AN-04 proof edition · CC0; exact source credit and separately licensed prerequisites

Normal extension and the complete boundary defect

Let $t\ge0$ be a product boundary coordinate, $D_t=-i\partial_t$, and let $u$ be an interior distribution which has some distribution extension across the boundary. If

has smooth finite matrix coefficients and $Pu=f$ in the interior, with $f$ in the dual conormal class $\mathcal A'$, then $u$ has exactly one extension $U\in\mathcal A'$. It satisfies $t^m(PU-f)=0$ as an ambient supported distribution. Moreover it is the only supported distribution extension satisfying that weighted equation. The tangential differential orders of the $a_j$ can be arbitrary finite integers. The last section proves the invariant version for differential operators of total order $m$ with an invertible principal normal coefficient.

The proof below retains Sections 5.1–5.5 of AN03-U034, Global boundary operators, compressed wave fronts, and normal extension, with corrected transpose terminology, explicit nested compact sets in the bootstrap, a direct construction of the supported distribution and the complete boundary defect. Original source credit: Codex, September 2026, CC0. Current source connections and clarifications: AN-04 course-writing task and OpenAI Codex, 5 October 2026, CC0.

The mathematical antecedent is the approved Hörmander III, 2007 eBook, ISBN 978-3-540-49938-1, Section 18.3; its use in the higher-order Cauchy problem is in Section 23.2. The proof is supplied here in full, using the exact earlier programme proofs below.

Definitions and exact prerequisites

The conormal amplitude and complete test-space component proves the dyadic Besov norms (C2), smooth coefficient and coordinate bounds at every real index, and conormal completeness. The dual conormal component proves the exact tangent generators (GA1)–(GA2), positive-support approximation (GS1)–(GS7), dual pairing and interior injectivity (GD1)–(GD4), actual boundary trace and intrinsic derivatives (GD5)–(GD12). We retain its conormal index convention $\kappa_k=-k-n/4$ and smooth-test threshold $m_0=-(n+2)/4$.

All distribution pairings are complex-linear, against the dual density: $D_t^{\mathrm t}=-D_t$. Hilbert adjoints are explicitly identified when used for Sobolev norm estimates. The full Hilbert duality proof, Fourier Sobolev embedding, finite distribution-order and coordinate proof, and locally finite partitions PS5 are included earlier components. Their component licences and credits remain attached to them. The proof map supplies exact source hashes and unique proof locators.

Here extendibility means existence of an ambient distribution with the stated interior restriction; no special boundary values of that temporary extension are assumed. The symbols $f,U$ denote supported representatives when used in ambient equations. The compressed wave-front and $\mathcal N$ conclusions require the further compressed operator calculus and are not asserted by this component.

1. The exact local conormal seminorms

Fix $K=K_0\times[0,c/2]$ compactly contained in the coordinate chart, and use a slightly larger compact $K'$. For $\kappa_k=-k-n/4$, the conormal topology on supported tests is generated by

Smooth chart cutoffs are inserted in each norm. Formula (GA2) proves that these weighted normal derivatives span exactly the same filtered family as all words in $D'$ and $x_nD_n$. No lower term of the triangular polynomial relation is dropped. Tangential differential operators $a(x,D')$, of any fixed finite order, and their formal transposes map each $\mathcal A^k_K$ continuously into $\mathcal A^k_{K'}$. To verify this, expand a tangent word through $a$: every resulting term is another finite tangent word with smooth coefficients, including the derivatives of those coefficients. Multiplication by a compact smooth coefficient is bounded on each $B^{\kappa_k}_{2,\infty}$ by (C2)–(C3). The number of seminorms needed depends on the actual tangential order; no relation to the normal order is assumed.

For later use, the dyadic Fourier estimates give, for every real $s$,

On the low-frequency block the first norm controls the left side. On the $l$-th high block, $2^l\asymp(\sum_j|\xi_j|^2)^{1/2}$; multiply its $L^2$ norm by $2^{l(s-1)}$, use Plancherel for each $D_j$, and take the supremum over $l$. This proves (GE2) without a boundary norm convention or an omitted tangential term.

2. The normal primitive gains one conormal order

For $\phi\in C_c^\infty(X^\circ)$ with support in $K$, put

Choose a fixed normal cutoff $\chi(t)$ equal to one on $[0,c/2]$ and smoothly supported in $(-c,c)$, with positive-side support in $[0,c)$. Then $\chi\psi$ has compact support in the interior for each such $\phi$, although the distance of that support from the boundary need not be uniform.

We first prove that the localized Volterra operator $J:\phi\mapsto\chi\psi$ is bounded on $B^s_{2,\infty}$ for every real $s$, with fixed chart cutoffs. For this ambient-space estimate, choose a smooth input cutoff equal to one on the original $K$, extending a short distance across $t=0$, and write the integral as $i\chi(t)\int_{-\infty}^{t}\theta(s)\phi(x',s)\,ds$. For the original interior-supported $\phi$ this is exactly (GE3) after multiplication by $\chi$. The smooth cutoff is necessary in the negative-order dual estimate; a sharp input cutoff at zero would create an unjustified boundary multiplier. For $s=0$, Cauchy–Schwarz on the finite normal interval gives $\|\psi\|_{L^2}\le c\|\phi\|_{L^2}$. For a nonnegative integer $p$, differentiate $\chi\psi$ at most $p$ times. Tangential derivatives commute with the integral; each positive normal derivative of $\psi$ is the corresponding derivative of $\phi$ of one lower order; derivatives of $\chi$ multiply the same expressions. Thus $J:H^p\to H^p$ is bounded. For real cutoffs the Hilbert adjoint has the reversed integral $g(s)\mapsto-i\theta(s)\int_s^c\chi(t)g(t)\,dt$. The complex-linear distribution transpose has $+i$ instead, without complex conjugation. This distinction fixes the sign convention without changing either norm estimate. Both reversed integrals have the same integer derivative bounds, including every derivative of $\theta$. Hilbert duality therefore gives $J:H^{-p}\to H^{-p}$. For a real $s$, choose integers $p<s<q$. The dyadic operator matrix estimate between the two Sobolev endpoints is the one proved from (C2)–(C3): after weighting by $2^{ls-js}$, it decays geometrically in $|l-j|$. Summation proves the claimed $B^s_{2,\infty}$ bound, including negative indices. The constants depend on the fixed cutoffs, not on how near the input support is to $t=0$.

Now set $s=\kappa_k=-k-n/4$. We prove the stronger precise primitive estimate

since $\kappa_{k+1}=s-1$. For a tangent word with no normal derivative, $D'^\beta(\chi\psi)=J(D'^\beta\phi)$, so the Volterra bound controls its $B^{s-1}$ norm by the corresponding $\phi$ seminorm. Apply (GE2) to gain the missing one degree. Its tangential derivatives are $J(D'^{\beta+e_j}\phi)$, and its normal derivative is $\chi D'^\beta\phi+(D_n\chi)D'^\beta\psi$. The same Volterra estimate controls all these $B^{s-1}$ norms by a finite set of (GE1) seminorms for $\phi$.

For a word with positive normal order $a\ge1$, preserve its entire normal weight:

The right side has a factor $x_n$ times a tangent word in $\phi$, so its $B^{s-1}$ norm is controlled. Its normal derivative is

This is the full product rule, including its $-ia$ term; when $a=0$ that term is absent and the earlier calculation applies. The tangential derivative is $x_n^aD_n^{a-1}D'^{\beta+e_j}\phi$, again a smooth multiple of a tangent word. A cutoff derivative contributes $(D_n\chi)x_n^aD_n^{a-1}D'^\beta\phi$, which has the same bound. Applying (GE2) proves the $B^s$ estimate for each word, and summing finitely many words proves (GE4).

The gain just proved also holds with any fixed smooth normal output cutoff $\eta$ in place of $\chi$; its being one on the input support was needed only for the later equation, not for this bound. Indeed expand

For $b\ge1$, use (GE5)–(GE6) at order $b$, with the smooth additional coefficient $t^{a-b}D_t^{a-b}\eta$. For $b=0$, use the zero-normal-order Volterra estimate with output cutoff $t^aD_t^a\eta$. Smooth coefficients preserve the same Besov index. This treats every derivative on the cutoff, including the case $\eta=D_t\chi$ needed in (GE9).

3. First-order normal equations at every test order

Let $A_0(x,D')$ be a finite matrix of tangential differential operators with smooth coefficients of arbitrary finite orders, and let the extendible interior vector distribution $u$ solve

We prove that $u$ has a unique extension in $\mathcal A'$. For interior compact smooth tests $\phi$ supported in $K$, we first establish the bound

for every real $k$, with a finite $L$ depending on $k,K$. Since $u$ is extendible, choose one ambient distribution extension temporarily. On a fixed compact set it has finite order $M$, so its action on interior $\phi$ is bounded by finitely many $C^M$ seminorms. Choose $s>M+n/2$. Fourier inversion and Cauchy–Schwarz bound those seminorms by $H^s$, and a slightly larger $B^{s+\epsilon}_{2,\infty}$ norm bounds $H^s$ by a geometric dyadic sum. Choose $k_0$ sufficiently negative that $-k_0-n/4>s+\epsilon$. The $\alpha=0$ term of (GE1) then proves (GE8) for $k_0$. The extension's boundary values do not enter this interior-test estimate.

There is no uniform-support assumption hidden in the induction. For a requested target order $k$, choose a fixed outer compact product collar $K_*$ containing the desired compact $K$ in its relative interior. The preceding finite-order argument on $K_*$ fixes $k_0$. Choose a finite integer $N\ge\max(0,k-k_0)$. Fit $N+1$ nested compact product collars between $K$ and $K_*$, with a positive margin at each step. The tangential compact may be enlarged once; normal primitives preserve its support. Start with the bound at $k_0$ on the largest collar, then prove the bound at $k_0+1$ on the next smaller collar, and continue inward. Every cutoff and its derivative has support in the next larger collar. All constants may depend on this finite chain. When $N=0$, only the initial estimate and inclusion are needed.

Here is the single induction step. Assume (GE8) at order $k$ on the larger collar, and take the input $\phi$ in the smaller one. Let $\psi$ be (GE3), $\chi$ the fixed cutoff, and use the bilinear distribution pairing. The formal tangential transpose $A_0^{\mathrm t}$ retains every coefficient derivative and reverses the matrix maps. Since $D_n(\chi\psi)=\phi+(D_n\chi)\psi$ everywhere (because $\chi\phi=\phi$), the exact equation (GE7) gives

All arguments of $u$ in this formula are smooth and supported in the interior. The functional $f$ is continuous on compact $\mathcal A^k$ tests for every $k$: for $k\ge m_0$ this is its defining property, and for $k<m_0$ the inclusion $\mathcal A^k\hookrightarrow\mathcal A^{m_0}$ is continuous. The full $A_0^{\mathrm t}$ is continuous on $\mathcal A^k$ by Section 1, regardless of its tangential order. The primitive bound with output cutoffs $\chi$ and $D_n\chi$ controls the two arguments in (GE9). Apply the induction bound on the larger collar, and the bound for $f$ there, followed by (GE4). This gives (GE8) at $k+1$ on the smaller collar, with a finite, possibly increased $L$. For the chosen finite chain this proves (GE8) at $k_0+N$ on $K$. The continuous inclusion $\mathcal A^k\hookrightarrow \mathcal A^{k_0+N}$ gives the bound at the requested real order. Since the requested order and inner compact were arbitrary, this proves (GE8) on every compact at every real order.

Construct the supported extension rather than identify it with the temporary ambient one. For a compactly supported smooth boundary test $\phi$, the notation $\mathcal Q_\varepsilon\phi$ below means $\mathcal Q_\varepsilon(H\phi)$, where $H\phi$ is zero extension. It is smooth and interior-supported. All small $\varepsilon$ have one fixed compact output enlargement; their distance from the boundary may tend to zero. By (GD2) and (GS7) they converge to $H\phi$ in $\mathcal A^{k_1}$ for every $k_1>m_0$. The bound (GE8) on that output compact makes their pairings Cauchy, so define

The same estimate on a common enlargement makes the limit independent of the smoothing family: subtract any two interior-supported smooth approximants converging to $H\phi$ in the same $\mathcal A^{k_1}$.

More generally for any compactly supported $v\in\mathcal A^k$, use convergence in $\mathcal A^{k+\delta}$, $\delta>0$, and the bound (GE8) at $k+\delta$ to define the same limit $L(v)=\lim_\varepsilon u(\mathcal Q_\varepsilon v)$. For continuity at the original order $k$, apply (GE8) at $k$ to these approximants. The full commutator formula (GS2), bounded Fourier multipliers and coordinate cutoffs bound every output $\mathcal A^k$ seminorm by finitely many input seminorms at that same order, uniformly in $\varepsilon$. Taking the limit therefore gives an actual finite $\mathcal A^k$ bound for $L$. On intersections of test orders these definitions agree by using a common larger order. This constructs one continuous pairing with the full filtered test space, including every endpoint, without assuming endpoint convergence.

To construct its ambient supported representative directly, for an ambient compact smooth test $\Phi$ set $U(\Phi)=L(H(\Phi|_{t\ge0}))$. The smooth-to-conormal bound (GD2) controls this pairing by finitely many ordinary smooth seminorms on a fixed compact, so it is an ambient distribution. It is supported in $t\ge0$, and its pairing with a smooth boundary test depends only on the restriction of $\Phi$. Thus it has precisely the required dual conormal bounds (GD3). On an interior test, positive mollification converges in every ordinary smooth seminorm with compact support still away from the boundary. The action of the original interior $u$ consequently converges to its original value, proving the claimed interior restriction. Interior injectivity (GD4) proves uniqueness in $\mathcal A'$. No additional supported-distribution duality theorem is invoked.

4. Full normal order by the actual companion system

Let $m\ge1$ and retain the original normal-monic equation

Each $a_j$ is an arbitrary finite-order tangential differential operator with smooth matrix coefficients; the orders of different $a_j$ are not required to be at most $m-j$. Set $u_j=D_n^ju$, $0\le j<m$. Each $u_j$ is extendible, because an ordinary derivative of an ambient extension remains an ambient extension of the interior derivative. The exact first-order companion system is

For $m=1$ this is just (GE11), with its single matrix entry $a_0$; the displayed larger companion matrix is read for $m\ge2$. Section 3 applies componentwise to its actual tangential matrix, giving a unique vector extension $U_j\in\mathcal A'$ for every $j$.

The corrected derivative (GD9) has the same interior restriction as $U_{j+1}$ for $j<m-1$. Both belong to $\mathcal A'$, so interior injectivity (GD4) gives the exact equality

Similarly the final companion equation gives

The tangential operators act on $\mathcal A'$ by (GD9), with no normal delta correction of their own.

We keep the powers of $x_n$ through the elimination. The full commutator is

Start with $U_0=D_n^0U_0$. If $x_n^j(U_j-D_n^jU_0)=0$, then (GE13) implies $x_n^{j+1}(U_{j+1}-D_nU_j)=0$, while (GE15) implies $x_n^{j+1}D_n(U_j-D_n^jU_0)=0$. Hence induction gives

Apply (GE15) once more at $j=m-1$ to replace the first term of (GE14) after multiplication by $x_n^m$. For each lower term $j<m$, $x_n$ commutes with $a_j(x,D')$, and $x_n^m a_j(U_j-D_n^jU_0) =x_n^{m-j}a_jx_n^j(U_j-D_n^jU_0)=0$. Therefore the complete weighted equation is

No original tangential coefficient or lower normal term was removed.

5. Uniqueness among all supported distribution extensions

Let $V$ be the difference of two supported distribution extensions of the same interior $u$, each satisfying (GE17). Then $\operatorname{supp}V\subset\{x_n=0\}$. We derive its finite normal structure here. Work in an open product subchart with a compact enlargement, and let $M$ bound the distribution order of $V$ on that enlargement. The equation is used for $V$ on the open subchart; we do not assume that an arbitrary localized distribution satisfies the same equation. Taylor-expand a test $h(x',t)$ through degree $M$ at $t=0$, with a fixed normal cutoff $\eta(t)=1$ near zero:

The remainder has every normal derivative through degree $M$ zero on $t=0$; multiplication by a cutoff supported in a shrinking normal neighborhood and the order-$M$ bound show that $V$ annihilates it: a derivative of total order at most $M$, including derivatives on the shrinking cutoff, is bounded by $C\varepsilon$ because the remainder has the factor $t^{M+1}$. Tangential derivatives leave that factor intact. Define tangential distributions $v_j(g)=(-1)^jV(\eta(t)t^jg(x')/j!)$. Then applying $V$ to (GE18a) gives, with no omitted coefficient,

Here $\mu\le M$ is the highest nonzero coefficient. The definition of $v_j$ is independent of the cutoff because $V$ is supported at $t=0$. The representation is an identity of distributions on the smaller open subchart. A cutoff may be used to prove it, but the following equation is applied to that local identity for $V$ itself. The coefficients are tangential distributions, possibly vector valued. The normal distribution formulas, with every factor, are

They follow by applying the distributions to a test function and differentiating $x_n^m$ exactly $m$ times at zero; no coefficient is normalized away. In $x_n^mPV$, the original leading term $x_n^mD_n^m(v_\mu\delta^{(\mu)})$ has the top normal coefficient

which is nonzero when $v_\mu\ne0$. Every lower normal term $a_j(x,D')D_n^j$, $j<m$, has normal order at most $\mu+j-m\le\mu-1$ after multiplication by $x_n^m$. Taylor coefficients of $a_j$ at the boundary can lower that order further, never raise it. Contributions from $v_l$ with $l<\mu$ also have order below $\mu$. The coefficient of $\delta^{(\mu)}$ in $x_n^mPV=0$ therefore forces $v_\mu=0$, a contradiction. Repeating downward gives $V=0$. This proves uniqueness among all supported distribution extensions satisfying (GE17), stronger than uniqueness only in $\mathcal A'$.

The boundary delta derivatives are linearly independent over tangential distributions: test against $\eta(t)t^r g(x')/r!$ and isolate the $r$-th coefficient. Thus the top-coefficient comparison is an equality of actual distributions. This local uniqueness on each chart proves uniqueness on the whole collar.

6. The full boundary defect with every coefficient in order

Write $a_m=I$, and let $U_r=(\nabla_t^{\mathrm{int}})^rU$ and $\gamma_r=U_r|_{t=0}$. All these vectors belong to $\mathcal A'$ by the intrinsic derivative proof. Their interior restrictions are the ordinary derivatives of $u$. The original equation and injectivity therefore give $\sum_{j=0}^m a_j(x',t,D')U_j=f$.

The identity $D_tU_r=U_{r+1}-i\gamma_r\otimes\delta(t)$ implies, by induction on $j$,

For $j=1$ it is the intrinsic derivative identity. Applying $D_t$ to the formula for $j$ differentiates each displayed delta and adds the term with $\gamma_j$ from $D_tU_j$, proving the next formula with all signs intact. Consequently

Every $a_j$ remains in its original position on the left. In particular a $t$-dependent coefficient acts on the delta derivatives; it has not been replaced by its boundary value. Its full action is determined by the product rule

To prove this, pair against a test, expand its $\ell$-th normal derivative after multiplication by $b$, and use $\delta^{(s)}(h)=(-1)^s h^{(s)}(0)$; the sign ratio is $(-1)^\ell/(-1)^{\ell-q}=(-1)^q$. For matrix coefficients this is the same entrywise identity with the indicated matrix order. Tangential derivatives act on $v$ in their original operator order. Equations (NE2)–(NE3) exhibit every lower boundary term and show again that $t^m(PU-f)=0$. The original companion elimination gives the same conclusion independently.

7. Leading coefficients, charts and the precise global scope

For the invariant conclusion let $P$ have total differential order $m$ between bundles of the same finite rank over a manifold with boundary, and assume its principal coefficient on the nonzero normal covectors is invertible. In a boundary chart with defining coordinate $t$, collect its ordered derivatives as $P=a_m(x',t)D_t^m+\sum_{j<m}a_j(x',t,D')D_t^j$. Its leading matrix remains invertible in a sufficiently small chart: its determinant is continuous and nonzero at the boundary point. The inverse is smooth there, since the cofactor formula divides polynomials in smooth entries by that nonvanishing determinant; ordinary differentiation proves every required local derivative bound. Left multiplication by $a_m^{-1}$ gives the proved monic equation with source $a_m^{-1}f\in\mathcal A'$. No derivative is moved across $a_m^{-1}$. Its supported solution satisfies $t^m(PU-f)=0$, by multiplying the normalized equation on the left by $a_m$, which commutes with $t^m$.

The local supported distributions agree on overlaps after the actual bundle and density coordinate maps. Indeed these maps preserve the conormal test spaces and their full dual pairings by the earlier coordinate proof, and the two local extensions have the same interior restriction. Their difference vanishes by interior injectivity. Choose the proved locally finite partition subordinate to the charts and the interior, and define the global functional by the sum of local functionals on the partitioned test. Only finitely many summands meet a compact test; the overlap equality makes the result independent of the partition and equal to each local functional. Finite local conormal bounds give its full dual conormal continuity on each compact. In the interior use the original $u$, so this constructs the required global supported extension, unique by the same local injectivity.

If $\rho$ is another positive defining function in a chart, then $\rho=a(x',t)t$, where $a(x',t)=\int_0^1\partial_t\rho(x',st)\,ds$ is smooth and positive near the boundary. Therefore $\rho^m(PU-f)=a^m t^m(PU-f)=0$, and conversely by multiplication by $a^{-m}$. This proves that the weighted equation is independent of defining function. It holds for any smooth choices of ambient coefficient extensions: their differences vanish to every order at the boundary and on the interior side, so multiplication with a supported finite-order distribution vanishes by the earlier flat-test argument. The same argument applies after the finitely many derivatives in each differential term. Existence and uniqueness among all supported extensions are thus well defined independently of these choices.

The arbitrary tangential-order statement in the opening theorem is a statement on a fixed product collar. The coordinate-invariant assertion here uses the stated total-order and noncharacteristic hypotheses. The remaining compressed wave-front statement is a separate theorem; the proof above establishes the extension, all intrinsic jets and the exact distributional boundary defect needed for it.