimate.

6. Three solved checks of the mechanism

Exercise 1. A nonunitary matrix gauge

Take $E=\begin{pmatrix}0&1\\0&0\end{pmatrix}$, $K=2iE$, $m_x=M=\begin{pmatrix}i&1\\0&-2\end{pmatrix}$ and $\ell_x=K-M$. Compute the gauge and the Robin matrix.

Solution. Since $E^2=0$, the series NC4 stops at its linear term: $S=I+xE$, $S^{-1}=I-xE$. Substitution gives $2D_xS+KS=-2iE+2iE=0$. Direct ordered multiplication gives

Exercise 2. Smoothing a Robin mode

On the circle with normalized measure take $\kappa_0(z)=e^{iz}$ and a boundary value $b(z)=e^{inz}$, $n\in\mathbb Z$. Find the defect of Gaussian smoothing and cancel it. The periodic Fourier proof is the same finite-mode calculation as SY:H1; no periodic pseudodifferential theorem is needed in this example.

Solution. Multiplication raises the frequency by one. Thus

Exercise 3. Check the normal orientation and the imaginary coefficient

Suppress $z$, take $P=D_x^2-(r+ib)$, $A=-1$, $C=c_0$, with real constants, and let $u(x)=e^{itx}f(x)$, where $t\in\mathbb R$ and $f$ is any real smooth rapidly decreasing profile. Find both sides of NC20 explicitly.

Solution. Here $B=c_0$, $S_0=-r$, $J_0=0$, $E_{10}=E_{01}=b$, $E_{00}=-2c_0b$, and $E_{11}=0$. Since $D_xu=e^{itx}(tf-if')$,

Left: the exact real and imaginary parts of the off-diagonal coefficient in NC29. Middle: the exact defect coefficient in NC30 for $n=2$, with its sign fixed by the order of the commutator. Right: the three boundary forms from Exercise 3 at $r=2,t=0$; the volume contribution is zero when $b=0$. These are exact coefficient and profile calculations, not numerical solutions of a diffraction problem. The reproducible figure is tied to N1–N8 and the three solutions.

7. Credit, validation and the remaining estimate

Lars Hörmander, The Analysis of Linear Partial Differential Operators III: Pseudo-Differential Operators, corrected 2007 printing, ISBN 978-3-540-49938-1, Section 24.4, pp. 443–444 and 449, is the source for the scalar normal-gauge reduction, normal energy identity and the use of regularization. The present proof writes out the ordered matrix gauge, the transformed weak form, its complex Robin condition, the boundary-preserving Gaussian correction, the exact Hermitian operator array and its weak graph extension. It does not attribute the full matrix Robin argument to a scalar Dirichlet statement in the book. The earlier source-lift proof and Hilbert coefficient calculus retain their own source and component notices.

The prerequisite proof index gives the earlier results used here. Independent new exposition, code and illustration are CC0-1.0. Linked components retain their own licences.

The uniform commutant estimate for the full weak problem requires bounds for the source and every regularization defect, the Robin boundary contribution, characteristic-square control of energy regularity, incoming terms and a justified higher-order limit on a common microlocal neighborhood. The characteristic-symbol, weak-quadratic and source-lift supplements provide the earlier pieces.