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

Dual conormal distributions and intrinsic boundary jets

This component retains the supported-test and duality proofs in AN03-U034, Global boundary operators, compressed wave fronts, and normal extension, Sections 2.5–2.6 and 3.1–3.4, with the exact current prerequisite connections stated below. The original source was written and dedicated to the public domain by Codex, September 2026 (CC0). The source connections and clarifications are by the AN-04 course-writing task and OpenAI Codex, 5 October 2026, also CC0. The linked conormal amplitude component retains its separate GFDL 1.2 licence.

All pairings here are the complex-linear distribution pairing against dual densities. Thus the transpose of $D=-i\partial$ is $-D$; Hilbert antidual conventions are used only where explicitly stated elsewhere. The full conormal amplitude, Besov and completeness proofs supply (C2)–(C17). The locally finite partition proof PS5 supplies the global compactly supported chart cutoffs. All arguments apply entry by entry to finite-rank bundles with their actual density changes.

For clarity, an ambient distribution supported in a closed half-space annihilates a smooth function which is zero on that half-space. On a fixed compact set the distribution has some finite order $L$, by the complete test-function argument T0. Such a function and all its derivatives vanish at the boundary. Multiply it by a cutoff supported within normal distance $2\epsilon$ of the half-space and equal to one within distance $\epsilon$; the pairing does not change by support. Taylor's formula to degree greater than $2L+1$ bounds its first $L$ derivatives, including all derivatives on that cutoff, by $C\epsilon$. The distributional finite-order bound makes its pairing tend to zero. Compact localization proves the stated annihilation and the independence of smooth extensions used below.

The mathematical antecedent is the approved Hörmander III, 2007 eBook, Section 18.3. The complete test and trace proofs follow. The further compressed operator calculus and the noncharacteristic $\mathcal N$ extension are separate subsequent obligations.

1. The supported conormal test class

For real $m$, let $\mathcal A^m(X)$ consist of distributions supported in the closed manifold which, in every boundary chart and its extension, are conormal of order $m$ to $x_n=0$. Explicitly put $\kappa=-m-n/4$ and require

The topology uses all these local seminorms. Multiplication, coordinate changes and finite frame transitions are continuous on those Besov spaces by (C2)–(C3), and the following local calculation identifies the intrinsic tangent fields: a smooth normal coefficient vanishing at zero equals $t\int_0^1\partial_t b(x',st)\,ds$, so it is a smooth multiple of $t\partial_t$. Moving a coefficient through a word in tangent fields leaves only shorter words with smooth coefficients. Hence (GA1) is intrinsic and agrees exactly with (C15), with the original shift $-m-n/4$.

The coordinate seminorms can use $D'$ and $X_n=x_nD_n$. The precise comparison with the original weighted derivatives is

where the first polynomial identity defines an invertible triangular change of basis and the second is its inverse. The first follows by $D_n x_n=x_nD_n-i$ and induction: multiplying $x_n^kD_n^k$ by $X_n+ik$ on the right gives $x_n^{k+1}D_n^{k+1}$. These identities retain every lower term and its factor $i$. Tangential derivatives commute with $X_n$. On compact sets all smooth-coefficient tangent words reduce to these generators. In particular (GA1) makes $u$ smooth in the interior by repeated ordinary derivatives there and the local Sobolev estimate.

2. Approximation which preserves the original closed support

Let $q\in C_c^\infty(\mathbb R^n)$ have integral one and support in the open positive half-space. Set $q_\varepsilon(z)=\varepsilon^{-n}q(z/\varepsilon)$ and $Q_\varepsilon u=q_\varepsilon*u$. For a distribution supported in $x_n\ge0$, this convolution is smooth and supported in $x_n\ge c\varepsilon$, where $c=\min_{\operatorname{supp}q}z_n>0$. Its support remains in one fixed compact enlargement when the input support is fixed and $0<\varepsilon\le\varepsilon_0$.

We prove convergence in the conormal topology, including its weighted normal derivatives. Write $X_n=x_nD_n$, $D_n=-i\partial_n$. Integration by parts in the distribution pairing gives the exact identity

Indeed $Q_\varepsilon X_nu$ has coefficient $y_nD_{z_n}q_\varepsilon+iq_\varepsilon$, whereas $X_nQ_\varepsilon u$ has coefficient $x_nD_{z_n}q_\varepsilon$. Their difference is $z_nD_{z_n}q_\varepsilon-iq_\varepsilon =D_{z_n}(z_nq_\varepsilon)$. Both terms and their sign are retained.

Define $q_0=q$, $q_{j+1}=D_{z_n}(z_nq_j)$, and let $Q_{j,\varepsilon}$ denote convolution by $\varepsilon^{-n}q_j(z/\varepsilon)$. Every $q_j$ is smooth, has the same compact positive normal support, and $\int q_j=0$ for $j\ge1$. Repeating (GS1) proves

Tangential derivatives commute with all these convolutions. These are equalities on distributions; no boundary derivative term is dropped.

The Fourier multipliers satisfy, for each fixed $j$,

For small arguments use the mean, the first-moment integral and $|e^{-iv}-1|\le|v|$; for all arguments use their finite $L^1$ norms. These multipliers commute with the dyadic projections of (C2). Let $\kappa'>\kappa$, $\delta=\kappa'-\kappa>0$, and $\tau=\min(1,\delta)$. The multiplier bound on the $l$-th block is at most $C\min(\varepsilon2^l,1)$. Since

each difference multiplier in (GS3) maps $B^{\kappa'}_{2,\infty}$ to $B^\kappa_{2,\infty}$ with norm at most $C\varepsilon^\tau$. To verify (GS4), split at $2^l=\varepsilon^{-1}$. Below it the expression is $\varepsilon2^{l(1-\delta)}$, at most $\varepsilon^\delta$ if $\delta\le1$, and at most $\varepsilon$ otherwise. Above it the expression is $2^{-l\delta}\le\varepsilon^\delta$.

For $u\in\mathcal A^{m'}$, $m'<m$, the actual indices are $\kappa'=-m'-n/4$, $\kappa=-m-n/4$, so $\delta=m-m'>0$. Subtract $X_n^\ell u$ from (GS2). Its $j=0$ term is $(Q_\varepsilon-I)X_n^\ell u$, and all its $j\ge1$ terms have the mean-zero bounds in (GS3). Every input $D'^{\alpha'}X_n^{\ell-j}u$ has the same Besov index $\kappa'$, by the full definition (GA1). Consequently

Equations (GA2) and the full product rule now give every original weighted derivative seminorm and every smooth-coefficient tangent word. Thus $Q_\varepsilon u\to u$ in $\mathcal A^m$, not merely as an ordinary weak distribution, with finite-seminorm control.

Finally choose the locally finite chart partition $\varphi_j$ and input cutoffs $\psi_j=1$ near their supports from the included PS5 partition construction. In each boundary chart use the positive convolution just proved; in interior charts use a compact mollifier whose small translations remain interior. Use the actual half-density and bundle coordinate maps on both sides. Define

Choose each chart's convolution radius no larger than its fixed distance from the cutoff support to the chart edge; a constant multiple of $\varepsilon$ in that chart suffices. On a fixed input compact set only finitely many $\psi_j$ occur, giving a single compact output set $K'$, independent of sufficiently small $\varepsilon$. Smooth multiplications and chart maps are continuous in (GA1). Since $\sum_j\varphi_j\psi_j u=u$, (GS5) and the full Leibniz rule prove

For compact input its output is compact and supported in the interior, including in boundary charts. The convergence is uniform on bounded sets of each fixed compact $\mathcal A^{m'}$ source, by the displayed finite-seminorm estimates. This proves the required support-preserving smoothing lemma with its original order indices.

3. Dual distributions and boundary traces

3.1. The lowest test order and the dual class

Put $m_0=-(n+2)/4$. In a boundary chart, let $\phi(x',t)$ be smooth up to $t=0$, compactly supported for $t\ge0$, and let $H\phi$ be its zero extension. Integrating its normal Fourier transform by parts $N$ times gives, for $|\tau|\ge1$,

Each displayed boundary jet, including its sign, follows from the lower endpoint in integration by parts. The remainder and all its tangential derivatives are bounded by the corresponding compact smooth seminorms. For any requested number of frequency derivatives, first multiply by powers of $t$ under the integral and repeat the same integration by parts with enough extra terms; this gives the full order-$-1$ symbol estimates. Multiplying by a cutoff equal to one for $|\tau|\ge2$ and absorbing the compact-frequency part into a smooth amplitude yields the exact reduced conormal form (C16). For codimension one its amplitude order is $m+(n-2)/4$; order $-1$ means exactly $m=m_0$. If $\phi(x',0)\ne0$, the first term in (GD1) is nonzero, so a uniform claim of a lower conormal order is false. Hence

where a smooth boundary function is represented by its zero extension. The continuous inclusion for $m>m_0$ is immediate from the symbol orders or the dyadic Besov weights. Interior-supported smooth functions belong to every order.

Define $\mathcal A'(X)$ as the ambient supported distributions $u$ whose action on every compactly supported smooth boundary function is continuous with respect to the $\mathcal A^m$ topology, for every $m\ge m_0$. Precisely, for each compact $K\subset X$ and each such $m$, finitely many defining seminorms $p_{m,K,l}$ and a constant give

The action is independent of an ambient smooth extension of $\phi$: two extensions agreeing on the closed half-space differ by a smooth function vanishing on its interior and to every order at its boundary; the supported distribution annihilates that difference. This is checked in each chart and patched by a partition. The topology on $\mathcal A'$ used here is the weak topology generated by all $u\mapsto|u(v)|$ for compactly supported $v\in\mathcal A$, where $\mathcal A=\bigcup_m\mathcal A^m$.

We prove that the pairing in (GD3) extends uniquely to every compactly supported $v\in\mathcal A^m$, $m\ge m_0$. Choose $m_1>m$. The support-preserving $\mathcal Q_\varepsilon v$ is smooth with support in a fixed compact $K'$, and (GS7) gives convergence in $\mathcal A^{m_1}$. The bound (GD3) at order $m_1$ makes $u(\mathcal Q_\varepsilon v)$ converge. If another smooth sequence converges to $v$ in the same order, its difference has pairing tending to zero by that bound, so the extension is unique. To show continuity on $\mathcal A^m_K$, use (GD3) at $m_1$ on the approximants and take the limit. The inclusion $\mathcal A^m_K\hookrightarrow\mathcal A^{m_1}_{K'}$ is continuous, which supplies the finite $\mathcal A^m$ bound. No convergence in the same endpoint order $m$ has been assumed. This constructs the full pairing and the stated weak topology.

3.2. Interior restriction, absence of boundary-supported elements, density

If $u\in\mathcal A'$ vanishes on all tests in the interior, then for each compactly supported $v\in\mathcal A^m$ choose $m_1>\max(m,m_0)$. Each $\mathcal Q_\varepsilon v$ is smooth and supported in the interior, so $u(\mathcal Q_\varepsilon v)=0$. The convergence in $\mathcal A^{m_1}$ and the extended continuity prove $u(v)=0$. Smooth boundary tests are among these $v$, hence $u=0$ as an ambient distribution. Therefore

In particular no nonzero distribution supported only on $\partial X$ belongs to $\mathcal A'$. This follows from the exact density argument, not from a mistaken identification of the two distribution spaces.

Smooth boundary functions are weakly dense in $\mathcal A'$. First every smooth boundary function defines an element of $\mathcal A'$: on a compact test support, multiply that function by a smooth compact cutoff and pair it with the supported conormal distribution. In the normal form (C16), the compact smooth factor has rapidly decreasing normal Fourier transform. Integrating the full symbol amplitude against that transform bounds the pairing by finitely many $S^{m+(n-2)/4}$ seminorms for every real $m$; the normal-form topology comparison in (C16) gives the corresponding $\mathcal A^m$ bound. Interior terms use the ordinary distribution pairing. Thus the smooth approximants below really belong to $\mathcal A'$. For a compactly supported $v\in\mathcal A$, define $u_\varepsilon(v)=u(\mathcal Q_\varepsilon v)$. The adjoint of each local term in (GS6) is convolution with a compact smooth reflected kernel, followed by smooth cutoffs and coordinate/half-density maps. For a distribution $u$, this adjoint is a smooth function of the remaining variable, including at its boundary; on every compact set only finitely many chart terms occur. Thus $u_\varepsilon$ is represented by a smooth function on $X$. For each fixed $v$ of order $m$, choose $m_1>\max(m,m_0)$. By (GS7), $\mathcal Q_\varepsilon v\to v$ in $\mathcal A^{m_1}$, and the extended continuity gives

This is weak density with the actual test topology, for every compact conormal test. Together with (GD4), it identifies $\mathcal A'$ with its image of interior distributions; it does not make all interior distributions members of $\mathcal A'$.

3.3. The invariant boundary delta and trace

Let $\varphi$ be a compactly supported smooth test density on $\partial X$, with the dual bundle coefficient when appropriate. In a boundary chart define the supported distribution density

For a new defining function $\bar t=\alpha(x',t)t$, $\alpha(x',0)>0$, the exact distribution density law is $\delta(\bar t)|d\bar t|=\delta(t)|dt|$: the delta coefficient contributes $\alpha(x',0)^{-1}$ and the normal density contributes $\alpha(x',0)$. The tangential density and bundle transitions are the usual ones. Hence $T$ is intrinsic, not a choice of boundary coordinate or a half-density shortcut.

The original Fourier amplitude in (GD6) is constant in $\tau$, with the exact coefficient $(2\pi)^{-1}\varphi(x')$. Formula (C16) for codimension one therefore gives $m+(n-2)/4=0$, that is,

The direct Besov estimate uses the same dyadic amplitude bound, so the target topology is included. Define the boundary restriction by

The exact conormal order in (GD7) lies above $m_0$, so the extended pairing is available. Its continuity in $\varphi$ follows from (GD3) and (GD7), and its weak continuity in $u$ is one of the defining seminorms of $\mathcal A'$. Thus (GD8) is a distribution on the boundary.

For a smooth function $u$ up to the boundary, use the positive normal convolution on $T\varphi$. It is a unit-mass smooth approximate delta supported at positive normal distance $O(\varepsilon)$. Consequently $u(\mathcal Q_\varepsilon T\varphi)$ tends to $\int_{\partial X}u(x',0)\varphi(x')$, with the full chart density factor. The extension of the pairing in Section 3.1 gives the same limit for $u(T\varphi)$. Hence (GD8) agrees with ordinary smooth restriction. Its uniqueness among weakly continuous trace maps follows from (GD5).

3.4. Corrected differentiation with its exact sign

Let $u\in\mathcal A'(\overline{\mathbb R}{}^n_+)$, represented as an ambient distribution supported in the closed half-space. Write $u_\partial=u|_{x_n=0}$ from (GD8). Define

This is the original ambient derivative plus its full boundary delta correction, with no suppression of either term. The sign follows first for a smooth $u$ from

Both identities are direct distributional product rules, since $D_nH=-i\delta$. Adding the term in (GD9) makes $\nabla_j^\mathrm{int}u$ the supported representative of the ordinary interior derivative for smooth $u$.

We prove membership in $\mathcal A'$ for general $u$. Let $\phi$ be a compact smooth boundary test, and distinguish its ambient smooth extension from its supported zero extension $H\phi$. In the $\mathcal A'$ pairing, $u(-D_j\phi)$ is the pairing with the zero extension of $-D_j\phi$. The full distribution identity is

Therefore the two terms in (GD9) combine exactly to

The ordinary differential map $D_j:\mathcal A^m_K\to\mathcal A^{m+1}_{K}$ is continuous: the normal derivative multiplies the full conormal amplitude by $\tau$ and differentiates its smooth base coefficient, while tangential derivatives differentiate that coefficient; (C4)–(C5) control all remainders. Equivalently it is the exact order-one conormal map (C17) with compact support. If $m\ge m_0$, then $m+1\ge m_0$, so (GD12) and the defining estimate at order $m+1$ show that $\nabla_j^\mathrm{int}u\in\mathcal A'$. The map is weakly continuous: for each compact conormal test $v$, its defining functional is $u\mapsto u(-D_jv)$, another test in $\mathcal A$. In the interior the delta term vanishes, so $(\nabla_j^\mathrm{int}u)|_{X^\circ}=D_j(u|_{X^\circ})$.

The uncorrected derivative can fail to lie in $\mathcal A'$. For example choose a smooth $u$ with nonzero boundary value. By (GD10) the ambient derivative contains the nonzero boundary-supported term $-i u_\partial\delta$, while $H(D_nu)$ is in $\mathcal A'$. If $D_n(Hu)$ were in $\mathcal A'$, subtracting $H(D_nu)$ would put a nonzero boundary-supported distribution in $\mathcal A'$, contradicting (GD4). This proves the distinction, rather than treating the two derivatives as identical presentations.

4. Multiplication, intrinsic jets and their distributional meaning

Multiplication by a smooth coefficient preserves every local supported conormal order: commute the coefficient through each tangent word and use the Besov multiplier bound. It consequently preserves the dual class, since the dual pairing with a conormal test $v$ becomes the original pairing with the same coefficient times $v$. This action is weakly continuous and agrees with ordinary interior multiplication. The boundary trace satisfies $(au)|_{\partial X}=a|_{\partial X}\,u|_{\partial X}$, since multiplication of $a$ with $T\varphi$ in (GD6) evaluates its boundary value.

The intrinsic derivatives commute, and obey the ordinary product rule with these coefficients. Both assertions hold after interior restriction by distributional differentiation; each side belongs to the dual class by (GD12) and the multiplier statement, so injectivity (GD4) proves the equalities of their actual supported representatives. The same argument proves that a smooth vector field acts intrinsically in any chart and transforms by the ordinary chain rule. Its iterates have the weakly continuous traces just constructed. None of these assertions identifies a raw ambient derivative with the corrected derivative (GD9).