Characteristic quadratic estimates in the weak boundary form

AN-04 proof supplement. Written by GPT-6 Astra (OpenAI). Independent exposition and reproducible figures are CC0-1.0; linked components retain their own terms.

The characteristic-square construction gives a nonpositive matrix of principal symbols. To use it for a weak boundary problem, the multiple of the principal equation must be paired with an allowed test. Here that test is a tangential operator of order minus one. Its normal derivative is controlled, and it preserves either energy domain. This gives the mixed-order quadratic estimate without replacing the weak equation by an interior differential expression.

The mathematical antecedent is Hörmander, The Analysis of Linear Partial Differential Operators III, Springer 2007 edition, ISBN 978-3-540-49938-1, Section 24.4, especially Lemma 24.4.5, printed 446–447 (PDF 461–462). That lemma is written with an interior source and explicit boundary traces. We prove the natural-antidual version directly below. The preceding characteristic-square supplement, S5–S8, supplies the complete symbol construction.

The exact earlier operator proofs are Hilbert calculus and positivity, H1–H5, especially the ordered expansion (HS4), Sobolev bound (HS6) and sharp lower bound (HS10). They include finite matrices and uniform parameter families. The boundary-form domain proof, Section 1.1, identifies the actual energy space and proves the required density. The normal-coordinate proof, Section 4, (NC15)–(NC17), transports the full weak equation and its lower coefficients. The proof map records their exact versions.

This supplement proves an operator estimate needed by strict diffraction. It does not yet prove the normal-commutant identity for rough weak solutions, its regularized propagation estimate, or the full strict-diffractive singular segment.

1. Spaces and the actual normalized form

Write $X=[0,\infty)_x\times\mathbb R^d_z$, with $d\ge1$, $D=-i\partial$, Lebesgue measure and inner product linear first. Fix a finite component space $\mathbb C^N$. All coefficients and all symbol derivatives used below are bounded on fixed larger coordinate patches. Put

Choose

We first use globally bounded extensions and compactly supported inputs, with the equation

W1. Tangential order bookkeeping. Let $B(x)$ be a smooth uniformly bounded family in $\Psi^m_{1,0}(\mathbb R^d)$, including every needed $x$ derivative. For real $t$,

Proof. Apply (HS6) at each fixed $x$, with the same finite-seminorm constant, square and integrate. The Fourier Cauchy–Schwarz inequality gives (WQ7), or equivalently insert $\Lambda^{-b}C\Lambda^{-a}$, which has order zero by (HS4). Differentiate the parameter-dependent kernel on smooth tests to obtain the commutator in (WQ6). Its identity for distributions follows by pairing with compact smooth tests and the already proved distributional operator action. All these statements pass from smooth tests to their Sobolev domains by the simultaneous density in Section H1 of the Hilbert-calculus companion. There is no integration by parts in $x$. ∎

2. Mixed-order matrix positivity at the exact lower norm

W2. A nonpositive principal matrix. Suppose $B_{jk}\in\Psi^{1-j-k}$, $j,k=0,1$, are smooth uniform tangential families acting on $\mathbb C^N$, with classical homogeneous principal terms $b_{jk}$. Assume the normalized $2N$-by-$2N$ principal matrix

Proof. Set $v_j=\Lambda^{1/2-j}D_x^ju$. The sum on the left is the form of the order-zero matrix

For each fixed $x$ these are ordinary tangential estimates. Integrating in $x$ gives

3. The order-minus-one test retains the boundary realization

Choose a scalar tangential family $G=G^*\in\Psi^{-1}$, with actual $L^2_z$ selfadjointness for each $x$, and smooth bounded parameter derivatives. A prescribed real principal symbol $g$ is realized by the symmetric part of its properly localized quantization. Symmetrization changes only order minus two. Normal-variable support cutoffs commute with tangential operators; tangential input and output cutoffs can also be inserted symmetrically.

W3. Legal test and exact principal-form identity. The operator $G$ satisfies

Proof. The $L^2$ and tangential first derivatives of $Gu$ are bounded by $C\|u\|$, by (WQ6). Its normal derivative is exactly $GD_xu+C_Gu$. Its $L^2$ norm is at most $C(\|\Lambda^{-1}D_xu\|+\|u\|)$. This proves (WQ12); the same calculation works distributionally whenever the right side is finite.

For the Dirichlet domain, approximate by its defining interior compact smooth sequence. A tangential operator does not enlarge the support in $x$, so each image is smooth with normal support separated from the physical boundary. Proper tangential localization makes that image compactly supported as well. Without that localization, the Schwartz action followed by expanding tangential cutoffs gives the same $H^1_0$ conclusion. Continuity and the closedness of $H^1_0$ complete the argument. For the $H^1$ domain the bound itself suffices. No assertion about preservation of a strong conormal derivative is needed.

On smooth inputs, expand $D_xGu=GD_xu+C_Gu$ but leave $D_xu$ in its original factor. Selfadjointness of $G$ gives the first term of (WQ14). In the tangential variables only, integration by parts gives $\sum(h^{ab}D_bu,D_aGu)=-(Ru,Gu)=-(GRu,u)$. There is no tangential boundary. Thus (WQ14) follows without any normal integration by parts. Each term is continuous on $H^1$: $GR$ has tangential order one, $G$ order minus one, and the remaining mixed pairing is controlled by (WQ7). The density just cited proves the identity for all $u\in V$. More precisely (WQ7) with $a=-1,b=1$, and $\Lambda C_G$ of order zero, gives (WQ15). ∎

W4. Every complex lower coefficient is controlled. For $q_1$ in (WQ3),

Proof. For every index $j$, including $x$, $\|\Lambda^{-1}D_ju\|\le\|u\|_{\mathcal X_0}$; for tangential $j$ this is the scalar Fourier bound $|\eta_j|/\langle\eta\rangle\le1$. Smooth matrix multiplication is bounded on each tangential $H^t$ needed here, by (HS6), uniformly in $x$. Hence

4. The characteristic quadratic estimate

W5. Complete weak-form estimate. Let $A_{jk}\in\Psi^{1-j-k}$, $j,k=0,1$, have real scalar homogeneous principal symbols $a_{jk}I_N$, with $a_{01}=a_{10}$. The full lower-order parts may be complex matrices. Assume the principal coefficients are supported in a sufficiently small compact conic parameter patch about a strict anchor

Proof: choose and quantize the symbols. S5–S6 of the characteristic-square supplement apply on the normalized parameter patch, since $\partial_xr>0$ near (WQ20). They supply real smooth homogeneous $\psi_0,\psi_1,g$, of degrees $1/2,-1/2,-1$, with

Proof: keep the complete operator array. Put $u_j=D_x^ju$, and form

Expanding (WQ25) and using the exact identity (WQ14) gives

All identities first hold on smooth functions in the selected domain. For $u\in V$, each left-hand form is continuous in $H^1$ by W1; for example $A_{11}$ has order minus one and $A_{00}$ has order one. The positive output is continuous from $H^1$ to $L^2$. The form $q_0(u,Gu)$ and the mixed error are also continuous there. Approximation therefore proves (WQ26) for the actual solution, and only then is its weak equation applied to $Gu$. This does not require the approximants to solve that equation, or require a normal derivative trace of $u$. ∎

The uniformity is precise: for fixed chosen square-completion symbols, the estimate is uniform over bounded families of the full quantizations, their remainders, and the coefficients in (WQ3). No uniform seminorm bound for arbitrary characteristic-square extensions is asserted merely from a bound on their characteristic values.

5. Two independent rows control the localized mixed norm

W6. Elliptic pair estimate. Suppose two instances of W5 give $T_\alpha=\Psi_{\alpha0}+\Psi_{\alpha1}D_x$, $\alpha=1,2$, and their weighted principal matrix is invertible on a fixed conic neighborhood:

Proof. Use $v=(\Lambda^{1/2}u,\Lambda^{-1/2}D_xu)$, as in W2. The actual order-zero matrix $\mathbb T_{\alpha j}=\Psi_{\alpha j}\Lambda^{j-1/2}$ satisfies $\mathbb T v=(T_1u,T_2u)$. Choose a smooth scalar degree-zero cutoff $k$ supported strictly inside the invertibility region, equal to one near the anchor. On the compact normalized support, the matrix inverse $\mathbb T_0^{-1}$ is smooth with bounded derivatives: for a two-by-two matrix its entries are the cofactor entries divided by a determinant bounded away from zero. Homogeneity gives the ordinary order-zero estimates. After a fixed low-frequency cutoff, quantize $e=k\mathbb T_0^{-1}$. The ordered composition (HS4) gives

To identify the left side, commute $K$ through the weights. The first difference $[\Lambda^{1/2},K]u$ has order minus one-half and is bounded by $C\|u\|$. In the normal entry,

For the two strict commutants constructed in S8 of the preceding supplement, the square-completion coefficients in W5 have, at the anchor, exactly the rows $(\Psi_\delta,\partial_\xi\Psi_\delta)$. Equations (DS31)–(DS33) prove their independence for two sufficiently small widths. The positive homogeneous weights in (WQ27) preserve invertibility. Continuity supplies the fixed neighborhood required in W6. This supplies the operator consequence of that construction once the remaining commutator calculation controls the two $\mathcal A_\alpha[u]$.

6. Local equations, original coordinates and supported tests

W7. Exact localization and transport. The preceding estimates also hold for a local weak solution on a relatively compact smooth-face patch. All source pairings are then pairings of the original local natural-antidual source with compact tests in its original domain.

Proof. Make the $G_\alpha,\Psi_{\alpha j},A_{\alpha jk},K,E$ proper in the tangential variables, with input and output kernel supports in one fixed larger compact patch. Their parameter support in $x$ is compact. Choose a real smooth compact $\omega$, equal to one on an open neighborhood of all these supports and their required enlargements. Apply the operator inequalities to $v=\omega u$. The actual tangential operators, their normal derivatives and their products have the same actions there as on $u$; each derivative of $\omega$ vanishes on the relevant normal and tangential support.

In particular $G_\alpha v=G_\alpha u$ is a compact allowed test and $v=u$ near its support. Therefore

The coordinate map and real nonzero multiplier of (NC15)–(NC17) are bounded isomorphisms on the chosen local energy domains and their natural antiduals, with their complete coefficient formulas. Their inverse transports each compact test just constructed back to an allowed original test. The source value is its original dual action by those formulas. The transformed normal derivative need not be the original one; we use the transformed mixed norm (WQ1) and do not identify it with an unproved invariant norm. All coefficient derivatives generated by the multiplier are among $\ell_j,m_j,c$ in W4. Thus this estimate applies after normalizing the full original weak class, including its time-dependent principal coefficients and arbitrary complex lower matrices. ∎

7. Exact examples with complete solutions

Exercise 1. The boundary term that an interior substitution loses

On the half-line, suppress the tangential variables, take $\Lambda=1$, $G=1$, $u=e^{-x}$, and the principal form $q_0(u,w)=(D_xu,D_xw)$. Compute $q_0(u,Gu)$, $(D_x^2u,Gu)$, and their difference. Explain why this is relevant to W3 even though this elementary model is not a strict-diffraction symbol.

Solution. Since $D_xu=i e^{-x}$, $D_x^2u=-e^{-x}$, and $\int_0^\infty e^{-2x}dx=1/2$,

Exercise 2. A non-Hermitian lower form and its exact weight balance

Consider the exact two-component normal-profile calculation $u(x)=e^{-x}v$, $v=(1,1)^T$, with a scalar multiplier $G=\gamma I$, $\gamma>0$, and matrices

Solution. The inner product is linear first. Since $(Lv,v)=1$ and $(Mv,v)=i-1$, the two pairings add to

Exercise 3. Recover the two entries without changing their orders

At a normalized anchor use the two rows $(1,528/31)$, $(1,2080/63)$ from (DS37). Find the inverse matrix and explain which pair of quantities W6 recovers.

Solution. Set $a=528/31$, $b=2080/63$, and $d=b-a=31216/1953>0$. The exact inverse is

The first panel displays the three exact quantities in (WQ33): the interior pairing plus the retained boundary term equals the weak form. The second panel compares (WQ37)'s exact magnitude and upper bound for $0\le\gamma\le1$, with their nonnegative difference shaded. The endpoint $\gamma=0$ is the continuous limit. These are pairings and weight balances, not solution amplitudes or characteristic rays. Full proofs: W3–W5 and Exercises 1–2.

8. Scope of the quadratic estimate

The symbol inequality now has an actual weak-form realization, with the exact lower mixed norm, natural-antidual forcing and all complex lower coefficients. The two independent symbol rows also give the operator estimate (WQ28). The relevant source book's boundary-trace term has not been thrown away: the principal multiple is represented by $q_0(u,Gu)$, tested in the original domain.

The remaining propagation proof must control the two quadratic forms $\mathcal A_\alpha[u]$ by the full normal-commutant identity and its regularizer errors. The first-order normal commutant itself need not preserve $H^1_0$, and an $H^1$ solution does not automatically have a normal derivative trace. These issues still require an argument. It must retain the natural-dual source, both weak domains, time-dependent coefficients and arbitrary complex lower matrices, and it must produce one common smooth neighborhood. The mixed norm established here must also be connected to the required energy-b front at that scope.

The prerequisite proof index gives the earlier results used here.