Matrix damping in the boundary commutator estimate

The normal-gauge supplement proves the exact commutator identity on the actual weak graph. This supplement absorbs its complex matrix lower coefficients and a uniform class of regularization errors. It proves a mixed half-order estimate with the incoming term and the full boundary form retained. For Dirichlet data the boundary square has a favorable sign. For Robin data the order-one boundary form remains an explicit term; its control is a further part of strict propagation.

The proof uses the scalar characteristic completion with a positive matrix weight. It does not infer positivity of a block matrix merely from positivity of its matrix-valued polynomial on real normal covariables. Exercise 2 gives a counterexample to that inference.

1. The equation, the error family and the earlier proofs

D0. Exact setting. Use $X=[0,\infty)_x\times\mathbb R^d_z$, finite coefficients $\mathbb C^N$, $D=-i\partial$, and inner products linear first. All operators have fixed localization and uniform coefficient and symbol bounds, including the required $x$ derivatives. Define

Let $Z_\epsilon\in\Psi^1$, $0<\epsilon\le1$, be a bounded family, and suppose

The proof uses SDC:S5–S8, the complete scalar characteristic completion and two commutant construction; SDQ:W1–W6, the weighted block estimate, legal order-minus-one test and elliptic pair; NCQ:N6–N8, the full matrix normal identity and its weak graph extension; and SY:H7–H10/H14, the ordered symbol calculus, Sobolev bounds and matrix sharp Gårding estimate. Exact sources and proof locators are in the proof map.

2. A positive matrix weight with uniform derivatives

D1. Uniform square root. Choose a Hermitian symbol $k_\epsilon\in S^1$ representing $R_a+Z_{\epsilon a}$ modulo a bounded $\Psi^0$ family. Such a representative is obtained by taking the Hermitian part of a full left symbol: the actual self-adjointness and SY:H7 show that the difference has order zero. On $|\eta|\ge1$, put $\lambda=|\eta|$ and choose a fixed $L$ with $\|k_\epsilon\|\le L\lambda$. Take $M\ge4L+1$, and set

Here is a direct proof of smoothness and the uniform symbol bounds. Put $E_\epsilon=2k_\epsilon/(M\lambda)$, so $\|E_\epsilon\|\le1/2$, and use

A derivative of fixed total order $j$ of $E_\epsilon^n$ is a sum of at most $C_jn^j$ ordered products. At most $j$ factors are differentiated; the others have norm at most $1/2$. The differentiated factors obey their uniform $S^0$ bounds, with one inverse power of $\langle\eta\rangle$ per frequency derivative. The resulting majorant $C_jn^j2^{-(n-j)}$ is summable. Termwise differentiation proves every claimed bound, without commuting different derivatives of $E_\epsilon$. A fixed smooth low-frequency extension preserves these bounds; low-frequency changes contribute uniformly smoothing terms in subsequent calculations.

3. The scalar completion carries the matrix weight

Fix one of SDC:S8's scalar symbols $q=a\xi+c$, with $a$ of degree $-1$, $c$ of degree zero, and $q=V^2$ on $p=\xi^2-r=0$. Choose actual scalar operators

D2. Weighted damping inequality. There are bounded families of matrix operators $\Phi_{\epsilon j}\in\Psi^{1/2-j}$ and $G_\epsilon=G_\epsilon^*\in\Psi^{-1}$ such that

Apply SDC:S5–S6 to the scalar polynomial $-\lambda q$, whose value on the characteristic roots is $-(\lambda^{1/2}V)^2$. It supplies real scalar $\phi_j$ and $g_0$, of degrees $1/2-j$ and $-1$, such that

Quantize $s_\epsilon\phi_j$ to obtain $\Phi_{\epsilon j}$, and symmetrize the quantization of $g_0h_\epsilon$ to obtain $G_\epsilon$. The full damping form $\mathcal D_\epsilon[u]=\operatorname{Re}(H_\epsilon u,Qu)$ has the exact Hermitian array

The order-minus-one test argument W3–W4 also works for a matrix $G_\epsilon$. Its proof uses actual self-adjointness, scalar principal coefficients and ordered matrix Sobolev bounds; it never interchanges $G_\epsilon$ with a lower matrix. It gives $G_\epsilon:V\to V$, $\|G_\epsilon u\|_{H^1}\le C\|u\|_{\mathcal X_0}$, the exact principal-form identity WQ14, and $|q_1(u,G_\epsilon u)|\le C\|u\|_{\mathcal X_0}^2$. The graph trace inequality WSL:L6 applied to $G_\epsilon u$, or NC8, bounds its $H^{1/2}$ trace by the same norm.

Apply MD5 to this legal test. The $g$ and $\beta$ terms have the displayed bound in MD9. Also

4. Combine damping with the weak normal identity

D3. The complete estimate with the incoming term. SDC:S8 constructs $q$ for the fixed $M$ above, with

Let $\mathcal E[u]$ be the full volume array NC19. The exact weak identity gives $\mathcal E[u]=2\operatorname{Im}(g+Z_\epsilon u,Qu)-\mathcal B(u)$. The correction $\mathcal Z_\epsilon[u]=2\operatorname{Im}(Z_\epsilon u,Qu)$ has the exact Hermitian array

Apply SDQ:W5 to this complete array and the actual form equation MD5. It gives scalar $\Psi_j\in\Psi^{1/2-j}$, with symbols $\psi,\partial_\xi\psi$ at the strict anchor, and a legal scalar test $G_1\in\Psi^{-1}$. Its source term is bounded exactly as in MD13 and the preceding trace estimate. Moreover $\|Qu\|\le C\|u\|_{\mathcal X_0}$, from $A$ of order minus one and $B$ of order zero. Substituting the exact normal identity and then D2 yields

For Dirichlet data $\mathcal B(u)=-\|T b_1\|^2$, exactly. Move this term to the left in MD17. For the two independent commutants in SDC:S8, SDQ:W6 also bounds $\|Ku\|_{\mathcal X_{1/2}}^2$ by the sum of the two $\Psi$ squares and the lower $\mathcal X_0$ norm. This is a uniform half-order mixed estimate with the incoming terms retained. Identification with the required higher energy-b norm and propagation from incoming data require their separate proofs.

5. The Robin defect is separate from its homogeneous boundary form

D4. Exact defect expansion and its bound. In the natural domain write $b_1=\beta-\kappa_0b_0$. The full boundary form is

There is a lower trace bound that is useful for regularization:

D5. The scalar weight's Robin defect is uniformly lower order. For fixed real $s$, put

6. The actual conjugation error of the weight

D6. Uniform symbol, leading sign and complete remainder. For a tangential differential $R$ of order two with smooth bounded coefficients and real scalar homogeneous principal symbol $r$, define on Schwartz functions

Here are the uniform remainder details. Every ratio $(\partial_\eta^\alpha a_\epsilon)/a_\epsilon$ has uniform order $-|\alpha|$. For the rational factor this follows by repeated differentiation of its logarithm and products of derivatives; the displayed first logarithmic derivative has order minus one, and its successive derivatives have the corresponding negative orders. The factor $\langle\eta\rangle^s$ obeys the same rule. Also

More explicitly, if $r_{\rm full}(x,z,\eta)$ is the full differential symbol, the exact product-minus-$R$ amplitude is

The degree-one part uses only the real scalar principal $r$; differentiating the order-one part of $r_{\rm full}$ contributes order zero. Since $D_z=-i\partial_z$, this proves MD24. Its leading symbol is purely imaginary scalar. The adjoint expansion SY:H7 changes it to its complex conjugate modulo order zero, proving MD2. The exact identity $PA_\epsilon u=A_\epsilon Pu+Z_\epsilon A_\epsilon u$ holds on Schwartz functions by construction and on every compatible Sobolev space by SY:H8–H10 and density. Any additional localization commutator remains a separate term.

D7. The actual weighted input and every cutoff term. Suppose $u\in\mathcal X_s$, $Pu\in L^2_xH^{s-1}_z$, and the original local solution is in the energy domain. Choose a fixed scalar product cutoff $\chi(x,z)$, with $\partial_x\chi(0,z)=0$, supported inside the normalized patch. Extend the coefficients smoothly with bounded derivatives there, and put $u_\epsilon=A_\epsilon(\chi u)$. Its exact equation is

For a zero original trace, $\gamma u_\epsilon=0$ by weighted graph trace commutation. Such an $H^2$ function belongs to $H^1_0$: $u_\epsilon(x)=\int_0^x\partial_xu_\epsilon(t)\,dt$ in $L^2_z$, so its $L^2$ norm on $0<x<\delta$ is bounded by $\delta\|\partial_xu_\epsilon\|_{L^2(0,\delta)}$. A cutoff vanishing for $x<\delta/2$ and equal to one for $x>\delta$ therefore converges in $H^1$, including its derivative term; ordinary compact approximation away from the boundary proves the defining closure. For the Robin condition, multiplication by the scalar $\chi$, whose normal derivative is zero at the face, leaves the original homogeneous condition intact. Its weighted defect is exactly $\beta_\epsilon=[\kappa_0,A_\epsilon]\gamma(\chi u)$.

The two bounds needed for D4 are uniform:

7. Three solved checks

Exercise 1. A positive matrix weight without commuting derivatives

For $-1\le t\le1$, take $k(t)=\begin{pmatrix}t&i\\-i&-t\end{pmatrix}$, $\lambda=1$, and $M=8$. Verify a uniform positive bound for $8I+2k(t)$, and explain why matrix order still matters in derivatives of MD7.

Solution. Direct multiplication gives $k(t)^2=(1+t^2)I$. Its characteristic polynomial is $\mu^2-(1+t^2)$; the two eigenvalues of the Hermitian matrix are therefore $\pm\sqrt{1+t^2}$. Thus

Exercise 2. A matrix polynomial is not its block form

Let $A_0=-\operatorname{diag}(1,0)$, $A_2=-\operatorname{diag}(0,1)$, and $A_1=\begin{pmatrix}0&1\\1&0\end{pmatrix}$. Compare $A_0+tA_1+t^2A_2$ with the block matrix whose diagonal blocks are $A_0,A_2$ and whose two cross blocks are $A_1/2$.

Solution. For every real $t$,

Exercise 3. The regularizer's leading sign and size

In one tangential dimension use $r(z,\eta)=z\eta^2$ on a fixed coordinate patch. Compute the imaginary leading coefficient in MD24 and a bound independent of $\epsilon$.

Solution. On this patch it is $z_\epsilon=i e_\epsilon$, where

The panels show the exact eigenvalues in MD31, the nonpositive polynomial eigenvalues in MD32 alongside its positive block witness, and the rational transition $2t^2/(1+t^2)$ with $t=\epsilon|\eta|$. These are symbol and finite-matrix calculations, not numerical propagation results. Full proofs: D1–D7 and Exercises 1–3.

8. Sources and remaining propagation work

The primary source is Lars Hörmander, The Analysis of Linear Partial Differential Operators III: Pseudo-Differential Operators, Springer 2007, ISBN 978-3-540-49938-1, Section 24.4, especially Lemma 24.4.7, pp. 449–451. Its scalar Dirichlet argument supplies the normal-energy and regularization strategy. The matrix weight, legal matrix test, explicit Robin defect and uniform full-symbol formulation above are independent receiving proofs, with the exact earlier programme arguments linked in the proof map.

MD17 proves the uniform estimate with its actual incoming and boundary contributions. To conclude the full weak strict-propagation theorem, the remaining work must control the homogeneous Robin form, prove the incoming estimate from the earlier interior/reflection results, bound the retained cutoff-source pairings, identify the mixed gain with the energy-b front, and pass to higher-order supported limits on one common smooth neighborhood. No such conclusion is inferred from a finite check or from the scalar source theorem. Independent new text, code and illustration are CC0-1.0; linked components retain their licences.