# Global boundary operators, compressed wave fronts, and normal extension

*Written and dedicated to the public domain by Codex, September 2026 (CC0).*

The local half-space calculus already tells us how a totally characteristic operator acts near one boundary chart. On a manifold, the same calculation must survive changes of coordinates, corners in the operator kernel, and the distinction between an ordinary normal derivative and a vector field tangent to the boundary. This lesson builds that global calculus, proves its wave-front rules, and then uses the exact boundary trace to extend solutions of noncharacteristic equations. A final argument shows when conormal regularity is genuine smoothness up to the boundary.

We keep the original Fourier convention \(D=-i\partial\), with inverse factor \((2\pi)^{-n}\). A supported distribution is always represented on the closed half-space; a restricted interior distribution is a different object. Matrix products retain their input and output order. All orders of symbols and conormal distributions are real unless a theorem specifies an integer normal order.

The named prerequisites are [Singularities along a submanifold and smooth boundary passage](conormal-transmission.md), [Totally characteristic operators on the half space](totally-characteristic-operators.md), [From symbol estimates to operators on every Sobolev scale](euclidean-symbol-calculus.md), and [Detecting regularity without choosing coordinates](geometric-microlocal-calculus.md). The proofs below reproduce their particular formulas when used, so the global construction remains readable from these exact entry points.

## 1. Stretched kernels and compressed covectors

### 1.1. The real projective blowup

Let \(Y\) be a smooth embedded submanifold of codimension \(k\ge1\) in \(X\).
In coordinates \((z,y)\), \(Y=\{y=0\}\), \(y\in\mathbb R^k\).
Replace the normal origin by its lines:
\[
 \{(z,L,v):L\in\mathbb{RP}^{k-1},\ v\in L\},
 \qquad \beta(z,L,v)=(z,v).
 \tag{GL1}
\]
Here \(v\) is the actual normal vector, with both signs. Equivalently the model
is \((z,s,\omega)\), \(\omega\in S^{k-1}\), modulo
\((s,\omega)\sim(-s,-\omega)\), with \(v=s\omega\).
For the chart in which the \(i\)-th component of the line is nonzero, represent
that line by a vector \(w\) with \(w_i=1\). Coordinates are
\((z,s,w_j:j\ne i)\), and \(y=s w\). On overlaps,
\[
 s_j=s_i w_j,\qquad w_\ell^{(j)}=w_\ell^{(i)}/w_j^{(i)} .
 \tag{GL2}
\]
These are smooth invertible changes where \(w_j\ne0\), including \(s_i=0\).
They give a smooth manifold; the exceptional set is its projectivized normal
bundle, and the projection is a diffeomorphism away from \(Y\).
For \(k=1\), there is one line and the local projection is the identity.

To prove independence of coordinates, let \(\bar y(z,y)\) be another normal
coordinate system vanishing on \(Y\). Hadamard's formula gives
\[
 \bar y(z,y)=A(z,y)y,\qquad
 A(z,y)=\int_0^1\partial_y\bar y(z,\lambda y)\,d\lambda ,
 \quad A(z,0)\in GL(k,\mathbb R).
 \tag{GL3}
\]
In a chart \(y=s w\), choose an index \(j\) for which
\((A(z,0)w)_j\ne0\). The new coordinates are
\(\bar s=s(A(z,s w)w)_j\),
\(\bar w_\ell=(A(z,s w)w)_\ell/(A(z,s w)w)_j\), and
the actual transformed tangential coordinate \(\bar z(z,s w)\).
The denominator stays nonzero locally; all these functions are smooth at
\(s=0\). The inverse coordinate change has the same construction. The maps
agree away from the exceptional set, and therefore everywhere by continuity,
so their cocycle identities hold. This proves coordinate independence and
identifies the exceptional transition with the actual normal derivative.

If \(f,g\) vanish on \(Y\), their pullbacks have the form \(sF,sG\) by the
same integral formula. Wherever the normal derivative of \(g\) on the line
is nonzero, \(G(z,0,w)\ne0\), and \(f/g=F/G\) extends smoothly. These ratio
charts are thus intrinsic. No positive-ray quotient replaces the projective
quotient in this construction.

### 1.2. The positive corner and its exact coordinates

For boundary charts on two manifolds, keep \(x_n,y_n\ge0\) and their positive
normal rays. The projective interior normal cone at the corner is
\((\mathbb R_+^2\setminus\{0\})/\mathbb R_{>0}\). Its coordinate and the
radial coordinate are
\[
 t=\frac{x_n+y_n}{2},\qquad
 r=\frac{2(x_n-y_n)}{x_n+y_n},\qquad
 x_n=t(1+r/2),\quad y_n=t(1-r/2),
 \quad t\ge0,\ -2\le r\le2 .
 \tag{GL4}
\]
At \(t=0\), \(r\) records the ray; the projection collapses this interval
to the original corner. The two side faces are \(r=-2\), where \(x_n=0\),
and \(r=2\), where \(y_n=0\).

Boundary coordinate changes have normal parts
\(\bar x_n=\alpha(x)x_n\), \(\bar y_n=\gamma(y)y_n\), with
\(\alpha,\gamma>0\). Set \(a=1+r/2\), \(b=1-r/2\).
Their exact lifted law is
\[
 \bar t=\frac t2\,[\alpha(x)a+\gamma(y)b],\qquad
 \bar r=\frac{2[\alpha(x)a-\gamma(y)b]}
                  {\alpha(x)a+\gamma(y)b}.
 \tag{GL5}
\]
The denominator is strictly positive on the entire closed interval:
\(a,b\ge0\), \(a+b=2\), and both coefficients are positive.
The tangential coordinates are the original coordinate changes evaluated
at \(x_n=t a,y_n=t b\). Thus the full lift is smooth up to every face and
corner, and its inverse is the lift of the inverse changes. It preserves
each side face and multiplies \(t\) by a smooth positive function.
These laws glue the stretched product intrinsically.

![The original positive normal quadrant and its stretched rectangle](../figures/global_compressed_corner_geometry.png)

Figure GL-F1. The two panels retain the exact coordinates in (GL4).
The new face retains the ray when both normal variables vanish. The lifted
diagonal crosses that face. Only the two normal variables are drawn;
tangential coordinates and covectors retain their full dimensions in
(GL7)–(GL10). Reproducible source: ../figures/global_compressed_corner_geometry.py.

For the square of one manifold the interior diagonal lifts to
\[
 \widehat\Delta=\{x'=y',\ r=0,\ t\ge0\}.
 \tag{GL6}
\]
The projection restricts to \((x',t)\mapsto(x',x_n=t)\), a diffeomorphism
with the original diagonal. The lifted diagonal avoids \(r=\pm2\), and
is transverse to \(t=0\) because its tangent includes the \(t\) direction.
This proves all these assertions at boundary points as well as in the
interior.

### 1.3. The compressed bundle and both natural maps

Pull \(N^*\widehat\Delta\) back by (GL6). In local coordinates its covectors
have the form
\(\xi'\cdot d(x'-y')+\rho\,dr\).
The normal differential of the projection at the lifted diagonal is
\[
 (\delta(x'-y'),\delta r)\longmapsto
       (\delta(x'-y'),\,t\,\delta r).
 \tag{GL7}
\]
This follows by differentiating \(x_n-y_n=t r\) at \(r=0\).
Dualizing gives the natural map from the ordinary cotangent bundle:
\[
 \lambda:T^*X\longrightarrow\widetilde T^*X,\qquad
 (\tau',\tau_n)\longmapsto(\xi'=\tau',\,\rho=x_n\tau_n).
 \tag{GL8}
\]
It is an isomorphism for \(x_n>0\). At \(x_n=0\) its kernel is precisely
the ordinary conormal line to the boundary, and its image is the hyperplane
\(\rho=0\), canonically \(T^*\partial X\). The compressed fibre itself
still has dimension \(n\); its other covectors have not been discarded.

The dual anchor is
\[
 \widetilde TX\longrightarrow TX,\qquad
 (v',v_n)\longmapsto
       \sum_{j<n}v_j\partial_{x_j}+x_n v_n\partial_{x_n}.
 \tag{GL9}
\]
Its smooth sections map bijectively onto smooth vector fields tangent to
the boundary. Indeed a tangent normal coefficient \(b(x',x_n)\) vanishes
at \(x_n=0\) and equals
\(x_n\int_0^1\partial_{x_n}b(x',s x_n)\,ds\).
This constructs its smooth inverse coefficient. Uniqueness follows in
the interior and hence at the boundary by continuity.

For an exact coordinate law write
\(\bar x'=F(x',x_n)\), \(\bar x_n=\alpha(x',x_n)x_n\), \(\alpha>0\).
In the interior a compressed covector is
\(\xi'\cdot dx'+\rho\,dx_n/x_n\).
Differentiating the original coordinate functions, without dropping terms,
gives
\[
 \begin{aligned}
 \xi'&=(\partial_{x'}F)^T\bar\xi'
               +(\partial_{x'}\log\alpha)\bar\rho,\\
 \rho&=x_n(\partial_{x_n}F)^T\bar\xi'
               +(1+x_n\partial_{x_n}\log\alpha)\bar\rho .
 \end{aligned}
 \tag{GL10}
\]
These expressions extend smoothly to the boundary. There their determinant
is \(\det\partial_{x'}F\ne0\); locally in the collar the matrix is invertible,
and its inverse is furnished by the inverse boundary coordinate change.
The law agrees with the pulled-back conormal law in the interior by
(GL7)–(GL8), and hence agrees everywhere. It proves the intrinsic bundle
identification, both anchors, and the invariant hyperplane \(\rho=0\).

### 1.4. The original symplectic form and density

In the interior substitute the full formula \(\tau_n=\rho/t\) into the
ordinary cotangent form \(\sum_j d\tau_j\wedge dx_j\). Since
\(d(\rho/t)=t^{-1}d\rho-\rho t^{-2}dt\), its exact expression is
\[
 \omega=\sum_{j<n}d\xi_j\wedge dx_j+t^{-1}d\rho\wedge dt .
 \tag{GL11}
\]
It is nondegenerate for \(t>0\). The \(t^{-1}\) factor is a boundary
singularity; there is no smooth symplectic form asserted there.
In the indicated order of coordinates,
\[
 \frac{\omega^n}{n!}
 =(-1)^{n(n+1)/2}t^{-1}
       dx'\wedge dt\wedge d\xi'\wedge d\rho,\qquad
 |\omega^n/n!|^{1/2}
 =t^{-1/2}|dx'\,dt\,d\xi'\,d\rho|^{1/2}.
 \tag{GL12}
\]
For the sign, move the original ordered pairs
\((d\xi_1,dx_1,\ldots,d\rho,dt)\) to the displayed base-then-fibre order;
the number of transpositions is \(n(n+1)/2\).
The density law is intrinsic because (GL11) was pulled from the original
cotangent form. Its singular factor remains explicit.

### 1.5. The full stretched kernel for every symbol order

Let \(a\in S_{\mathrm{la}}^m\), \(m\in\mathbb R\), in the local half-space
calculus. Keep its exact inverse Fourier distribution
\[
 A(x,z)=(2\pi)^{-n}\int e^{iz\cdot\xi}a(x,\xi)\,d\xi,\qquad
 K(x,y)=x_n^{-1}
       A\left(x',x_n,x'-y',\frac{x_n-y_n}{x_n}\right)
       \quad(x_n>0).
 \tag{GL13}
\]
Oscillatory integrals are defined by the cutoff-independent distribution
construction in (C4)–(C5). No residual assumption is made.
Differentiation in \(z\) raises the amplitude order by its exact degree;
integration by parts in \(\xi\) to a degree exceeding that order plus
\(n\) proves that \(A\) is smooth off \(z=0\), with arbitrary decay in
\(|z|\ge1\), uniformly with every base derivative on compact sets.
Near \(z=0\) its full phase and amplitude exhibit a conormal distribution
of order \(m\) by (C16), with ambient dimension \(2n\) and codimension \(n\).

The Fourier support in the original definition implies
\(\operatorname{supp}_{z_n}A\subset(-\infty,1]\). In fact the partial
Fourier transform of \(a\) is supported in \([-1,\infty)\); inverse
transformation evaluates it at \(-z_n\), with the original inverse
Fourier factor. At \(z_n=1\) the distribution is already smooth, since
that point is away from \(z=0\). Its vanishing on \(z_n>1\) therefore
makes every derivative vanish on \(z_n=1\).

The determinant of (GL4) is \(-t\), so a kernel half-density pulls back
with coefficient \(k=t^{1/2}K\). Put \(H=t^{1/2}k=tK\). The entire formula is
\[
 k=t^{-1/2}(1+r/2)^{-1}
 A\left(x',t(1+r/2),x'-y',\frac{r}{1+r/2}\right),\qquad
 H=(1+r/2)^{-1}
 A\left(x',t(1+r/2),x'-y',\frac{r}{1+r/2}\right).
 \tag{GL14}
\]
Thus both square-root factors, the normal Jacobian and the complete
argument of \(A\) are retained.

Near the lifted diagonal \(1+r/2>0\). In (GL13) change only the integration
variable \(\xi_n=(1+r/2)\rho\). Its positive Jacobian cancels exactly
the displayed \((1+r/2)^{-1}\), giving
\[
 H=(2\pi)^{-n}\int
 e^{i[(x'-y')\cdot\xi'+r\rho]}\,
 a\bigl(x',t(1+r/2),\xi',(1+r/2)\rho\bigr)\,d\xi'\,d\rho .
 \tag{GL15}
\]
The amplitude has all order-\(m\) estimates on compact base sets: an
\(r\)-derivative gives \(t\partial_{x_n}a/2\) or
\(\rho\partial_{\xi_n}a/2\); the latter has the original order \(m\).
A \(t\)-derivative has the bounded factor \(1+r/2\); frequency derivatives
lower the order normally. The same argument handles any iterated mixed
derivative. Equations (GL14) and (GL15) are exact descriptions of the same
kernel, not a substitution omitting a density factor. They prove that
\(H|dx'\,dy'\,dt\,dr|^{1/2}\) is conormal along \(\widehat\Delta\),
smooth in \(t\ge0\).

At \(r=2\), the last argument of \(A\) is \(1\), so the preceding support
argument gives side flatness, including all \(t\) and tangential derivatives.
At \(r=-2\), that argument tends to minus infinity. Its derivatives and the
displayed prefactor grow only as fixed powers of \((2+r)^{-1}\).
The arbitrary large-\(|z|\) decay absorbs each such power, proving flatness
there as well. These estimates are uniform for compact \(t,x',y'\) ranges.
Off \(\widehat\Delta\), the same Fourier argument proves smoothness.

### 1.6. The exact principal half-density and inverse reconstruction

In the full amplitude of (GL15), the difference from its \(r=0\) value is
\[
 \frac r2\int_0^1
 [\,t\partial_{x_n}a+\rho\partial_{\xi_n}a\,]
 \bigl(x',t(1+s r/2),\xi',(1+s r/2)\rho\bigr)\,ds .
 \tag{GL16}
\]
Retain this entire integral. Integration by parts in \(\rho\), using
\(r e^{ir\rho}=i^{-1}\partial_\rho e^{ir\rho}\), changes it to the exact
order-\((m-1)\) amplitude
\[
 \frac i2\,\partial_\rho\int_0^1
 [\,t\partial_{x_n}a+\rho\partial_{\xi_n}a\,]
 \bigl(x',t(1+s r/2),\xi',(1+s r/2)\rho\bigr)\,ds .
 \tag{GL17}
\]
A compact cutoff in \(r\) on this chart is independent of \(\rho\) and
does not change that calculation. The defining cutoff limit justifies
the integration by parts, by (C4)–(C5).
Hence the principal conormal half-density of \(H\), in the parametrization
\((x',x',t,0;\xi',-\xi',0,\rho)\), is precisely the class of
\[
 a(x',t,\xi',\rho)\,|dx'\,dt\,d\xi'\,d\rho|^{1/2}
 \quad\pmod{\text{one lower amplitude order}} .
 \tag{GL18}
\]
For \(k\), the additional \(t^{-1/2}\) is still present. Comparing
(GL18) with (GL12) makes its scalar coefficient \(a\) a principal
symbol on the compressed cotangent bundle. The full remainder is
(GL16)–(GL17); it has not been removed from the original kernel.
The coordinate invariance follows from the full determinant law
(C19)–(C20) and the actual conormal coordinate change (GL10).
In the intrinsic convention of (C21), the half-density in (GL18) has
symbol order \(m+n/2\): its fibre half-density has dilation degree \(n/2\).
Dividing the original principal half-density of \(k\) by (GL12) leaves the
ordinary scalar or matrix symbol order \(m\). Neither convention changes
the amplitude or the original operator.

Conversely, take a compactly supported conormal family \(H\) along
\(\widehat\Delta\), of this order and smooth parameter type, flat at
both side faces. For \(z_n<1\) its inverse kernel is
\[
 A(x,z)=\frac{2}{2-z_n}
 H\left(x',x'-z',\,\frac{x_n(2-z_n)}2,\,
                         \frac{2z_n}{2-z_n}\right),\qquad
 A(x,z)=0\quad(z_n\ge1).
 \tag{GL19}
\]
These are the inverse of the full coordinate and prefactor formulas,
not just their diagonal restrictions. Near \(z=0\), pull back an
oscillatory conormal amplitude for \(H\) by this smooth map. Its normal
phase is \(z'\cdot\xi'+[z_n/(1-z_n/2)]\rho\). The change
\(\rho=(1-z_n/2)\eta_n\) has positive Jacobian \(1-z_n/2\), which cancels
the prefactor \(1/(1-z_n/2)\) in (GL19). The resulting amplitude has order
\(m\) with all base derivatives, including \(x_n\), by the same product
and chain rules used for (GL15). Compact localization and the full
amplitude reduction (C5) therefore give a smooth \(x\)-dependent inverse
Fourier symbol of order \(m\), with all finite-seminorm bounds.

Away from \(z=0\), (GL19) is smooth. Flatness at \(r=-2\) gives arbitrary
decay as \(z_n\to-\infty\): each \(x_n\)-derivative introduces at most
one additional power of \(2-z_n\), absorbed by another flatness order.
Flatness at \(r=2\) gives a smooth zero extension at \(z_n=1\).
Compact tangential support controls \(z'=x'-y'\); all these far
contributions are Schwartz in \(z\), uniformly with every \(x\)-derivative.
If \(H\) is supported in \(t\le T\), then \(A\) is supported in
\(x_n\le2T\), since \(x_n=t(1+r/2)\le2t\). Its base derivatives thus
have every original \((1+x_n)^{-\nu}\) estimate.

Define the exact symbol
\[
 a(x,\xi)=\int e^{-iz\cdot\xi}A(x,z)\,dz .
 \tag{GL20}
\]
The near contribution is order \(m\) by the preceding reduction and the
far contribution is residual. Since \(A=0\) for \(z_n>1\),
\[
 \mathcal F_{\xi_n}a(x,\xi',s)
 =2\pi\int e^{-iz'\cdot\xi'}A(x,z',-s)\,dz',
 \qquad \operatorname{supp}_s\mathcal F_{\xi_n}a\subset[-1,\infty).
 \tag{GL21}
\]
This proves lacunarity with its exact sign and Fourier constant.
Thus \(a\in S_{\mathrm{la}}^m\), and (GL13)–(GL15) recover the original
half-density kernel exactly. Fourier inversion proves uniqueness of the
symbol on \(x_n>0\), and smoothness in \(x_n\) gives uniqueness at zero.
This proves the compact local converse as well as the forward statement
for every real \(m\).

### 1.7. The exact boundary action

Let \(u\) be smooth up to the boundary with compact support. Write
\(s=x_n>0\), \(a_r=1+r/2\). At fixed \(s\), the actual changes of
integration variables are
\[
 y_n=s\frac{1-r/2}{1+r/2},\qquad
 t=s/a_r,\qquad |dy_n|=s a_r^{-2}|dr|,
 \qquad K(x,y)|dy_n|=a_r^{-1}H(x',y',s/a_r,r)|dr|.
 \tag{GL22}
\]
Thus the full operator action, interpreted as a distributional pairing at
the diagonal, is
\[
 (T_a u)(x',s)=\int_{-2}^{2}\int
 \frac{H(x',y',s/a_r,r)}{a_r}
 u\left(y',s\frac{1-r/2}{a_r}\right)\,dy'\,dr .
 \tag{GL23}
\]
For a forward symbol, compactly localize the base variables on the
output region; (GL14) retains the same uniform side estimates there.
For an inverse kernel its given compact support provides this localization.
Choose a partition in \((x'-y',r)\) which is one near its origin and
supported away from \(r=\pm2\). On that part the smooth conormal family
and the compact smooth test function in (GL23) converge, with every
\(x'\) derivative, as \(s\) decreases to zero. Oscillatory formula (GL15)
justifies the convergence: integrate by parts in its normal base variables
to an order exceeding the amplitude order plus \(n\), exactly as in (C4).
The resulting integrable frequency majorant is uniform in \(s\).

On the complementary part the kernel is a smooth function. Near \(r=-2\),
every \(s\), \(x'\) or integration-variable derivative of (GL23) introduces
only a fixed negative power of \(a_r\). The arbitrary side-flatness order
in (GL14) absorbs that power. Near \(r=2\), the same statement follows
from smooth flatness at that side. Away from those sides all factors have
ordinary compact smooth bounds. Dominated convergence, including each
derivative, proves the boundary limit
\[
 \begin{aligned}
 (T_a u)(x',0)
 &=\int_{-2}^{2}\int a_r^{-1}H(x',y',0,r)u(y',0)\,dy'\,dr\\
 &=\int\!\int_{-\infty}^{1}
       A(x',0,x'-y',z_n)u(y',0)\,dz_n\,dy'\\
 &=(2\pi)^{-(n-1)}\int e^{ix'\cdot\xi'}
       a(x',0,\xi',0)\widehat{u(\cdot,0)}(\xi')\,d\xi'.
 \end{aligned}
 \tag{GL24}
\]
The second equality uses the full change \(z_n=r/a_r\),
\(dz_n=a_r^{-2}dr\), and \(H(x',y',0,r)=a_r^{-1}A(x',0,x'-y',z_n)\).
The last equality is exact partial Fourier inversion at normal frequency
zero. The preceding localization and integration by parts justify both
equalities even when \(A\) is not a function at \(z=0\). When \(n=1\),
the tangential integrals have dimension zero and the factor is one.
This is the full boundary action, with all orders and all kernel factors,
and it agrees with the original jet formula (5.2) for \(k=0\).

## 2. Proper global operators and conormal smoothing

### 2.1. The global kernel and its actual pushforward

Choose a smooth function \(t\) on the stretched square, positive off its
new face and vanishing simply on that face. Near a boundary diagonal point
it can be the original \((x_n+y_n)/2\). Define the order-\(m\) class by
kernels represented as
\[
 k=t^{-1/2}H,\qquad
 H\in I^m(X\widehat\times X,\widehat\Delta;\Omega^{1/2}),
 \qquad H\text{ flat on both original side faces}.
 \tag{GC1}
\]
Here conormality at the new face has precisely the smooth family meaning
specified before (GL1). A bundle kernel additionally has values in
\(\operatorname{Hom}(E_y,F_x)\). Multiplication by its smooth frame
changes is included in the conormal topology.

There is a well-defined pushforward despite the separate factor
\(t^{-1/2}\). Indeed, in the original normal chart, let a test
half-density have coefficient \(p(x,y)\). Its pullback has coefficient
\(t^{1/2}p(\beta(x',y',t,r))\), since the absolute normal determinant is
\(t\). Thus the actual pairing is
\[
 \langle\beta_*k,p\rangle
 =\int_0^\infty
   \left\langle H(x',y',t,r),
       p(x',t(1+r/2),y',t(1-r/2))\right\rangle_{x',y',r}\,dt.
 \tag{GC2}
\]
The conormal family is a smooth distribution-valued function of \(t\).
On a compact interval its pairing with the smooth compact test family is
bounded by finitely many test derivatives and symbol seminorms, by (C4).
The integral is consequently well defined and continuous. The blowdown
has compact projective fibres, so the inverse image of a compact test
support is compact; away from the corner it is a diffeomorphism. Those
facts reduce the general pairing to finitely many such charts. Equation
(GC2) is the complete meaning of pairing \(k\) with the pulled-back test
half-density. Neither square-root factor is separately discarded, and no
pairing of an arbitrary distribution with a nonsmooth test coefficient is
asserted.

If \(\bar t\) is another admissible defining function, Hadamard's formula
gives \(\bar t=c t\), with \(c>0\) smooth near the new face. Away from
that face both functions are positive. Hence
\[
 \bar H=\bar t^{1/2}k=c^{1/2}H.
 \tag{GC3}
\]
Multiplication by \(c^{1/2}\), and by its smooth inverse, preserves the
conormal class and side flatness. The kernel, its pairing and its order
are independent of this choice.

Every compactly localized kernel of (GC1) is the original half-space
kernel of a lacunary order-\(m\) symbol by (GL19)–(GL21). Away from the
lifted diagonal the localized resolved coefficient is smooth, and the
same inverse gives a residual symbol. The latter assertion also applies
between distinct boundary charts: coordinates in the two charts may be
used as the two independent tangential variables in the smooth kernel;
there is no conormal singularity to align. Near an actual diagonal point
use the same coordinate chart on both factors. Interior charts give the
ordinary conormal pseudodifferential kernel by (C18).

The local theorem on smooth inputs therefore proves a continuous map
\[
 A:C_c^\infty(X;\Omega^{1/2}\otimes E)
       \longrightarrow C^\infty(X;\Omega^{1/2}\otimes F).
 \tag{GC4}
\]
For fixed input support and a fixed output compact set, only finitely many
product charts occur. Each local estimate uses finitely many derivatives
of the input, and the sum of these estimates is finite. This proves the
stated LF-to-Fréchet continuity. Formula (GL24) supplies its boundary
value, and the original formula (5.2) supplies every normal jet. Conversely
the local quantizations have exactly (GC1) by (GL13)–(GL18), so this kernel
description and the patched local definition determine the same class.
We denote it by \(\Psi_b^m(X;E,F)\).

To check coordinate invariance at all orders, the stretched coordinate
change is (GL5), its conormal change is (GL10), and its complete amplitude
and determinant are (C19). Reducing that entire amplitude by (C5) gives
the transformed symbol, with the full order-\((m-N)\) remainder after
\(N\) terms. All frequency factors created by a base derivative pair
with a frequency derivative and retain order \(m\). Thus the operator
class, not only its leading term, is coordinate invariant. The local
construction and the pushforward above agree in the interior and then
as distributions by the same conormal family pairing (GC2).

### 2.2. The symbol quotient and actual realization of every symbol

For a compressed symbol \(a\), the original principal half-density of
the kernel is the class of
\[
 a(x',t,\xi',\rho)t^{-1/2}
       |dx'\,dt\,d\xi'\,d\rho|^{1/2}.
 \tag{GC5}
\]
The singular half-density multiplying \(a\) is the invariant absolute
symplectic half-density in (GL12). The complete determinant law (C20)
therefore gives a scalar symbol, or a section of
\(\operatorname{Hom}(E,F)\), on \(\widetilde T^*X\), of ordinary order
\(m\), modulo order \(m-1\). Both (GL16) and (GL17) remain its actual
lower-order kernel remainder. Define \(S^m(\widetilde T^*X)\) using every
local compact base set, all base derivatives, and the frequency bound
\(\langle(\xi',\rho)\rangle^{m-|\alpha|}\). The fibre transformation
(GL10) and its inverse have smooth bounded coefficients on each compact
chart; the chain rule proves that this definition is intrinsic. The
leading symbol map has kernel exactly \(\Psi_b^{m-1}\), by the local
normal-form kernel assertion (C21).

We prove surjectivity while retaining the original prescribed symbol.
Choose a locally finite cover by precompact boundary or interior charts
\(U_j\), with locally finite closures. Choose a partition \(\varphi_j\)
with support in these charts and \(\psi_j\in C_c^\infty(U_j)\) equal to
one on a neighborhood of \(\operatorname{supp}\varphi_j\). For a given
symbol \(a\), compactly localize a full representative \(a_j\) in the
\(j\)-th frame so that it equals \(a\) on that neighborhood. In a
boundary chart take exactly the lacunary modification
\[
 (a_j)_\rho(x,\xi)=\int a_j(x,\xi',\xi_n-v)\rho(v)\,dv,
 \quad \widehat\rho\in C_c^\infty((-1/2,1)),\quad
 \widehat\rho=1\text{ near }0.
 \tag{GC6}
\]
Keep \(a_j=(a_j)_\rho+[a_j-(a_j)_\rho]\): the bracket is residual by
the full Taylor integral (4.8), and every original symbol seminorm
remains controlled. Quantize the first term and set
\[
 A=\sum_j\varphi_j T_{(a_j)_\rho}\psi_j
 \tag{GC7}
\]
in boundary charts, using ordinary quantization for interior charts.
The sum is locally finite and has order \(m\). Its two support projections
are proper: above a compact base set only finitely many chart closures
occur, and both factors in each term have compact support in that chart.
The leading term is \(\sum_j\varphi_j a=a\). Multiplication on the input
has its complete product remainder, of order \(m-1\), from (8.3);
\(\psi_j=1\) near the working diagonal. The residual bracket in (GC6)
has not become an equality between the original full symbol and its
modification; it is explicitly the error in this realization. This gives
both quotients
\[
 \Psi_b^m/\Psi_b^{m-1}\simeq
 S^m(\widetilde T^*X)/S^{m-1}(\widetilde T^*X),
 \tag{GC8}
\]
with the same isomorphism when the operator spaces are restricted to
proper support. The bundle version follows in the same charts, with the
actual frame transitions and (GC5).

### 2.3. Proper composition, adjoints and all asymptotic remainders

Let \(A\in\Psi_b^m(X;F,G)\) and
\(B\in\Psi_b^{m'}(X;E,F)\) be properly supported. For every compact
output set the support of \(A\) meets only a compact set of intermediate
variables, and that compact set meets only a compact input set under
\(B\). The reverse argument starts at a compact input set. This proves
proper support of the composed operator and permits all intermediate
partitions to be finite on the compact sets under consideration.

We give the localization argument, since a residual kernel at the new
face need not be a smooth kernel on the original square. Localize near
an output-input diagonal point. A sufficiently small common chart
contains both variables. Split the intermediate variable into a slightly
larger part of that chart and its complement. On the complementary part
both kernels are separated from their actual diagonal, so their resolved
coefficients are smooth with side flatness. On the chart part use the
original local ordered product theorem (8.1)–(8.3). Any localization of
either factor off its own diagonal gives a residual symbol by (GL19)–(GL21).
That theorem then gives a residual product: its finite-order bound holds
for every real order assigned to the residual factor, while its complete
far term (8.2) is residual as well.

For clarity, this also handles two different boundary charts. For a
localized rectangular smooth kernel \(R(x,y)\), with output and input
coordinates taken from their respective charts, the literal inverse is
\[
 A_R(x,z)=x_nR\bigl(x,(x'-z',x_n(1-z_n))\bigr),
 \qquad a_R(x,\xi)=\int e^{-iz\cdot\xi}A_R(x,z)\,dz.
 \tag{GC9a}
\]
This is the same (GL19) inverse written at fixed \(x_n>0\), with all
chart half-density factors included in \(R\). Resolved smoothness,
side flatness, compact chart cutoffs and the full far estimates of
(GL19)–(GL21) make \(a_R\) residual and lacunary. The two coordinate
tuples both range in Euclidean half spaces, so the original local product
theorem applies to these rectangular symbols with the intermediate
coordinate tuple used as its common integration variable. If only one
factor has a diagonal singularity, use its chart for the intermediate
variable and its adjacent variable; the other is represented by
(GC9a). If neither factor has a singularity, either choice works. Near
a fixed output-input point off the diagonal, use disjoint neighborhoods
of those two points and split the intermediate variable into their
neighborhoods and the complement. At least one factor in each term is
then separated from its diagonal, and the preceding argument proves a
residual result. This proves the required smoothness off the lifted
diagonal, its family regularity at the new face, and its side flatness;
it does not replace a corner residual by an ordinary smoothing kernel.

In the common chart the complete product symbol is \(c=c_1+c_2\), where
\(c_1\) and \(c_2\) are the actual (8.1) and (8.2), with the original
factor order \(a\) before \(b\). The full expansion is
\[
 c\sim\sum_\alpha\frac1{\alpha!}\partial_\xi^\alpha a(x,\xi)
 D_{x'}^{\alpha'}D_v^{\alpha_n}
 b(x',v x_n,\xi',v\xi_n)\big|_{v=1},
 \qquad
 c-\sum_{|\alpha|<N}(\cdots)\in S^{m+m'-N}.
 \tag{GC9}
\]
The exact far contribution \(c_2\in S^{-\infty}\) is part of \(c\).
All its seminorms and the remainder seminorms are controlled by finitely
many seminorms of the two original factors, by that local proof. Hence
\[
 AB\in\Psi_b^{m+m'},\qquad
 \sigma(AB)=\sigma(A)\sigma(B).
 \tag{GC10}
\]
For matrices this is \(E\xrightarrow{\sigma(B)}F
\xrightarrow{\sigma(A)}G\); no order is exchanged. The same argument
proves that the residual class is a two-sided ideal among properly
supported operators of any finite order.

Transposition of the stretched square exchanges \(x,y\) and sends
\(r\) to \(-r\), keeping \(t=(x_n+y_n)/2\). Its kernel half-density
law is
\[
 H_{A^*}(x',y',t,r)=H_A(y',x',t,-r)^*.
 \tag{GC11}
\]
Conormality and side flatness are preserved. The local adjoint (7.3),
including the residual adjoint of \(a-a_\rho\), gives
\(a^\dagger-a^*\in S^{m-1}\). Thus
\[
 A^*\in\Psi_b^m(X;F,E),\quad
 \sigma(A^*)=\sigma(A)^*,\quad
 (\lambda A+\mu B)^*=\overline\lambda A^*+\overline\mu B^*.
 \tag{GC12}
\]
Proper support is preserved by this exchange. All adjoints here use the
fixed metrics and the original half-density pairing.

The calculus admits full asymptotic sums. Here is the construction needed
later for parametrices. In each chart let \(a_j\in S^{m-j}\) be the
actual localized symbols to be summed. Choose \(f\) zero on the unit
frequency ball and one outside twice that ball. Choose \(R_j\) increasing
so that \(f(\xi/R_j)a_j\) has each of the first \(j\) seminorms in
\(S^{m-j+1}\) at most \(2^{-j}\), on the first \(j\) compact base sets.
This is possible because the support has \(|\xi|\ge R_j\), giving the
extra factor \(R_j^{-1}\); frequency derivatives of the cutoff have the
same bound after the product rule. The exact sum
\[
 a_{\mathrm{sum}}=\sum_{j=0}^\infty f(\xi/R_j)a_j,
 \qquad
 a_{\mathrm{sum}}-\sum_{j<N}a_j\in S^{m-N}
 \tag{GC13}
\]
converges in the stated seminorms. For the remainder, its infinite tail
lies in \(S^{m-N}\); each of its finitely many low-frequency differences
is residual. Lacunarize the full sum by (GC6), retaining the new residual
difference, and patch as in (GC7). Each finite off-diagonal discrepancy
is residual by the localization argument above. Consequently the global
operator differs from every prescribed finite operator sum by exactly
the asserted lower-order class. This proves asymptotic completeness,
including on the proper-support subspace, without omitting any finite
term or its remainder.

### 2.4. The supported and restricted Sobolev maps

For \(m\ge0\) and every real \(s\), the actual local theorem (PS1)–(PS17)
gives the supported bound with loss \(m\). Smooth coordinate changes
and frame multiplications on compact sets are bounded on every real
Sobolev scale: integer bounds follow by the chain and product rules,
negative integers by their exact adjoints with the Jacobian, and
intermediate orders by the Fourier interpolation already proved in the
local prerequisite. On a fixed compact input set, proper support and a
partition reduce \(A\) to finitely many of those bounds. A rectangular
residual term has every order, so use its original order-zero bound and
the continuous inclusion \(H^s\subset H^{s-m}\), valid for \(m\ge0\).
Thus, for each compact \(K\), there is a compact \(K'\) and a constant
depending on finitely many localized full-symbol seminorms such that
\[
 \operatorname{supp}u\subset K\quad\Longrightarrow\quad
 \operatorname{supp}Au\subset K',\qquad
 \|Au\|_{\dot H^{s-m}(K')}
       \le C\|u\|_{\dot H^s(K)}.
 \tag{GC14}
\]
The supported action is the original transpose action (9.1), so it
includes distributions supported at the boundary. Both localizations
and coordinates keep those terms; they are not zeroed by a chosen
extension. For a nonproper operator the same proof gives local output
bounds for compact input.

The corresponding restricted map follows from the exact local quotient
map (PS16)–(PS17). Local supported representatives give the bounded map;
the local action preserves the full ideal of distributions supported
only on the boundary by (9.2), so different representatives have the
same interior output. Taking the infimum over representatives in each
fixed chart gives the restricted norm bound. Patch the actual quotient
maps using the same finite cutoffs. This proves
\(\overline H^s_{\mathrm{comp}}\to\overline H^{s-m}_{\mathrm{loc}}\),
and the compact-output assertion under proper support. No gain of
\(-m\) is asserted when \(m<0\); the original residual examples prohibit
such a claim.

### 2.5. The supported conormal class and the residual receiving map

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
\[
 Pu\in B^\kappa_{2,\infty,\mathrm{loc}}
 \quad\text{for every }P\in\operatorname{Diff}_b(X),
 \qquad \operatorname{supp}u\subset X.
 \tag{GA1}
\]
The topology uses all these local seminorms. Multiplication, coordinate
changes and finite frame transitions are continuous on those Besov
spaces by (C2)–(C3), and (GL9) identifies the intrinsic tangent fields.
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
\[
 x_n^kD_n^k=\prod_{j=0}^{k-1}(X_n+ij),\qquad
 X_n^k=\sum_{j=0}^k c_{kj}\,x_n^jD_n^j,
 \tag{GA2}
\]
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.

If \(A\in\Psi_b^d\) is proper, its local conormal preservation theorem
11.2, with the actual index \(\kappa=-m-n/4\), gives
\[
 A:\mathcal A^m(X;E)\longrightarrow\mathcal A^m(X;F)
 \tag{GA3}
\]
continuously, for every real \(m,d\). Finite localized product estimates
prove the global continuity as in (GC14). An off-diagonal resolved term
is residual and obeys the same theorem in the rectangular chart.
The order \(d\) does not change the conormal index: the local proof
factorizes each full tangent derivative of the original operator through
an even-order totally characteristic differential operator and applies
the order-zero Besov bound to its two complete factors. In particular
neither the original frequency nor a positive-order remainder is omitted.

There is a stronger receiving statement for a residual operator
\(R\in\Psi_b^{-\infty}\). Fix an input compact set and \(N>0\).
For every tangent word \(P\), the composition \(PR\) is still residual,
by the local product formulas and (GC10). Its order-zero bound therefore
gives
\[
 \|PRu\|_{H^{-N}(K')}\le C_{P,N,K}\|u\|_{\dot H^{-N}(K)},
 \qquad
 Ru\in\mathcal A^{N-n/4}(X).
 \tag{GA4}
\]
The inclusion \(H^{-N}\subset B^{-N}_{2,\infty}\) is immediate from
the dyadic \(\ell^2\) and \(\ell^\infty\) norms. Any compactly supported
distribution of order \(L\) belongs to \(H^{-N}\) for
\(N>L+n/2\): its Fourier transform is bounded by
\(C\langle\xi\rangle^L\), and the weighted square integral converges
for exactly that strict inequality. The same estimate is uniform on a
family with fixed support and bounded distribution-order seminorm.
Thus
\[
 R:\mathcal E'(X)\longrightarrow
       \mathcal A(X):=\bigcup_{m\in\mathbb R}\mathcal A^m(X),
 \tag{GA5}
\]
with the explicit fixed-Sobolev-source continuity in (GA4). This is a
conormal receiving map, not an assertion that a residual operator
produces a smooth function at the boundary.

### 2.6. 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
\[
 [X_n,Q_\varepsilon]u
     =D_{z_n}(z_nq_\varepsilon)*u.
 \tag{GS1}
\]
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
\[
 X_n^\ell Q_\varepsilon
       =\sum_{j=0}^{\ell}\binom\ell j
          Q_{j,\varepsilon}X_n^{\ell-j}.
 \tag{GS2}
\]
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 \(j\),
\[
 |\widehat q(\varepsilon\xi)-1|
       \le C\min(\varepsilon|\xi|,1),\qquad
 |\widehat q_j(\varepsilon\xi)|
       \le C_j\min(\varepsilon|\xi|,1)\quad(j\ge1).
 \tag{GS3}
\]
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
\[
 \sup_{l\ge0}2^{-l\delta}\min(\varepsilon2^l,1)
       \le C_\delta\varepsilon^{\tau}\quad(0<\varepsilon\le1),
 \tag{GS4}
\]
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
\[
 \|D'^{\alpha'}X_n^\ell(Q_\varepsilon u-u)\|_{B^\kappa_{2,\infty}}
 \le C_{\alpha',\ell}\varepsilon^{\min(1,m-m')}
       \sum_{j=0}^\ell
       \|D'^{\alpha'}X_n^{\ell-j}u\|_{B^{\kappa'}_{2,\infty}}.
 \tag{GS5}
\]
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 as in (GC7). 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
\[
 \mathcal Q_\varepsilon u
    =\sum_j\varphi_j Q^{(j)}_\varepsilon(\psi_j u).
 \tag{GS6}
\]
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
\[
 \mathcal Q_\varepsilon:\mathcal E'(X)\to C^\infty(X),\qquad
 \mathcal Q_\varepsilon u\longrightarrow u\text{ in }\mathcal A^m(X)
 \quad(u\in\mathcal A^{m'},\ m'<m).
 \tag{GS7}
\]
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 \(|\tau|\ge1\),
\[
 \widehat{H\phi}(x',\tau)
 =\sum_{j=0}^{N-1}\frac{\partial_t^j\phi(x',0)}{(i\tau)^{j+1}}
  +\frac{1}{(i\tau)^N}
       \int_0^\infty e^{-it\tau}\partial_t^N\phi(x',t)\,dt .
 \tag{GD1}
\]
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
\[
 C_c^\infty(X)\hookrightarrow\mathcal A^m_c(X)
 \quad\text{continuously for every }m\ge m_0,
 \tag{GD2}
\]
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
\[
 |u(\phi)|\le C_{m,K}\sum_{l=1}^{L}p_{m,K,l}(H\phi)
 \quad(\operatorname{supp}\phi\subset K).
 \tag{GD3}
\]
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
\[
 \mathcal A'(X)\longrightarrow\mathcal D'(X^\circ),
 \qquad u\longmapsto u|_{X^\circ}
 \quad\text{is injective}.
 \tag{GD4}
\]
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
\[
 u_\varepsilon(v)=u(\mathcal Q_\varepsilon v)
       \longrightarrow u(v).
 \tag{GD5}
\]
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
\[
 T\varphi=\varphi(x')\otimes\delta(t),\qquad
 \delta(t)=(2\pi)^{-1}\int_{\mathbb R}e^{it\tau}\,d\tau.
 \tag{GD6}
\]
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,
\[
 T:C_c^\infty(\partial X;E^*\otimes\Omega_{\partial X})
       \longrightarrow
       \mathcal A^{(2-n)/4}_c(X;E^*\otimes\Omega_X)
 \quad\text{continuously}.
 \tag{GD7}
\]
The direct Besov estimate uses the same dyadic amplitude bound, so the
target topology is included. Define the boundary restriction by
\[
 \langle u|_{\partial X},\varphi\rangle=u(T\varphi),
 \qquad u\in\mathcal A'(X).
 \tag{GD8}
\]
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
\[
 \nabla_j^\mathrm{int}u
   :=D_j u+i\delta_{jn}\,u_\partial\otimes\delta(x_n).
 \tag{GD9}
\]
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
\[
 D_n(Hu) = H(D_nu)-i\,u(x',0)\otimes\delta(x_n),
 \qquad
 D_j(Hu)=H(D_ju)\quad(j<n).
 \tag{GD10}
\]
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
\[
 -D_j(H\phi)
     =H(-D_j\phi)+i\delta_{jn}\,
                         \phi(x',0)\otimes\delta(x_n).
 \tag{GD11}
\]
Therefore the two terms in (GD9) combine exactly to
\[
 (\nabla_j^\mathrm{int}u)(\phi)
   =u\bigl(-D_j(H\phi)\bigr).
 \tag{GD12}
\]
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.

### 3.5. Proper totally characteristic operators on the dual class

Let \(B\in\Psi_b^d(X;E,F)\) be properly supported. Its formal adjoint
\(B^*\) preserves \(\mathcal A^m\) continuously for every \(m\), by
(GC12) and (GA3). Given a compact output test \(v\in\mathcal A^m\),
proper support places \(B^*v\) in a compact set depending only on the
support of \(v\); its full conormal seminorms are bounded by finitely
many input seminorms. To use these bounds with complex-linear distribution evaluation, take
a compact dual-density test \(\phi\) and put
\(v=\iota_F^{-1}\phi\), where the exact metric-and-density map
\(\iota_F\) is defined in Section 3.6. The bilinear transpose is
\(B^{\mathrm t}=\iota_E B^*\iota_F^{-1}\). The actual linear dual action is
\[
 (Bu)(\phi)=u(B^{\mathrm t}\phi).
 \tag{GD13}
\]
The two conjugate-linear maps in this transpose make it complex-linear.
Section 3.6 proves the identity with the full bundle and density factors
and proves that its conormal-test seminorms have the required finite bounds.
For a smooth boundary test, (GD3) therefore proves
\(Bu\in\mathcal A'\). For smooth \(u\), the local adjoint identity
(7.6), transported by those exact maps, gives the original kernel action.
For general \(u\), (GD5) and weak continuity extend that equality; in the
interior it matches the usual distributional operator. The coefficient order
and bundle maps remain those of the original \(B\), with no scalar
commutation of matrices.

The boundary jet formula extends as well. In a local half-space chart,
let \(a\in S^d_{\mathrm{la}}\), and define the \(k\)-th interior normal
derivative of \(u\) by iterating (GD9), then taking (GD8). For smooth
\(u\), the exact formula is
\[
 \begin{aligned}
 \bigl(\nabla_n^{\mathrm{int},k}T_au\bigr)|_{x_n=0}
   &=\sum_{j=0}^k\binom{k}{j}
       a_{kj}(x',D')
       \bigl(\nabla_n^{\mathrm{int},j}u|_{x_n=0}\bigr),\\
 a_{kj}(x',\xi')
   &=\sum_{i=0}^j\binom{j}{i}
       \bigl(D_{x_n}^{k-j}D_{\xi_n}^{i}a\bigr)
                   (x',0,\xi',0).
 \end{aligned}
 \tag{GD14}
\]
Every coefficient in the inner sum and the outer binomial factor is
retained. Its \(i\)-th summand has order \(d-i\). The tangent boundary
operator \(a_{kj}(x',D')\) acts continuously on boundary distributions
after compact localization; this is the ordinary local symbol action.
The left and right sides of (GD14) are weakly continuous in \(u\) by
(GD8), (GD9), (GD13) and the boundary operator continuity. Smooth
functions are weakly dense by (GD5), so (GD14) holds for every
\(u\in\mathcal A'\). It is a statement about their actual boundary
traces, not about an arbitrary supported representative's raw
distributional normal derivatives.

### 3.6. Linear distribution tests and Hermitian adjoints

The distributions in (GD3), (GD11), (GE9), and (GT6) evaluate their
dual-density tests complex-linearly. A Hermitian pairing, with inner
product linear in its first argument, is conjugate-linear in its test.
We construct the exact map between these evaluations. This also fixes
which operator must be transposed in (GD13).

Let \(\Omega_X\) be the density bundle, let \(\mu\) be a smooth positive
density, and choose smooth positive Hermitian metrics \(h_E,h_F\).
The test for an \(E\)-valued distribution is a compact section of
\(E^*\otimes\Omega_X\). Define
\[
 \iota_E:E\longrightarrow E^*\otimes\Omega_X,\qquad
 \iota_E(v)(w)=h_E(w,v)\mu .
 \tag{DT1}
\]
This is conjugate-linear in \(v\), and it is bijective in each fibre:
in a local frame its matrix is invertible because the Hermitian metric
is positive and \(\mu\) is nowhere zero. Its inverse is smooth and
conjugate-linear. Both maps retain the density and the full metric matrix.
Define the antilinear-test evaluation of the same distribution \(U_{\mathrm L}\)
by
\[
 U_{\mathrm H}(v)=U_{\mathrm L}(\iota_Ev),\qquad
 U_{\mathrm L}(\phi)=U_{\mathrm H}(\iota_E^{-1}\phi).
 \tag{DT2}
\]
Thus \(U_{\mathrm L}\mapsto U_{\mathrm H}\) is complex-linear in the
distribution and bijective. Supports agree, since \(\iota_E\) and its
inverse preserve the support of every test. Evaluating fixed tests on
either side proves continuity in both weak dual topologies.
This is the pairing used in the linked local Theorem 9.1(a),(b).

Here are the density factors when the original kernel acts on half-densities.
For
\(\mathbf B:E\otimes\Omega_X^{1/2}\to F\otimes\Omega_X^{1/2}\),
write its function-section presentation and its Hermitian adjoint as
\[
 B_\mu=\mu^{-1/2}\mathbf B\mu^{1/2},\qquad
 B_\mu^*=\mu^{-1/2}\mathbf B^*\mu^{1/2}.
 \tag{DT3}
\]
Indeed \(w\mapsto w\mu^{1/2}\) carries
\(\int h_E(w,v)\mu\) exactly to the intrinsic half-density pairing.
Applying the defining adjoint identity for \(\mathbf B\), then this
map and its inverse, proves the second equality. The factors
\(\mu^{-1/2}\) and \(\mu^{1/2}\) stay in the operator product.
For an operator \(B\) already acting on function sections, \(B^*\)
means its adjoint for the displayed \(h_E,h_F,\mu\).

For compact smooth \(w,\phi\), set \(v=\iota_F^{-1}\phi\). Then
\[
 \begin{aligned}
 \int (Bw)\mathbin{\cdot}\phi
 &=\int h_F(Bw,v)\mu\\
 &=\int h_E(w,B^*v)\mu\\
 &=\int w\mathbin{\cdot}
               \bigl(\iota_E B^*\iota_F^{-1}\phi\bigr).
 \end{aligned}
 \qquad
 B^{\mathrm t}=\iota_E B^*\iota_F^{-1}:
 F^*\otimes\Omega_X\longrightarrow E^*\otimes\Omega_X .
 \tag{DT4}
\]
The dot here is evaluation of a dual-density section on its vector,
with no complex conjugation. Two conjugate-linear maps surrounding
the linear \(B^*\) give a complex-linear \(B^{\mathrm t}\).
The integral identity characterizes this transpose on smooth tests.
Proper support supplies its compact test domain, so it also defines the
distributional action in (GD13). Applying (DT2) and (DT4) gives
\[
 \begin{aligned}
 (B U_{\mathrm L})(\phi)&=U_{\mathrm L}(B^{\mathrm t}\phi),\\
 (B U)_{\mathrm H}(v)
   &=(B U_{\mathrm L})(\iota_Fv)
     =U_{\mathrm L}(\iota_E B^*v)
     =U_{\mathrm H}(B^*v).
 \end{aligned}
 \tag{DT5}
\]
Every equality has its stated test space. The same original operator
therefore acts in both evaluations.

For completeness these maps preserve every original conormal-test order.
Locally write \(\iota_Ev=M(x)\overline v\,|dx|\), where \(M\)
includes the full positive density coefficient and the metric matrix.
For \(D=-i\partial\), the complete derivative formula is
\[
 D^\alpha(M\overline v)
 =\sum_{\beta\le\alpha}\binom{\alpha}{\beta}
       (D^{\alpha-\beta}M)(-1)^{|\beta|}
                       \overline{D^\beta v}.
 \tag{DT6}
\]
Complex conjugation reflects the full Fourier variable
\(\xi\mapsto-\xi\), preserving the original radial dyadic Besov weights.
Smooth compact multiplication obeys the finite seminorm bounds of (GA3).
The product rule for every boundary-tangent word gives the same finite
sum, with all derivatives of \(M\); the smooth inverse matrix has
the same property. Hence \(\iota_E,\iota_E^{-1}\), and their \(F\)
versions preserve every filtered \(\mathcal A^m\) continuously.
Combining this fact with the adjoint estimate preceding (GD13) proves
the required transpose estimate, with its actual input and output compacts.
(GD3) proves that \(B U_{\mathrm L}\in\mathcal A'\), and evaluation
of each fixed transposed test proves weak continuity. The maps (DT2)
are also bijections of the corresponding conormal dual classes.

The linear action is independent of the auxiliary metric and density.
For two choices let
\(S_E=\iota_{E,1}^{-1}\iota_{E,2}\) and
\(S_F=\iota_{F,1}^{-1}\iota_{F,2}\); these are smooth linear
invertible test maps. Applying (DT4) twice gives exactly
\[
 B^*_2=S_E^{-1}B^*_1 S_F,\qquad
 U_{\mathrm H,2}(v)=U_{\mathrm H,1}(S_Ev),\qquad
 \iota_{E,2}B^*_2\iota_{F,2}^{-1}
      =\iota_{E,1}B^*_1\iota_{F,1}^{-1}.
 \tag{DT7}
\]
For a fixed function-section operator \(B\), these equalities follow
by multiplying by the stated inverse maps in their displayed order.
If one changes a half-density presentation, (DT3) supplies its full
additional transport. No density or bundle is silently identified
with another.

Two scalar computations test the signs. On the positive half-line take
\(B=iI\), real \(\chi\in C_c^\infty((1,2))\), \(\chi\ne0\),
\(U_{\mathrm L}=H\chi\), and \(\phi=\chi(t)\,dt\).
Then
\[
 B^{\mathrm t}=iI,\quad B^*=-iI,\qquad
 (B U_{\mathrm L})(\phi)=i\int\chi^2\,dt,\qquad
 U_{\mathrm L}(B^*\phi)=-i\int\chi^2\,dt .
 \tag{DT8}
\]
The last two values differ. In (DT5) the test evaluation
\(U_{\mathrm H}\) is antilinear, and its factor \(-i\) gives
\(+iU_{\mathrm H}(\chi)\), exactly the original action.
In a flat scalar chart let \(C\) be complex conjugation. Since
\(CDC=-D\), one has \(D^{\mathrm t}=-D\) and \(D^*=D\).
The corrected normal action (GD9)--(GD12) therefore reads
\[
 \begin{aligned}
 (\nabla_n^{\mathrm{int}}U)_{\mathrm L}(\phi)
    &=U_{\mathrm L}(-D_n(H\phi)),\\
 (\nabla_n^{\mathrm{int}}U)_{\mathrm H}(v)
    &=U_{\mathrm H}(D_n(Hv))
      =U_{\mathrm H}(H D_nv)+iU_{\mathrm H}(v|_{t=0}\otimes\delta),\\
 D_n(Hv)&=H D_nv-i(v|_{t=0})\otimes\delta .
 \end{aligned}
 \tag{DT9}
\]
The \(+i\) in the second line uses the antilinearity of
\(U_{\mathrm H}\); the normal delta in the third line retains
its factor \(-i\). For a variable metric or density the exact test
map is
\(\iota_E^{-1}[-D_n(H\iota_Ev)]\), obtained from (DT2).
Its full product derivatives of the density and metric remain;
the flat formula is asserted only in the stated flat chart.

Finally the normal primitive in Section 5.2 is
\(Jf(t)=i\chi(t)\int_{-\infty}^{t}\theta(s)f(s)\,ds\).
Fubini on the compact support of \(\chi,\theta\) gives its full
bilinear transpose and flat Hermitian adjoint:
\[
 \begin{aligned}
 (J^{\mathrm t}g)(s)
   &=i\theta(s)\int_s^\infty\chi(t)g(t)\,dt,\\
 (J^*g)(s)
   &=-i\overline{\theta(s)}
                  \int_s^\infty\overline{\chi(t)}g(t)\,dt .
 \end{aligned}
 \tag{DT10}
\]
To see the first line, interchange the integrals in
\(\int i\chi(t)\int_{s\le t}\theta(s)f(s)\,ds\,g(t)\,dt\);
the coefficient of \(f(s)\) is exactly the first displayed expression.
Replacing \(g\) by its conjugate and conjugating that coefficient
proves the second line. For the real cutoffs chosen in Section 5.2,
and \(\chi\) supported below \(c\), this is exactly its stated
\(-i\theta(s)\int_s^c\chi(t)g(t)\,dt\).
All derivatives of both cutoffs appear in the integer Sobolev bounds.
The Hermitian adjoint supplies the negative-order Sobolev estimate
there. The bilinear transpose supplies the raw linear evaluations
in (GE9) and (GT6), including every coefficient derivative.
The boundary jet formula (GD14), weighted extension (GE23),
and trace comparison (SC4) therefore use one consistent original
distributional action.

![The exact transpose and Hermitian-adjoint maps](../figures/linear-and-hermitian-duals.png)

The two arrows \(\iota_F,\iota_E\) include the complete metric and
density. Their square commutes by (DT4), and (DT5) transports the
same operator to its two dual evaluations. The half-density factors
are (DT3), and the scalar phase test is (DT8).

## 4. Compressed wave fronts

### 4.1. Ellipticity and the characteristic set at every symbol order

Let \(B\in\Psi_b^m(X;E,F)\) and let \(b\) be its complete local
symbol. At a nonzero compressed covector \(q=(x_0,\zeta_0)\), call
\(B\) elliptic if the two bundle ranks agree and there are a base
neighborhood \(U\), an open cone \(\Gamma\) containing
\(\zeta_0\), constants \(c,R>0\), and local frames such that
\[
 b(x,\zeta):E_x\to F_x\text{ is invertible},\qquad
 \|b(x,\zeta)^{-1}\|\le c^{-1}\langle\zeta\rangle^{-m}
 \quad(x\in U,\ \zeta\in\Gamma,\ |\zeta|\ge R).
 \tag{GW1}
\]
The set of covectors where this fails is \(\operatorname{Char}B\).
Changing the representative by \(S^{m-1}\) does not change (GW1):
\(b_0^{-1}(b-b_0)\) is \(O(\langle\zeta\rangle^{-1})\), so for
large \(|\zeta|\) its ordered Neumann series is invertible. Smooth
frame changes conjugate or left/right multiply by uniformly invertible
matrices on a smaller compact chart. Thus the definition is intrinsic.
The complement of the characteristic set is open and conic; the set
itself is closed and conic in \(\widetilde T^*X\setminus0\).

The inverse in (GW1) has the exact order \(-m\) on a smaller cone.
Differentiate \(b^{-1}b=I_E\):
\[
 \partial_{\zeta_j}b^{-1}
   =-b^{-1}(\partial_{\zeta_j}b)b^{-1},\qquad
 \partial_{x_j}b^{-1}
   =-b^{-1}(\partial_{x_j}b)b^{-1}.
 \tag{GW2}
\]
Repeated product differentiation keeps the original matrix order.
Every frequency derivative lowers the order by one; every base
derivative leaves it unchanged. Multiplying by a conic cutoff gives a
global chart symbol of order \(-m\). This proves the symbol estimate
needed for an actual microlocal parametrix.

Here is the complete construction. Choose nested cones
\(\Gamma_0\Subset\Gamma_1\Subset\Gamma\) and base neighborhoods
\(U_0\Subset U_1\Subset U\), with a smooth large-frequency cutoff
\(\chi\) supported in \(U_1\times\Gamma_1\) and equal to one on
\(U_0\times\Gamma_0\) for \(|\zeta|\) large. Set
\(c_0=\chi b^{-1}\), with the original map \(F\to E\).
The lacunary realization (GC6)–(GC7) changes this by a residual
symbol only. The ordered product gives
\[
 c_0\# b=\chi I_E+e_1,\qquad e_1\in S^{-1},
 \quad e_1\text{ residual away from the elliptic working cone}.
 \tag{GW3}
\]
Suppose the sum \(c_0+\cdots+c_{N-1}\) has error
\(e_N\in S^{-N}\), residual away from that cone. Its next correction is
\[
 c_N=-\chi_1e_N b^{-1}\in S^{-m-N},
 \qquad (e_N+c_N b)=0
 \quad\text{where }\chi_1=1\text{ on the nonresidual support of }e_N.
 \tag{GW4}
\]
The actual product \(c_N\# b-c_Nb\) is one order lower by (GC9),
so the new error is in \(S^{-N-1}\), still residual off the
elliptic cone. The cutoff \(\chi_1\) is supported inside the cone
where \(b^{-1}\) exists. At each stage the part not cancelled by
\(\chi_1=1\) was already residual; include it in the final residual
instead of dividing it by \(b\). Asymptotic summation (GC13) produces
a single properly supported \(C\in\Psi_b^{-m}(F,E)\) with the
global localized identity
\[
 CB=Q+R,
 \qquad Q=\operatorname{Op}(\chi I_E)\in\Psi_b^0(E,E),
 \quad R\in\Psi_b^{-\infty},
 \tag{GW5}
\]
and \(Q\) is elliptic on \(U_0\times\Gamma_0\). The low-frequency
part and all coordinate-patching errors are residual, and every
finite error is retained until its correction in (GW4).
The same construction on \(b c=I_F\) yields a right parametrix
when needed; it is a separate ordered calculation, not inferred by
commuting the matrices in (GW4).

### 4.2. The compressed wave-front set

For a supported distribution \(u\in\mathcal D'_X\), define
\[
 \operatorname{WF}_b(u)
 :=\bigcap_{\substack{B\in\Psi_b^0(X;E,E)\text{ proper}\\
                         Bu\in\mathcal A(X;E)}}
       \operatorname{Char}B
 \quad\subset\widetilde T^*X\setminus0 .
 \tag{GW6}
\]
The zero operator is always an admissible test, with characteristic
set equal to the whole nonzero compressed bundle. The identity is an
admissible test exactly when \(u\in\mathcal A(X)\). Thus the family
of tests is never empty. Every characteristic set is closed and conic by (GW1), and so
is its intersection. At a covector outside \(\operatorname{WF}_b(u)\)
there is one properly supported order-zero \(B\), elliptic there,
with \(Bu\in\mathcal A\). The target cannot be changed to
\(C^\infty(X)\) for arbitrary supported distributions: (GA5)
receives a residual operator into \(\mathcal A\), and its original
corner kernel may fail to smooth at the boundary.

For later use we prove the finite-cover consequence of (GW6). If
\(\operatorname{WF}_b(u)\) is empty above a compact \(K\subset X\),
the compressed unit cosphere above \(K\) has a finite cover by cones
where order-zero operators \(B_1,\ldots,B_N\) are elliptic and
\(B_ju\in\mathcal A\). Choose a smooth partition of unity
\(\chi_j\) in those cones at \(|\zeta|\ge R\), summing to a spatial
cutoff \(\varphi=1\) near \(K\). Apply (GW3)–(GW4) separately to the
symbols \(\chi_j b_j^{-1}\), preserving their factor order.
Asymptotic summation gives properly supported \(C_j\) and a residual
\(R\) such that
\[
 \sum_{j=1}^N C_jB_j=\varphi+R.
 \tag{GW7}
\]
The missing compact-frequency part is a residual kernel and is included
in \(R\). Choose the output supports of all \(C_j\) in one compact
neighborhood of \(\operatorname{supp}\varphi\). Then \(R\) has
compact output support; proper support gives one compact input set.
Choose a compact smooth \(\psi\) equal to one on that input set.
The exact identity \(Ru=R(\psi u)\) lets (GA5) apply to the compact
distribution \(\psi u\), even when the original \(u\) is not compact.
Acting on \(u\), every \(C_j(B_ju)\) belongs to
\(\mathcal A\) by (GA3), and \(Ru\in\mathcal A\) by (GA5).
Thus \(\varphi u\in\mathcal A\). On a noncompact manifold the
order obtained from this finite cover may depend on the compact set.
Retain the original class \(\mathcal A=\bigcup_m\mathcal A^m\), and
define its exact local enlargement by

\[
 \mathcal A_{\mathrm{loc}}(X)
 =\{u\in\mathcal D'_X:
       \chi u\in\mathcal A(X)\text{ for every }\chi\in C_c^\infty(X)\}.
 \tag{GW8a}
\]

Multiplication and restriction give the injective map
\(\mathcal A\hookrightarrow\mathcal A_{\mathrm{loc}}\). The preceding
finite-cover argument proves the exact replacement for the global-order
assertion:

\[
 \operatorname{WF}_b(u)=\varnothing
       \quad\Longleftrightarrow\quad u\in\mathcal A_{\mathrm{loc}}(X).
 \tag{GW8}
\]

For the converse, at any compressed covector over \(x\), choose a
compact smooth \(\chi\) equal to one near \(x\). The multiplication
operator \(\chi I\) is properly supported, elliptic at that covector,
and maps \(u\) into the original \(\mathcal A\), by (GW8a). These
testers exclude every covector. For the forward implication, apply
(GW7) on a compact neighborhood of \(\operatorname{supp}\chi\), then
multiply its conormal output by \(\chi\). The maximum of the finitely
many conormal orders is one valid order for that compact output.
If \(u\) has compact support, choose \(\chi=1\) on its support;
then \(\chi u=u\) and (GW8) does imply \(u\in\mathcal A\).
In particular the original global-order equivalence holds on compact
\(X\). No single order is asserted after an infinite exhaustion.

The distinction is necessary. On
\(X=\mathbb R_y\times[0,\infty)_t\), of dimension two, choose
nonzero \(\theta\in C_c^\infty((-1/4,1/4))\) and form the actual
locally finite supported distribution

\[
 u(y,t)=\sum_{j=1}^\infty\theta(y-j)\otimes\delta^{(j)}(t).
 \tag{GW8b}
\]

Each compact set meets only finitely many summands. The \(j\)-th
summand has normal amplitude \((i\tau)^j\theta(y-j)\) with inverse
factor \((2\pi)^{-1}\), so its conormal order is exactly \(j\), by
the original codimension-one shift \(m+(n-2)/4\) with \(n=2\).
It belongs to \(\mathcal A^j\), including all tangent derivatives.
It does not belong to \(\mathcal A^m\) when \(m<j\). To verify the
last statement directly, select a bounded interval in tangential
frequency on which the squared Fourier transform of \(\theta\) has
positive integral. On the full sharp dyadic annulus, restrict the
normal frequency to \(c_1 2^l<|\tau|<c_2 2^l\) strictly inside
that annulus. Its squared Fourier integral is bounded below by
\(c_j2^{l(2j+1)}\). The (GA1) Besov factor is
\(2^{l(-m-1/2)}\), so the resulting norm is at least
\(c'_j2^{l(j-m)}\), which diverges for \(m<j\).
Multiplying (GW8b) by a compact cutoff equal to one around its
\(j\)-th boundary support isolates that summand. Thus (GW8b) lies
in \(\mathcal A_{\mathrm{loc}}\) and has empty compressed wave front,
but lies in no \(\mathcal A^m\). This proves strictness of the
displayed injection. The quotient
\(\mathcal A_{\mathrm{loc}}/\mathcal A\) records precisely this
failure of one globally bounded order, with kernel of the quotient
map equal to the original \(\mathcal A\).

The stronger smoothness conclusion for \(u\in\mathcal A'\) needs
the separate local intersection theorem proved below. It is not
being inferred from conormality alone.

### 4.3. Residual localization and elliptic inclusion

Let \(A\in\Psi_b^m\) have a full symbol of order \(-\infty\)
in a conic neighborhood of a closed conic set \(\Gamma\). At each
\(q\in\Gamma\) choose an order-zero cutoff \(D_q\) elliptic at
\(q\) whose large-frequency symbol is supported in a smaller cone
inside that neighborhood. The full ordered product expansion (GC9)
has every term residual there: derivatives of the cutoff stay in the
smaller cone, derivatives of the full symbol of \(A\) have arbitrary
negative order there, and the exact far product term is residual.
The coordinate-invariant remainders (GC13) therefore give
\(D_qA\in\Psi_b^{-\infty}\), after harmless compact spatial
localization. By (GA5), \(D_qAu\in\mathcal A\). Hence
\[
 \operatorname{WF}_b(Au)\cap\Gamma=\varnothing.
 \tag{GW9}
\]
This proves precisely the conormal residual target; it makes no
unsupported boundary smoothness claim.

For the elliptic inclusion, let \(q\) be outside both
\(\operatorname{Char}B\) and \(\operatorname{WF}_b(Bu)\), where
\(B\in\Psi_b^m\) is proper. Select an order-zero tester \(D\)
elliptic at \(q\) with \(DBu\in\mathcal A\). The product \(DB\)
has invertible principal symbol \(d b\) near \(q\) in the original
order; its inverse is \(b^{-1}d^{-1}\). Apply (GW3)–(GW5) to
\(DB\), choosing \(\chi\) supported in the common elliptic cone
and equal to one near \(q\). There is an order-zero tester \(Q\)
elliptic at \(q\) and a residual \(R\) with the exact identity
\(Q=E DB+R\). The first term is conormal by (GA3), and the residual term by
(GA5). Thus \(Qu\in\mathcal A\), giving
\[
 \operatorname{WF}_b(u)
   \subset\operatorname{WF}_b(Bu)\cup\operatorname{Char}B.
 \tag{GW10}
\]
Every inverse and product has retained the bundle map order.

For a properly supported \(B\in\Psi_b^m\), the forward inclusion
follows by the complementary microlocal division. If
\(q\notin\operatorname{WF}_b(u)\), take an order-zero elliptic
\(C\) there with \(Cu\in\mathcal A\). Construct its right local
parametrix \(P_C\) as above, so \(P_CC=Q_0+R_0\), where \(R_0\)
is residual and \(Q_0\) has full symbol equal to the identity on a
smaller cone about \(q\) at high frequency. Choose \(D\) of order
zero, elliptic at \(q\), with full symbol supported inside that
smaller cone. Put \(E=DBP_C\). Since \(D\) is supported where the
full symbol of \(Q_0\) is the identity, the exact product (GC9)
and its far residual give \(DB(I-Q_0)\in\Psi_b^{-\infty}\).
The product \(DBR_0\) is residual by (GC10). Therefore
\[
 DB=E C+R,
 \qquad E\in\Psi_b^m,\quad R\in\Psi_b^{-\infty}
 \tag{GW11}
\]
with \(R=DB(I-Q_0)-DBR_0\). Thus
\(DBu=E(Cu)+Ru\in\mathcal A\), so
\[
 \operatorname{WF}_b(Bu)\subset\operatorname{WF}_b(u)
 \quad(B\in\Psi_b^m\text{ proper}).
 \tag{GW12}
\]
The same inclusion holds for an arbitrary ordinary smooth differential
operator, including the unweighted normal derivative. We prove this
separately because \(D_n\) itself is not a totally characteristic
operator. For \(q\notin\operatorname{WF}_b(u)\), choose \(C\)
elliptic at \(q\) with \(Cu\in\mathcal A\). The left localized
parametrix gives \(P_CC=Q+R\), where the full symbol of \(Q\)
is exactly one on a high-frequency cone about \(q\) and \(R\) is
residual. Thus
\[
 Qu=P_C(Cu)-Ru\in\mathcal A.
 \tag{GW12a}
\]
Choose an order-zero tester \(D\), elliptic at \(q\), with full
symbol supported in a smaller cone where \(Q=I\) microlocally.
Then \(D(I-Q)\) is residual by (GC9), including its exact far term.
The ambient supported distribution \(D_j u\) is again supported in
the closed half-space. The local commutators are exactly
\[
 [D_j,T_a]=T_{D_{x_j}a}\quad(j<n),\qquad
 [D_n,T_a]=T_{D_{x_n}a}+T_{D_{\xi_n}a}D_n .
 \tag{GW12b}
\]
These are the unmodified (5.1), with \(D=-i\partial\) and all signs
retained. They first hold on interior compact smooth functions.
Theorem 9.1(e) in the local prerequisite approximates every supported
distribution weakly by such functions; each term in (GW12b) is a
composition of weakly continuous operators on supported distributions.
Taking that limit proves the same identity for the actual supported
representatives, including any boundary deltas.

Write \(w=(I-Q)u\). Since the full symbol of \(Q\) is constant one
on the working cone, the symbols \(D_{x_j}q\) and
\(D_{\xi_n}q\) vanish there to every symbol order. Apply \(D\)
on the left in (GW12b) and use (GC9): each resulting product is
residual. The exact identity
\[
 D D_jw
   =D(I-Q)D_ju+D[D_j,I-Q]u
 \tag{GW12c}
\]
is therefore a sum of residual operators applied to supported
distributions, \(u\) or \(D_nu\). It belongs to \(\mathcal A\)
by (GA5). On the other hand \(Qu\in\mathcal A\) by (GW12a), and
the ordinary differential map (C17) gives \(D_jQu\in\mathcal A\);
applying \(D\) preserves that class by (GA3). Hence
\(DD_ju\in\mathcal A\), so
\[
 \operatorname{WF}_b(D_ju)\subset\operatorname{WF}_b(u)
 \quad(j=1,\ldots,n).
 \tag{GW12d}
\]
Smooth coefficient multiplication is in \(\Psi_b^0\), and (GW12)
applies to it. Finite sums and products of the \(D_j\) with such
coefficients therefore give the same inclusion for every ordinary
smooth differential operator. The proof has kept the normal
\(T_{D_{\xi_n}a}D_n\) term in (GW12b); omitting it would make the
boundary claim unjustified.

### 4.4. Interior comparison and a noncharacteristic boundary

In the interior, (GL8) is the ordinary cotangent identification.
Localized global \(b\)-operators are ordinary pseudodifferential
operators there by (GL13)–(GL15), and every ordinary properly
supported local operator can be realized with the same interior
kernel in the global class, using (GC7). Also
\(\mathcal A\) restricts to \(C^\infty\) in the interior by (GA1):
every derivative is a combination of tangent derivatives on a compact
interior chart, and the local Sobolev estimates give all smooth
derivatives. Both implications in the tester definition therefore
give the exact equality
\[
 \operatorname{WF}_b(u)|_{T^*X^\circ}
       =\operatorname{WF}(u|_{X^\circ}).
 \tag{GW13}
\]

Let \(P=\sum_{|\alpha|\le m}a_\alpha(x)D^\alpha\) be a smooth
ordinary differential operator of positive integer order \(m\), and
let \(\phi\) vanish simply at the boundary, with \(\phi=c x_n\),
\(c(x',0)>0\), in a chart. No original coefficient is removed.
The complete differential expression
\[
 \phi^mP
   =\sum_{|\alpha|\le m}
      c(x)^m x_n^{m-\alpha_n}
      a_\alpha(x)D'^{\alpha'}
       \bigl(x_n^{\alpha_n}D_n^{\alpha_n}\bigr)
 \tag{GW14}
\]
retains every lower-order and tangential term; the displayed factor
order is valid because \(D'\) commutes with \(x_n\). Thus
\(\phi^mP\in\operatorname{Diff}_b^m\). At \(x_n=0\), all principal
terms except \(\alpha=(0,m)\) contain a positive power of \(x_n\).
Its complete boundary principal compressed symbol is
\[
 \sigma_m(\phi^mP)(x',0,\xi',\rho)
     =c(x',0)^m a_{(0,m)}(x',0)\rho^m.
 \tag{GW15}
\]
If the boundary is noncharacteristic, the original leading normal
coefficient \(a_{(0,m)}(x',0)\) is invertible. Equation (GW15) is
invertible exactly when \(\rho\ne0\); its boundary characteristic
set is precisely the embedded tangential hyperplane
\(T^*\partial X=\{\rho=0\}\), with the zero section excluded.
Applying the full inclusion (GW10) to the actual operator
\(\phi^mP\) yields
\[
 \operatorname{WF}_b(u)|_{\partial X}
  \subset
  \operatorname{WF}_b(\phi^mPu)|_{\partial X}
        \cup(T^*\partial X\setminus0).
 \tag{GW16}
\]
For \(m=0\), \(P\) is multiplication by its original invertible
coefficient under the corresponding noncharacteristic hypothesis;
the same parametrix gives (GW16) with an empty boundary
characteristic set. Formula (GW14) is not used with a negative power.

## 5. Noncharacteristic normal extension

### 5.1. 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
\[
 p_{k,K,L}(\phi)
 =\sum_{|\alpha|\le L}
       \|x_n^{\alpha_n}D^\alpha\phi\|_{B^{\kappa_k}_{2,\infty}}
 \quad(L=0,1,2,\ldots).
 \tag{GE1}
\]
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
\(s\),
\[
 \|w\|_{B^s_{2,\infty}}
 \le C_s\left(\|w\|_{B^{s-1}_{2,\infty}}
       +\sum_{j=1}^{n}\|D_jw\|_{B^{s-1}_{2,\infty}}\right).
 \tag{GE2}
\]
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.

### 5.2. The normal primitive gains one conormal order

For \(\phi\in C_c^\infty(X^\circ)\) with support in \(K\), put
\[
 \psi(x',t)=i\int_0^t\phi(x',s)\,ds,
 \qquad D_n\psi=\phi,
 \quad \psi=0\text{ below the normal support of }\phi.
 \tag{GE3}
\]
Choose a fixed normal cutoff \(\chi(t)\) equal to one on
\([0,c/2]\) and supported 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.
Choose these fixed cutoffs real valued. The Hermitian adjoint has the
reversed integral
\(g(s)\mapsto-i\theta(s)\int_s^c\chi(t)g(t)\,dt\).
The bilinear transpose has the full factor \(+i\); both identities and
their exact relation are proved in (DT10), Section 3.6.
Its derivatives obey the same integer estimates, including the
derivatives of the smooth \(\theta\), so duality
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
\[
 p_{k,K',L}(\chi\psi)
 \le C_{k,K,L}\,p_{k+1,K,L'}(\phi)
 \quad\text{for a finite }L'=L'(L),
 \tag{GE4}
\]
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:
\[
 x_n^aD_n^aD'^\beta\psi
   =x_n^aD_n^{a-1}D'^\beta\phi.
 \tag{GE5}
\]
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
\[
 D_n(x_n^aD_n^aD'^\beta\psi)
  =x_n^aD_n^aD'^\beta\phi
      -ia\,x_n^{a-1}D_n^{a-1}D'^\beta\phi.
 \tag{GE6}
\]
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).

### 5.3. 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
\[
 (D_n+A_0(x,D'))u=f\quad\text{in }X^\circ,
 \qquad f\in\mathcal A'(X).
 \tag{GE7}
\]
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
\[
 |u(\phi)|\le C_{k,K}p_{k,K,L}(\phi)
 \tag{GE8}
\]
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.

Assume (GE8) at some \(k\). 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
\(D_n(\chi\psi)=\phi+(D_n\chi)\psi\) on the support of
\(\phi\), the exact equation (GE7) gives
\[
 u(\phi)
  =-f(\chi\psi)
   +u\bigl(A_0^{\mathrm t}(\chi\psi)
              -(D_n\chi)\psi\bigr).
 \tag{GE9}
\]
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 operator \(A_0^{\mathrm t}\) and the normal cutoff multiply
\(\mathcal A^k\) continuously by Section 5.1, regardless of the
tangential order of \(A_0\). Apply the induction hypothesis and
then the *full* primitive estimate (GE4). It yields (GE8) at
\(k+1\), with finite constants and a possibly larger finite \(L\).
By induction (GE8) holds at \(k_0+j\) for every integer \(j\ge0\).
For an arbitrary real \(k\), choose \(j\) with \(k_0+j\ge k\);
the continuous inclusion \(\mathcal A^k\hookrightarrow
\mathcal A^{k_0+j}\) gives the desired bound.

Construct the supported extension rather than silently identifying
it with the temporary ambient one. For a smooth boundary test
\(\phi\) supported in \(K\), let \(\mathcal Q_\varepsilon\phi\)
be (GS6); it is smooth, supported in a fixed compact subset of the
interior, and converges to \(\phi\) in
\(\mathcal A^{k_1}\) for every \(k_1>m_0\), by (GD2) and (GS7).
The bounds (GE8) make
\[
 U(\phi):=\lim_{\varepsilon\downarrow0}
                   u(\mathcal Q_\varepsilon\phi)
 \tag{GE10}
\]
exist: use the bound at such a \(k_1\) on differences of
approximants. The limit is independent of the particular one-sided
smoothing family because both approximations converge in the same
\(\mathcal A^{k_1}\) topology. The uniform (GS2) multiplier
estimate and (GE8), followed by the continuous inclusion from
\(\mathcal A^k\) to a larger test order, prove for every
\(k\ge m_0\) a finite estimate of the form (GD3) for \(U\).
The smooth-test topology bounds those conormal seminorms by finitely
many ordinary smooth seminorms, by (GD1); thus \(U\) is a
distribution on the closed chart. The supported-distribution duality
of local Theorem 9.1(a),(c) gives its ambient supported
representative. Its interior restriction is \(u\), since
\(\mathcal Q_\varepsilon\) is an approximate identity there.
By (GD4), no second \(\mathcal A'\) extension of \(u\) exists.
This proves the first-order existence and uniqueness in the exact
dual topology.

### 5.4. Full normal order by the actual companion system

Let \(m\ge1\) and retain the original normal-monic equation
\[
 P=D_n^m+\sum_{j=0}^{m-1}a_j(x,D')D_n^j,
 \qquad Pu=f\text{ in }X^\circ,
 \quad f\in\mathcal A'(X).
 \tag{GE11}
\]
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
\[
 D_n\begin{pmatrix}u_0\\u_1\\\vdots\\u_{m-1}\end{pmatrix}
 +\begin{pmatrix}
 0&-I&0&\cdots&0\\
 0&0&-I&\cdots&0\\
 \vdots&\vdots&\vdots&\ddots&\vdots\\
 a_0&a_1&a_2&\cdots&a_{m-1}
 \end{pmatrix}
 \begin{pmatrix}u_0\\u_1\\\vdots\\u_{m-1}\end{pmatrix}
 =\begin{pmatrix}0\\0\\\vdots\\f\end{pmatrix}.
 \tag{GE12}
\]
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 5.3 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
\[
 \nabla_n^{\mathrm{int}}U_j=U_{j+1},
 \qquad D_nU_j-U_{j+1}
          =-i\,(U_j|_{x_n=0})\otimes\delta(x_n),
 \quad x_n(D_nU_j-U_{j+1})=0.
 \tag{GE13}
\]
Similarly the final companion equation gives
\[
 \nabla_n^{\mathrm{int}}U_{m-1}
        +\sum_{j=0}^{m-1}a_j(x,D')U_j=f,
 \quad x_n\left(D_nU_{m-1}
        +\sum_{j=0}^{m-1}a_j(x,D')U_j-f\right)=0.
 \tag{GE14}
\]
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
\[
 D_n(x_n^{j+1}W)
   =x_n^{j+1}D_nW-i(j+1)x_n^jW.
 \tag{GE15}
\]
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
\[
 x_n^jU_j=x_n^jD_n^jU_0\qquad(0\le j<m).
 \tag{GE16}
\]
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
\[
 x_n^m(PU_0-f)=0.
 \tag{GE17}
\]
No original tangential coefficient or lower normal term was removed.

### 5.5. 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. Localize to a compact chart and let \(M\)
bound the distribution order of \(V\). Taylor-expand a test
\(h(x',t)\) through degree \(M\) at \(t=0\), with a fixed normal
cutoff \(\eta(t)=1\) near zero:
\[
 h(x',t)=\eta(t)\sum_{j=0}^{M}
       \frac{t^j}{j!}\partial_t^jh(x',0)+t^{M+1}r(x',t)
 \quad\text{near }\operatorname{supp}V.
 \tag{GE18a}
\]
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. Define tangential distributions
\(v_j(g)=(-1)^jV(\eta(t)t^jg(x')/j!)\). Then applying \(V\) to
(GE18a) gives, with no omitted coefficient,
\[
 V=\sum_{j=0}^{\mu}v_j(x')\otimes\delta^{(j)}(x_n),
 \qquad v_\mu\ne0\text{ unless }V=0.
 \tag{GE18}
\]
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 local representations agree on overlapping charts,
so this argument applies to every compactly supported piece of \(V\).
The coefficients are tangential distributions, possibly vector
valued. The normal distribution formulas, with every factor, are
\[
 D_n^k\delta^{(j)}=(-i)^k\delta^{(j+k)},\qquad
 x_n^m\delta^{(j+k)}
 =\begin{cases}
 (-1)^m\dfrac{(j+k)!}{(j+k-m)!}\delta^{(j+k-m)},&j+k\ge m,\\
 0,&j+k<m.
 \end{cases}
 \tag{GE19}
\]
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
\[
 i^m\frac{(\mu+m)!}{\mu!}
       v_\mu(x')\otimes\delta^{(\mu)}(x_n),
 \tag{GE20}
\]
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 theorem so far is local on the stated product collar. The
globalization and boundary wave-front consequence are proved next.

### 5.6. The noncharacteristic boundary wave-front class

Write \(\widetilde T^*X\) for the compressed cotangent bundle of
(GL11)–(GL18), and embed \(T^*\partial X\setminus0\) as its
boundary covectors with zero normal compressed component. Define
the exact class
\[
 \mathcal N(X)
  :=\{v\in\mathcal A'(X):
       \operatorname{WF}_b(v)|_{\partial X}
       \subset T^*\partial X\setminus0\}.
 \tag{GE21}
\]
This is a condition on the already-defined wave-front set (GW6),
not a replacement for the dual conormal requirement.

Let \(P\) be a smooth ordinary differential operator of normal
order \(m\ge1\) whose boundary is noncharacteristic, and let an
extendible interior distribution \(u\) solve \(Pu=f\) with
\(f\in\mathcal N(X)\). In one product chart retain the complete
normal expansion
\[
 P=a_m(x)D_n^m+
       \sum_{j=0}^{m-1}a_j(x,D')D_n^j,
 \qquad a_m(x',0)\text{ invertible}.
 \tag{GE22}
\]
Shrinking the chart makes \(a_m(x)\) invertible throughout it.
Multiplying the equation on the *left* by its inverse gives the
normal-monic operator
\(P'=a_m^{-1}P\) and source \(f'=a_m^{-1}f\).
The order of matrix multiplication is retained. Smooth
multiplication preserves \(\mathcal A'\) by (GD13) and preserves
the boundary wave-front condition by (GW12); thus
\(f'\in\mathcal N\). Sections 5.3–5.5 give the unique local
\(U\in\mathcal A'\) with
\[
 x_n^m(P'U-f')=0,
 \qquad x_n^m(PU-f)=0.
 \tag{GE23}
\]
The second equality follows by multiplying the first on the left
by \(a_m\), which commutes with the scalar \(x_n^m\). In (GE23)
the composition \(x_n^mP\) is also its actual totally
characteristic differential action on supported distributions:
the original coefficients, their product order, and the raw
ambient normal derivatives agree with (GD13) after transposition.

The full noncharacteristic estimate (GW16), with the original
defining function \(\phi=x_n\) in this chart, applies to this
\(U\):
\[
 \operatorname{WF}_b(U)|_{\partial X}
 \subset
 \operatorname{WF}_b(x_n^mPU)|_{\partial X}
       \cup(T^*\partial X\setminus0)
 =\operatorname{WF}_b(x_n^mf)|_{\partial X}
       \cup(T^*\partial X\setminus0).
 \tag{GE24}
\]
Multiplication by \(x_n^m\) is a proper order-zero
totally characteristic operator after localization. The full
forward inclusion (GW12), rather than an unsupported assertion
about its zero set, gives
\[
 \operatorname{WF}_b(x_n^mf)|_{\partial X}
       \subset\operatorname{WF}_b(f)|_{\partial X}
       \subset T^*\partial X\setminus0.
 \tag{GE25}
\]
Equations (GE24)–(GE25) prove \(U\in\mathcal N\).
On overlapping boundary charts, two such local extensions have the
same interior restriction and belong to \(\mathcal A'\), so (GD4)
makes them equal. The product coordinate changes and bundle maps
preserve \(\mathcal A'\) by (GA4) and the compressed wave-front
condition by (GL11)–(GL18) and (GW6). The local extensions therefore
glue to a global \(U\in\mathcal N(X)\), uniquely determined by the
interior \(u\):
\[
 Pu=f\text{ in }X^\circ,\quad f\in\mathcal N(X),\quad
 \partial X\text{ noncharacteristic for }P
 \quad\Longrightarrow\quad
 \exists!\,U\in\mathcal N(X),\ U|_{X^\circ}=u.
 \tag{GE26}
\]
The uniqueness in (GE26) is also immediate from (GD4). Its
existence uses the actual weighted equation (GE23); an arbitrary
ambient extension would not supply the conclusion.
For an order-zero \(P=a_0(x)\) invertible at the boundary, local
inversion gives \(U=a_0^{-1}f\in\mathcal N\) by (GD13) and
(GW12), with the same interior restriction and uniqueness by
(GD4). Thus (GE26) also covers this endpoint without applying
the positive-order companion construction to a zero-dimensional
system.

## 6. Tangential action at the boundary

### 6.1. The actual tangential action and the conormal topology

Let \(b(x',t,\xi')\in S^d\), \(d\in\mathbb R\), smooth down to
\(t=0\), with its original full family of tangential symbol
seminorms. Its left quantization, with the same Fourier convention
as (GL13), is
\[
 (B_bv)(x',t)=(2\pi)^{-(n-1)}
     \int e^{i(x'-y')\cdot\xi'}b(x',t,\xi')v(y',t)
                  \,dy'\,d\xi'.
 \tag{GT1}
\]
For \(n=1\), the tangential dimension is zero and (GT1) is smooth
multiplication. Insert the actual proper-support kernel cutoff of
\(B_b\) in (GT1) when needed. It maps compact smooth tests to
compact smooth tests, including in the normal variable. Its
transpose \(B_b^{\mathrm t}\) is another properly supported
tangential operator of order \(d\), with smooth \(t\)-dependent
coefficients; this follows directly by transposing the kernel and
Taylor-expanding \(b(y',t,\xi')\) in \(y'-x'\), retaining the exact
far kernel as a tangential smoothing term. Thus (GT1) acts by
transposition on every distribution for which proper support is
specified, before any wave-front restriction is imposed.

We prove the stronger topology statement needed below. With the
actual \(\mathcal A^k\) seminorms (GE1), for each compact chart
\(K\), real \(k\), and finite \(L\), there are a compact \(K'\),
finite \(L'\), and \(C\) such that
\[
 p_{k,K,L}(B_bv)\le C p_{k,K',L'}(v)
 \quad(v\in\mathcal A^k_{K'}),
 \qquad
 p_{k,K,L}(B_b^{\mathrm t}v)\le C p_{k,K',L'}(v).
 \tag{GT2}
\]
Here the compact sets are chosen to include the source and target
of the properly supported localized kernel. To prove the Besov
part, first localize both base variables to compact sets and extend
the resulting smooth \(x=(x',t)\)-dependence periodically on a
larger box. At \(t=0\), use the smooth collar extension of
Section 3 of the linked local lesson before periodic extension;
for each finite estimate its extension bounds involve only a
finite number of the original one-sided symbol seminorms.
Its Fourier series is
\[
 b(x,\xi')=\sum_{\ell\in\mathbb Z^n}
       e^{i\ell\cdot x}b_\ell(\xi'),\qquad
 |\partial_{\xi'}^\beta b_\ell(\xi')|
   \le C_{N\beta}\langle\ell\rangle^{-N}
                         \langle\xi'\rangle^{d-|\beta|}
 \quad(\forall N).
 \tag{GT3}
\]
This follows by integrating by parts \(N\) times in the compact
base Fourier coefficient; every base derivative of the original
symbol has the same order \(d\). Choose an even integer
\(M=2r\ge\max(d,0)\). The full Fourier multiplier
\(b_\ell(\xi')\langle\xi'\rangle^{-M}\) is bounded uniformly in
\((\xi',\xi_n)\) by \(C_N\langle\ell\rangle^{-N}\), so it is
bounded on \(B^s_{2,\infty}(\mathbb R^n)\) for every real \(s\):
it commutes with each full dyadic projection and Plancherel gives
the bound on each block. The factor
\(\langle\xi'\rangle^M=(1+|\xi'|^2)^r\) is the *complete*
finite polynomial in tangential derivatives, not an isotropic
replacement. Multiplication by \(e^{i\ell\cdot x}\) shifts full
frequency by \(\ell\); direct dyadic overlap shows its
\(B^s_{2,\infty}\) operator bound grows by at most a fixed
polynomial in \(\langle\ell\rangle\). Choosing \(N\) larger than
that degree plus \(n+1\), then summing (GT3), proves
\[
 \|B_bv\|_{B^s_{2,\infty}}
 \le C_{s,b}\sum_{|\beta|\le M}
          \|D'^\beta v\|_{B^s_{2,\infty}}.
 \tag{GT4}
\]
The same argument applies to every normal and tangential base
derivative of \(b\), including the transpose symbol and its exact
far smoothing kernel. Its properly supported kernel cutoffs are
smooth multiplication on the two sides and obey the same estimates.

The normal variable is unchanged by the kernel in (GT1), hence
\(t^aB_b=B_bt^a\). Differentiate the full expression rather than
identifying normal and tangential orders:
\[
 t^aD_n^aD'^\gamma(B_bv)
 =\sum_{j=0}^{a}\sum_{\beta\le\gamma}
       \binom aj\binom\gamma\beta
       t^j B_{D_n^jD_{x'}^\beta b}
        \bigl(t^{a-j}D_n^{a-j}D'^{\gamma-\beta}v\bigr).
 \tag{GT5}
\]
The formula also applies termwise to the proper kernel cutoff;
its derivatives are included in the differentiated symbol.
Every \(t^j\) is a smooth compact multiplier, and (GT4) controls
the remaining tangential operator by finitely many further
\(D'\) seminorms. This proves (GT2), with *each* original normal
weight and derivative retained. In particular \(B_b\) and
\(B_b^{\mathrm t}\) preserve every filtered \(\mathcal A^k\),
and the dual formula
\[
 (B_bu)(v)=u(B_b^{\mathrm t}v)
 \tag{GT6}
\]
defines a weakly continuous map \(\mathcal A'\to\mathcal A'\).
The stronger action statement follows from the explicit topology
estimate; it does not identify \(B_b\) with an isotropic
\(n\)-covariable pseudodifferential symbol.

### 6.2. The equatorial compressed cutoff

Choose a smooth conic symbol \(t_0(\xi',\rho)\) of order zero,
independent of \(x'\), with the original low-frequency cutoff, so
that
\[
 t_0=1\quad(2|\rho|<|\xi'|,\ |\xi'|>1),\qquad
 t_0=0\quad(|\rho|>|\xi'|).
 \tag{GT7}
\]
Multiply by a smooth normal base cutoff equal to one on
\(0\le t<1\) and supported in \(t<2\). The construction on the
unit sphere is possible because the closed equatorial band and
the normal caps are disjoint. Apply the *original* lacunarization
of Lemma 4.4 to this full symbol and write
\(t_\rho\in S^0_{\mathrm{la}}\). The difference
\(t_\rho-t_0\) is residual, with its original normal-base
decay; it does not change the principal compressed symbol.
Let \(T=T_{t_\rho}\), localized properly in a boundary chart.

For \(u\in\mathcal N\), put \(w=(I-T)u\). At every boundary
tangential covector \(q\), the full symbol of \(I-T\) vanishes
on a conic neighborhood of \(q\), so (GW9) removes \(q\) from
\(\operatorname{WF}_b(w)\). At every other boundary covector
\(q\), the definition of \(\mathcal N\) removes \(q\) from
\(\operatorname{WF}_b(u)\), and (GW12) removes it from
\(\operatorname{WF}_b(w)\). Thus the boundary portion is empty:
\[
 \operatorname{WF}_b((I-T)u)|_{\partial X}=\varnothing.
 \tag{GT8}
\]
On a compact boundary patch, closedness of \(\operatorname{WF}_b\)
on the compact cosphere gives a collar in which it remains empty.
The finite microlocal cover argument (GW7) then makes a spatially
localized \(w\) an element of \(\mathcal A\). Equation (GT2)
gives \(B_bw\in\mathcal A\). We have therefore proved, as a
local *conormal equality* rather than an unproved smoothness claim,
\[
 B_bu=B_bTu+v,
 \qquad v\in\mathcal A\text{ near the boundary}.
 \tag{GT9}
\]
The equality is between the actual distributional actions (GT6).

Because \(t_\rho\) is independent of \(x'\), the unlocalized
Kohn–Nirenberg composition in (GT1) is exact:
\[
 B_bT_{t_\rho}=T_a,
 \qquad a(x',t,\xi',\rho)=b(x',t,\xi')t_\rho(t,\xi',\rho).
 \tag{GT10}
\]
Indeed integration in the intermediate tangential variable gives
\((2\pi)^{n-1}\delta(\eta'-\xi')\); no derivative of
\(t_\rho\) in \(x'\) appears. The support of \(t_0\) in (GT7)
ensures that on the high-frequency nonresidual part
\(\langle\xi'\rangle\asymp\langle(\xi',\rho)\rangle\).
All \(\xi'\), \(\rho\), and base derivatives of the product
therefore obey the full order-\(d\) symbol bounds. The residual
difference from lacunarization remains residual after
multiplication by \(b\), using arbitrary residual order to
absorb its fixed order \(d\). Since normal Fourier convolution
does not change a tangential multiplier, the product retains
the original lacunarity. Thus \(a\in S^d_{\mathrm{la}}\).

Proper-support cutoffs make (GT10) an equality modulo a residual
*full* compressed operator. To verify the asserted residual class,
write each far tangential cutoff as a kernel factor vanishing near
\(x'=y'\) and integrate by parts in \(\xi'\) arbitrarily many
times. On the nonresidual support of \(t_0\), the entire normal
frequency is bounded by a constant times \(|\xi'|\), so this
gain is arbitrary in the full \((\xi',\rho)\) order. The
lacunarization remainder is already residual. Derivatives of the
cutoff and amplitude obey the same bounds, proving the full
residual assertion. Its action on supported distributions lies
in \(\mathcal A\) by (GA5), so it does not affect the boundary
wave-front conclusions.

### 6.3. Boundary action, microsupport, and elliptic comparison

The local boundary wave-front set of \(v\in\mathcal A\) is empty
by definition, and adding such a \(v\) does not change a
wave-front set: a regularizing tester for one summand works for
the sum, and subtracting the same \(v\) gives the reverse
inclusion. Equations (GT9)–(GT10) and (GW12) therefore give
\[
 \operatorname{WF}_b(B_bu)|_{\partial X}
   =\operatorname{WF}_b(T_au)|_{\partial X}
   \subset\operatorname{WF}_b(u)|_{\partial X}
   \subset T^*\partial X\setminus0.
 \tag{GT11}
\]
Since (GT6) also gives \(B_bu\in\mathcal A'\), this proves
\(B_b:\mathcal N\to\mathcal N\) with the original tangential
operator. It also proves that the action is continuous in the
weak topology of \(\mathcal A'\) tested on fixed conormal
functions.

Suppose \(b\) is of order \(-\infty\) on a conic neighborhood of
the complement of a closed tangential cone \(\Gamma\) at the
boundary. At a tangential covector \(q\notin\Gamma\), the full
symbol \(a=b t_\rho\) is of order \(-\infty\) on a compressed
cone about \(q\). The original residual localization (GW9)
excludes \(q\) from \(\operatorname{WF}_b(T_au)\). At normal
compressed directions (GT11) already excludes every covector.
Hence the exact boundary microsupport statement is
\[
 \operatorname{WF}_b(B_bu)|_{\partial X}
 \subset\operatorname{WF}_b(u)|_{\partial X}\cap\Gamma.
 \tag{GT12}
\]

Write \(b_0(x',\xi')=b(x',0,\xi')\). At a tangential boundary
covector \(q=(x',0,\eta',0)\), (GT7), (GL18), and (GC4) give
the actual principal compressed symbol of \(T_a\):
\[
 \sigma_d(T_a)(q)=\sigma_d(b_0)(x',\eta')\cdot 1.
 \tag{GT13}
\]
The boundary operator action (GL24) has the same principal
symbol, with the original \((2\pi)^{-(n-1)}\) Fourier factor.
If \(b_0\) is elliptic at \(q\), then \(T_a\) is elliptic
there. The exact elliptic inclusion (GW10), applied to
\(T_au=B_bu-v\) from (GT9), gives
\[
 \operatorname{WF}_b(u)|_{\partial X}
   \subset
   \operatorname{WF}_b(B_bu)|_{\partial X}
       \cup\operatorname{Char}(b_0).
 \tag{GT14}
\]
The factor order in (GT13) stays matrix order; for vector bundles,
ellipticity means the actual boundary matrix is invertible.

### 6.4. What the argument gives in the interior

At an interior point \((x',t)\), \(t>0\), the tangential action
is the family (GT1). The full kernel is a tangential
pseudodifferential kernel times \(\delta(t-s)\). Split it with a
cutoff in \(x'-y'\) that is one near zero. Off the tangential
diagonal the tangential kernel is smooth, by arbitrary integration
by parts in \(\xi'\). Its output can therefore be singular only
in the normal variable, so every covector there has \(\xi'=0\).
On the tangential diagonal, fix an output covector with
\(\xi'\ne0\) and take a conic cutoff on which
\(|\xi'|\ge c|(\xi',\xi_n)|\) for some \(c>0\). The full
symbol \(b(x,\xi')\) obeys the ordinary isotropic symbol
estimates on that cone, since its only frequency derivatives are
in \(\xi'\) and \(\langle\xi'\rangle\asymp\langle\xi\rangle\)
there. The complementary frequency cutoff has no output
wave-front in this cone by nonstationary integration in the
full oscillatory kernel. The standard local integration-by-parts
proof of pseudolocality therefore applies to the retained
kernel. Together with (GW13), this gives the exact interior
inclusion
\[
 \operatorname{WF}_b(B_bu)|_{T^*X^\circ\cap\{\xi'\ne0\}}
       \subset
       \operatorname{WF}_b(u)|_{T^*X^\circ\cap\{\xi'\ne0\}}.
 \tag{GT15}
\]
The region on which (GT15) is useful depends on the actual
wave-front covectors of \(u\). It does not claim an all-interior
inclusion: at \(\xi'=0\), a tangential smoothing kernel can
carry a normal singularity between distinct tangential points
at the same \(t\), precisely because the kernel still contains
\(\delta(t-s)\). For example, take a properly supported smooth
tangential kernel \(K(x',y')\) with
\(K(x'_1,y'_0)\ne0\) and \(x'_1\ne y'_0\), and
\(u=\delta(x'-y'_0)\otimes\delta(t-t_0)\) with \(t_0>0\).
Then \(B_bu=K(x',y'_0)\delta(t-t_0)\) has a pure-normal
wave-front covector at \((x'_1,t_0)\), while \(u\) has no
wave-front there. This proves that the exception is necessary,
rather than leaving an all-interior claim untested.

## 7. The Hardy weight and a singular test

### 7.1. The weighted norm with its complete order shift

Localize in a compact boundary product chart. An element
\(u\in\mathcal A^m\) has, modulo a smooth term, the exact
normal conormal oscillatory representation
\[
 u(x',t)=(2\pi)^{-1}\int_{\mathbb R}
         e^{it\tau}a(x',\tau)\,d\tau,
 \qquad
 |D_{x'}^\beta\partial_\tau^j a(x',\tau)|
     \le C_{\beta j}\langle\tau\rangle^{\mu-j},
 \quad \mu=m+\frac{n-2}{4}.
 \tag{HS1}
\]
This is the *original* codimension-one shift, including the
ambient dimension. The inverse coefficient and \(D=-i\partial\)
convention are retained. For a normal derivative of order
\(a\ge0\), the amplitude is \(\tau^aD_{x'}^\beta a\), of
order \(\mu+a\). Split its integral at \(|\tau|=t^{-1}\),
\(0<t<1\). The low-frequency absolute integral is bounded by
\(Ct^{-\mu-a-1}\) when \(\mu+a>-1\), by
\(C(1+|\log t|)\) at equality, and by \(C\) below it.
On the high-frequency part integrate by parts in \(\tau\)
\(N>\mu+a+1\) times, including the derivatives of a smooth
cutoff at \(|\tau|=t^{-1}\). Each resulting term is bounded
by \(Ct^{-\mu-a-1}\). The same estimate holds for every
tangential derivative, uniformly on a smaller compact chart.
Thus the complete safe weighted \(L^2\) implication is
\[
 \int_{K_0}\int_0^c
     t^{2\nu}|D'^\beta D_n^a u(x',t)|^2
                       \,dt\,dx'<\infty
 \quad\text{whenever }
 \nu\ge0,\quad \nu>m+\frac n4+a.
 \tag{HS2}
\]
The strict endpoint includes the logarithmic case. Equation
(HS2) states the weight *inside the norm* as \(t^\nu\);
the power inside the squared integral is \(2\nu\). A cached
target record that writes a single exponent \(N\) on the
squared integral cannot be substituted for the original formula
without checking whether its \(N\) names \(\nu\) or \(2\nu\).
This is a routing ambiguity, not a claim of an error in a
human source.

### 7.2. Hardy's exact coefficient and the failed direct iteration

Let \(v\in C_c^\infty((0,\infty))\) and \(\lambda\ge0\). The
boundary terms in integration by parts vanish at both ends.
Writing \(I=\int t^{2\lambda}|v|^2dt\),
\(J=\int t^{2\lambda+2}|v'|^2dt\), the *full* calculation is
\[
 \begin{aligned}
 (2\lambda+1)I
  &=-\int_0^\infty t^{2\lambda+1}
                     \partial_t|v|^2\,dt\\
  &=-2\operatorname{Re}\int_0^\infty
            t^{2\lambda+1}v'\overline v\,dt
   \le2\sqrt{IJ},\\
 I&\le\frac{4}{(2\lambda+1)^2}J.
 \end{aligned}
 \tag{HS3}
\]
The sign in the middle line is retained, and the last inequality
uses Cauchy–Schwarz. For a compactly supported \(v\) away from
zero, iteration at \(\lambda=0,1,\ldots,k-1\) gives
\[
 \int|v|^2dt
 \le\left[\prod_{j=0}^{k-1}
          \frac{4}{(2j+1)^2}\right]
       \int t^{2k}|v^{(k)}|^2dt.
 \tag{HS4}
\]
One cannot remove the cutoff at zero unless all cutoff-error
terms and the final weighted integral have actual bounds.
For \(v=D'^\beta D_n^a u\), using only (HS2) on the final
integral would require
\[
 k>m+\frac n4+(a+k),
 \quad\text{equivalently}\quad 0>m+\frac n4+a.
 \tag{HS5}
\]
When that last number is nonnegative, no choice of the number
of Hardy iterations fixes the estimate. This is the exact
remaining deficit of the *conormal-only* argument, including
both the original order and the derivative count. It identifies
where \(u\in\mathcal A'\) must enter; it is not a proof of
\(\mathcal A'\cap\mathcal A=C^\infty\).

### 7.3. A singular conormal test of the missing hypothesis

In one normal dimension choose \(\chi\in C_c^\infty([0,\infty))\)
equal to one near zero and put
\(u(t)=\chi(t)t^{-3/4}H(t)\). The Fourier amplitude has
normal order \(-1/4\), so (HS1) gives its exact conormal
order \(m=0\) when \(n=1\). It is locally integrable but
not in \(L^2\), since
\(\int_0^c t^{-3/2}dt=\infty\). Moreover
\(t^kD_n^ku\) has the same leading power for every \(k\),
so the final integral in (HS4) diverges for every \(k\),
exactly as (HS5) predicts.

Take a nonnegative \(q\in C_c^\infty((1,2))\) with integral
one and set \(q_\varepsilon(t)=\varepsilon^{-1}q(t/\varepsilon)\).
The boundary delta is conormal of order \((2-n)/4=1/4\)
by (GD7), and (GS7) makes \(q_\varepsilon=Q_\varepsilon\delta\)
converge to \(\delta\) in every \(\mathcal A^{m'}\) with
\(m'>1/4\). Yet the ordinary interior pairing is
\[
 u(q_\varepsilon)
   =\varepsilon^{-3/4}
       \int_1^2s^{-3/4}q(s)\,ds
   \longrightarrow+\infty.
 \tag{HS6}
\]
If this \(u\) belonged to \(\mathcal A'\), continuity on that
\(\mathcal A^{m'}\) would make the same pairings converge to
the finite value \(u(\delta)\). Thus
\(u\in\mathcal A^0\setminus\mathcal A'\). This example
proves that the dual requirement excludes at least the displayed
power singularity; it does not replace a proof for every
conormal amplitude.

## 8. Smoothness from the dual conormal condition

### 8.1. Every boundary delta jet and its precise conormal order

For a compact smooth tangential density \(h(x')\), the normal
delta derivative has the original Fourier representation
\[
 h(x')\otimes\delta^{(j)}(t)
  =(2\pi)^{-1}\int e^{it\tau}(i\tau)^j h(x')\,d\tau,
 \qquad
 m_j=\frac{2-n}{4}+j=\frac12-\frac n4+j.
 \tag{SP1}
\]
The exponent \(m_j\) follows from the *unmodified*
codimension-one order relation in (C16): the amplitude has
order \(j=m_j+(n-2)/4\). The factor \((2\pi)^{-1}\) and
\(i^j\) have not been absorbed. Formula (GA1) gives the
same order by dyadic estimation, including every tangent
derivative. For \(u\in\mathcal A'\) define
\(g_j=(\nabla_n^{\mathrm{int},j}u)|_{t=0}\) by (GD8)–(GD9).
These are tangential distributions at this stage. For a smooth
boundary \(u\), direct differentiation and the distributional
delta sign give
\[
 u(h\otimes\delta^{(j)})=(-1)^j
       \langle\partial_t^ju|_{t=0},h\rangle
       =(-i)^j\langle g_j,h\rangle.
 \tag{SP2}
\]
Both sides are weakly continuous on \(\mathcal A'\): the left
is one of its defining conormal-test pairings by (SP1), and
the right is a composition of the weakly continuous corrected
derivative and trace maps (GD8)–(GD9). Smooth functions are
weakly dense by (GD5), so (SP2) holds for *every*
\(u\in\mathcal A'\), with no assertion that the ambient
uncorrected derivative has the same trace.

### 8.2. Quantitative scaled test expansion in the actual topology

Fix \(J=[1,2]\) and a compact tangential chart set. For
\(\psi\in C_c^\infty(X_0\times(1,2))\), put
\[
 v_\varepsilon(x',t)=\varepsilon^{-1}
                 \psi(x',t/\varepsilon),\qquad
 M_j(x')=\int_1^2s^j\psi(x',s)\,ds.
 \tag{SP3}
\]
The test is smooth and supported in the interior for each
\(\varepsilon>0\). Taylor's formula at the *actual* boundary
gives the distributional expansion
\[
 v_\varepsilon
  =\sum_{j=0}^{N-1}
      \frac{(-1)^j\varepsilon^j}{j!}
          M_j(x')\otimes\delta^{(j)}(t)
       +R_{N,\varepsilon}.
 \tag{SP4}
\]
We need its quantitative conormal topology, not only
distributional convergence. For every integer \(N\ge1\),
real \(M>m_0+N\) with \(m_0=(2-n)/4\), compact output
set \(K\), and finite tangent seminorm index \(L\), there
are finite \(L'\), \(C\) such that
\[
 p_{M,K,L}(R_{N,\varepsilon})
       \le C\varepsilon^N
             \sum_{|\gamma|\le L'}
                   \|D_{x',s}^\gamma\psi\|_{L^\infty},
       \qquad0<\varepsilon\le1.
 \tag{SP5}
\]
Here the original Besov index is \(\kappa=-M-n/4\);
the strict condition is exactly \(\kappa+N+1/2<0\).

For completeness, Fourier transform (SP4) in \(t\). Its
normal factor is
\[
 \int_1^2e^{-i\varepsilon s\tau}\psi(x',s)\,ds
   -\sum_{j=0}^{N-1}
       \frac{(-i\varepsilon\tau)^j}{j!}M_j(x').
 \tag{SP6}
\]
For \(\varepsilon|\tau|\le1\), Taylor's integral
remainder bounds (SP6), with every tangential derivative,
by \(C\varepsilon^N|\tau|^N\); for
\(\varepsilon|\tau|\ge1\), the Schwartz integral and all
retained polynomial terms bound it by
\(C(\varepsilon|\tau|)^{N-1}\). The tangential Fourier
transform decays faster than any power of \(|\xi'|\), with
constants controlled by finitely many displayed \(\psi\)
derivatives. On a full dyadic block \(|(\xi',\tau)|\asymp2^l\),
the \(L^2\) size in the normal frequency contributes
\(2^{l/2}\). For \(2^l\le\varepsilon^{-1}\), multiplying
by \(2^{l\kappa}\) gives
\(C\varepsilon^N2^{l(\kappa+N+1/2)}\), bounded by
\(C\varepsilon^N\). For \(2^l\ge\varepsilon^{-1}\), it
gives
\(C\varepsilon^{N-1}2^{l(\kappa+N-1/2)}\), whose
maximum is \(C\varepsilon^{-\kappa-1/2}\le
C\varepsilon^N\) under the same strict condition.
The low block is bounded directly by the Taylor remainder.
Tangentially dominant blocks gain arbitrary decay from the
\(x'\)-Fourier transform and obey the same inequality.
Applying \(D'\) differentiates \(\psi\). Applying
\(tD_t\) to (SP4) rescales the same test and acts on each
\(\delta^{(j)}\) by its exact eigenvalue
\(tD_t\delta^{(j)}=i(j+1)\delta^{(j)}\); the Taylor
remainder still has the two bounds above. The triangular
identity (GA2) therefore supplies every weighted derivative
seminorm, proving (SP5) with all lower terms retained.

The dual estimate (GD3), used at this freely chosen high
order \(M\), turns (SP5) into
\[
 |u(R_{N,\varepsilon})|
       \le C_{u,N,K}\varepsilon^N
              \sum_{|\gamma|\le L'}
                     \|D_{x',s}^\gamma\psi\|_\infty.
 \tag{SP7}
\]
This is an estimate of distributions in \((x',s)\) of a
finite negative Sobolev order, since a sufficiently high
Sobolev norm controls the finite smooth-test seminorm.

### 8.3. The scaled Taylor series in distributions

An element \(u\in\mathcal A'\) is smooth in the open collar
only when it also lies in \(\mathcal A\). Assume that
intersection from now on, and set
\(F_\varepsilon(x',s)=u(x',\varepsilon s)\) for \(s\in J\).
For every \(\psi\) in (SP3), the change of variables gives
\(\langle F_\varepsilon,\psi\rangle=u(v_\varepsilon)\).
Insert (SP4), then use the exact sign (SP2). The two
\((-1)^j\) factors cancel, leaving
\[
 F_\varepsilon(x',s)
  =\sum_{j=0}^{N-1}
      \frac{i^j\varepsilon^j s^j}{j!}g_j(x')
        +O_{H^{-L_N}(K_0\times J)}(\varepsilon^N)
 \quad\text{for every }N\ge1.
 \tag{SP8}
\]
The remainder means (SP7) uniformly on compact tangential
sets; \(L_N\) is finite but may depend on \(N\). Formula
(SP8) is an *all-order distribution-valued* boundary Taylor
series, with the original \(i^j\) from \(D=-i\partial\).
No smoothness of \(g_j\) has been assumed.

### 8.4. One fixed conormal growth exponent for every derivative

Let \(u\in\mathcal A^m\), with the original normal amplitude
order \(\mu=m+(n-2)/4\). The frequency split in (HS1)–(HS2)
gives, after enlarging a fixed exponent slightly to absorb a
possible logarithm, a number
\(A>\max(\mu+1,0)\) such that for *every* normal derivative
order \(a\), tangential multiindex \(\beta\), and compact
subchart there is a constant \(C_{a\beta}\) with
\[
 \sup_{(x',s)\in K_0\times J}
  |D_{x'}^\beta\partial_s^a F_\varepsilon(x',s)|
       \le C_{a\beta}\varepsilon^{-A}.
 \tag{SP9}
\]
Indeed \(\partial_s^a F_\varepsilon
=\varepsilon^a\partial_t^au(x',\varepsilon s)\);
the conormal amplitude order rises by exactly \(a\), so
its high-frequency bound contains
\(\varepsilon^a\varepsilon^{-\mu-a-1}
=\varepsilon^{-\mu-1}\). The low-frequency and logarithmic
cases satisfy the same enlarged \(A\). Crucially the exponent
\(A\) is independent of \(a\) and \(\beta\), although the
constants depend on them. Thus for every integer \(K\),
\(\|F_\varepsilon\|_{H^K(K_0\times J)}
\le C_K\varepsilon^{-A}\).

We record the precise smoothing inference used below. Suppose
smooth \(h_\varepsilon\) on a fixed compact coordinate box obey
\(\|h_\varepsilon\|_{H^K}\le C_K\varepsilon^{-A}\)
for all \(K\), and for every \(N\) obey
\(\|h_\varepsilon-h\|_{H^{-L_N}}
\le C_N\varepsilon^N\), where \(h\) is a distribution.
Then \(h\) is smooth. Fix any one positive integer \(N\), and
retain its finite negative order \(L_N\). On a compact subbox insert
a fixed smooth cutoff before using full Fourier blocks; multiplication
preserves the displayed Sobolev bounds. For any \(a>0\), choose
\(\varepsilon=2^{-al}\). The low- and high-Sobolev block bounds give
\[
 \|\Delta_l h\|_2
 \le C_N 2^{l(L_N-aN)}
           +C_K2^{l(aA-K)}.
 \tag{SP10}
\]
For any requested \(r\ge0\), choose \(a>(L_N+r+2)/N\),
then an integer \(K>aA+r+2\). Both exponents in (SP10) are
strictly less than \(-r-2\), so it gives
\(\|\Delta_l h\|_2\le C_r2^{-l(r+2)}\) for every
\(r\). Summing their squared \(H^r\) weights proves membership
in every local Sobolev space, and the Fourier Sobolev estimate proves
smoothness. This argument uses the actual \(L_N\); it does not assert
one negative order valid for all Taylor degrees. The single positive
degree \(N=1\) would suffice for this smoothing inference.

### 8.5. Smoothness of every boundary coefficient

The expansion (SP8) includes lower powers of \(\varepsilon\),
so apply an exact finite scale cancellation. For any integer
\(N\ge1\), take distinct positive scales
\(\lambda_1,\ldots,\lambda_N\), for example \(1,\ldots,N\),
and the Lagrange interpolation coefficients at zero
\[
 c_l=\prod_{r\ne l}
          \frac{-\lambda_r}{\lambda_l-\lambda_r},
 \qquad
 \sum_{l=1}^{N}c_l\lambda_l^j
          =\begin{cases}1,&j=0,\\0,&1\le j<N.
            \end{cases}
 \tag{SP11}
\]
For \(g_0\), set
\(G_{N,\varepsilon}=\sum_lc_lF_{\lambda_l\varepsilon}\).
Equations (SP8) and (SP11) make
\(G_{N,\varepsilon}=g_0+O_{H^{-L_N}}(\varepsilon^N)\).
Equation (SP9) gives every high Sobolev norm bounded by
\(C_{K,N}\varepsilon^{-A}\). The dyadic argument (SP10),
with the adjustable scale specified there, proves
\(g_0\in C^\infty\).

Proceed by induction. If \(g_0,\ldots,g_{j-1}\) are smooth,
define on \(s\in J\)
\[
 H_{j,\varepsilon}(x',s)
  =j!i^{-j}s^{-j}\varepsilon^{-j}
      \left[F_\varepsilon(x',s)
       -\sum_{k=0}^{j-1}
          \frac{i^k\varepsilon^ks^k}{k!}g_k(x')\right].
 \tag{SP12}
\]
The factor \(s^{-j}\) is smooth on \(J\), and every
term is a smooth function there. Formula (SP8), taken to
order \(N+j\), gives
\(H_{j,\varepsilon}=g_j+\sum_{l=1}^{N-1}
\varepsilon^l h_l(x',s)+O_{H^{-L}}(\varepsilon^N)\),
where the \(h_l\) may still be distributions. The same
Lagrange weights (SP11) cancel all intermediate powers.
The high derivative bound (SP9), the smooth already-known
coefficients and (SP12) give
\(\|H_{j,\varepsilon}\|_{H^K}
\le C_{jK}\varepsilon^{-(A+j)}\). The dyadic
argument proves \(g_j\in C^\infty\). Induction proves
\[
 g_j\in C^\infty(\partial X)
       \quad\text{for every }j\ge0.
 \tag{SP13}
\]
The argument uses the same original \(u\) at every step;
it neither assumes nor constructs an unrelated boundary
extension.

### 8.6. Upgrading the whole series and removing the weight

Now every coefficient in (SP8) is smooth. Fix a desired
integer \(N\) and a smooth seminorm order \(r\). Expand (SP8)
through some \(N'>N\), so the remainder is
\(O_{H^{-L_{N'}}}(\varepsilon^{N'})\). Its high
\(H^K\) norm is at most \(C_K\varepsilon^{-A}\) by (SP9)
and the smooth polynomial terms. Interpolation between
\(H^{-L_{N'}}\) and \(H^K\) gives the required bound as follows.
Choose an integer \(d>r+n/2\), then \(N'>N+A\), retaining its
actual finite \(L=L_{N'}\). With \(K>d\), Fourier Hölder gives
the \(H^d\) exponent
\(\theta N'-(1-\theta)A\), where
\(\theta=(K-d)/(K+L)\). Choose \(K\) so large that
\(\theta>(N+A)/(N'+A)\); the exponent is then greater than
\(N\). The Fourier Sobolev bound \(H^d\hookrightarrow C^r\)
proves \(O_{C^r}(\varepsilon^N)\). The discarded coefficients
of degrees \(N,\ldots,N'-1\) contribute the same
\(O_{C^r}(\varepsilon^N)\). Hence the full exact Taylor
statement is
\[
 u(x',\varepsilon s)
  =\sum_{j=0}^{N-1}
       \frac{i^j\varepsilon^js^j}{j!}g_j(x')
       +O_{C^r(K_0\times J)}(\varepsilon^N)
 \quad\text{for every }N,r.
 \tag{SP14}
\]
Taking any fixed interior value, for example \(s=3/2\)
and \(\varepsilon=2t/3\), then differentiating in \(s\)
and \(x'\) before that restriction, shows that every ordinary mixed
derivative of \(u\) has a continuous boundary limit with
the precise Taylor coefficient in (SP14). This proves
\(u\in C^\infty(X)\). Conversely a smooth boundary function
belongs to \(\mathcal A\) by (GD1)–(GD2), and to
\(\mathcal A'\) by the smooth pairing and (GD3). Therefore,
as spaces of their actual supported representatives,
\[
 \mathcal A'(X)\cap\mathcal A(X)=C^\infty(X).
 \tag{SP15}
\]

The original Hardy inequality (HS3) now applies to each
ordinary derivative of a compactly localized \(u\). Insert a
normal cutoff equal to zero for \(t<\delta\) into (HS4).
The errors from its \(j\)-th derivative are supported in
\(\delta<t<2\delta\); the factor \(t^j\) in the final
weighted norm cancels its \(\delta^{-j}\) derivative size,
leaving an \(O(\delta^{1/2})\) \(L^2\) error because
every derivative of \(u\) is bounded there by (SP14).
Let \(\delta\downarrow0\). Thus the complete original
coefficient product survives for smooth boundary \(v\):
\[
 \int_0^\infty|v(t)|^2dt
 \le\left[\prod_{j=0}^{k-1}
         \frac4{(2j+1)^2}\right]
       \int_0^\infty t^{2k}|v^{(k)}(t)|^2dt,
 \tag{SP16}
\]
provided the far-end cutoff is retained. All ordinary
derivatives of the localized \(u\) are in \(L^2\), with
or without the safe weights of (HS2), by its proved
smoothness. The direct conormal-only attempt (HS5) remains
invalid; the all-order dual delta-jet estimate (SP5) is
the additional ingredient that completes the weight-removal
argument.

## 9. Smooth testers and boundary consequences

### 9.1. Smooth testers in the compressed definition

For \(u\in\mathcal A'(X)\), every properly supported
\(B\in\Psi_b^0\) satisfies \(Bu\in\mathcal A'\) by the
actual transpose action (GD13). Therefore (SP15) gives the
pointwise equivalence of admissible regularity tests
\[
 Bu\in\mathcal A(X)
       \quad\Longleftrightarrow\quad
 Bu\in C^\infty(X).
 \tag{SC1}
\]
Substituting *the same* family of operators and their unchanged
characteristic sets into the defining intersection (GW6)
yields the exact alternative definition
\[
 \operatorname{WF}_b(u)
   =\bigcap_{\substack{B\in\Psi_b^0\text{ proper}\\
                       Bu\in C^\infty(X)}}
                     \operatorname{Char}B
 \quad(u\in\mathcal A').
 \tag{SC2}
\]
No residual operator was asserted to produce a smooth
boundary function on an arbitrary supported distribution;
(SC1) uses both \(Bu\in\mathcal A'\) and the conormal
test result.

The finite cosphere argument (GW7) and (GW8) prove
\(\operatorname{WF}_b(u)=\varnothing\Rightarrow
u\in\mathcal A_{\mathrm{loc}}\) for any supported distribution.
If also \(u\in\mathcal A'\), compact smooth cutoffs preserve that
dual class by (GD13). Each cutoff output is in the original
\(\mathcal A\), so (SP15) makes it smooth. Cutoffs equal to one on
each compact neighborhood therefore prove the exact local comparison

\[
 \mathcal A'(X)\cap\mathcal A_{\mathrm{loc}}(X)
   =C^\infty(X)=\mathcal A'(X)\cap\mathcal A(X).
 \tag{SC2a}
\]

The reverse inclusion follows from (GD1)--(GD3), which place every
smooth boundary function in the single order \(m_0=-(n+2)/4\), with
local seminorm constants. Hence there is no global-order assumption
hidden in this smoothness conclusion. In particular,
\[
 \operatorname{WF}_b(u)=\varnothing,\quad u\in\mathcal A'
        \quad\Longrightarrow\quad u\in C^\infty(X).
 \tag{SC3}
\]
Conversely a smooth boundary function has its supported
representative in \(\mathcal A\) by (GD1)–(GD2), so the
identity operator is an order-zero tester with empty
characteristic set. Thus its compressed wave-front set is
empty. The implication and converse use the original
regularity class at the boundary, not interior-only
smoothness.

### 9.2. The boundary trace wave-front inclusion

Let \(g=u|_{\partial X}\) be the intrinsic trace (GD8) of
\(u\in\mathcal A'\), and let
\(q=(y',0,\eta',0)\) be a nonzero tangential boundary
compressed covector. Suppose \(q\notin\operatorname{WF}_b(u)\).
By the *definition* (GW6), some properly supported
\(B\in\Psi_b^0\) is elliptic at \(q\) and has
\(Bu\in\mathcal A\) on a neighborhood of \(y'\). Its
dual action remains in \(\mathcal A'\), so (SP15)
makes \(Bu\) smooth there. The original boundary jet
formula (GD14) at \(k=0\) gives the exact receiving map
\[
 (Bu)|_{\partial X}=B_0 g,
 \qquad
 \sigma_0(B_0)(y',\eta')
    =\sigma_0(B)(y',0,\eta',0).
 \tag{SC4}
\]
The first equality is initially (GL24) on smooth functions;
both sides are weakly continuous in \(u\) by (GD5),
(GD8), and (GD13), which proves it for the actual dual
distribution. The second equality retains the full
half-density comparison (GL18) and the original
\((2\pi)^{-(n-1)}\) tangential Fourier convention; no
normal factor has been set to one by a change of scale.

The ordinary boundary symbol in (SC4) is invertible at
\((y',\eta')\). Choose a conic cutoff \(\zeta\) there and
construct its ordered inverse symbol by
\(c_{-0}=\zeta\sigma_0(B_0)^{-1}\); at each lower order,
subtract the complete composition defect and multiply
on the correct side by that inverse. The asymptotic sum
gives a proper boundary operator \(C_0\) with
\(C_0B_0=\operatorname{Op}(\zeta)+R\), where \(R\)
is smoothing near \((y',\eta')\). Since \(B_0g\) is
smooth, the localized \(g\) is smooth there. This proves
\[
 \operatorname{WF}(u|_{\partial X})
   \subset
 \operatorname{WF}_b(u)|_{\partial X}
          \cap(T^*\partial X\setminus0).
 \tag{SC5}
\]
The boundary cotangent bundle is embedded by the exact
compressed anchor (GL8); pure normal compressed directions
have not been mistaken for trace covectors.

### 9.3. A tangential smooth tester for every regular boundary covector

Let \(u\in\mathcal N(X)\), and
\(q=(y',0,\eta',0)\ne0\) on the embedded boundary
cotangent bundle. If a properly supported tangential
operator \(B_b=b(x,D')\) is elliptic at
\((y',\eta')\) and \(B_bu\) is smooth on \(X\),
then its boundary wave-front set is empty. The exact
elliptic inclusion (GT14) immediately gives
\(q\notin\operatorname{WF}_b(u)\).

For the converse assume \(q\notin\operatorname{WF}_b(u)\).
Closedness of the boundary wave-front set on the compact
cosphere gives a small tangential base patch \(V\) about
\(y'\) and a conic tangential frequency patch \(\Gamma\)
about \(\eta'\) whose product closure misses
\(\operatorname{WF}_b(u)|_{\partial X}\). Choose an
order-zero tangential symbol \(b(x',t,\xi')\) supported
in that base and cone, with normal cutoff supported in a
small collar, equal to a nonzero scalar or the identity
matrix in a smaller patch about \((y',0,\eta')\), and
properly support its kernel. Its boundary symbol \(b_0\)
is elliptic at \(q\). By the full tangential theorem
(GT11)–(GT12), \(B_bu\in\mathcal N\) and
\[
 \operatorname{WF}_b(B_bu)|_{\partial X}
       \subset\operatorname{WF}_b(u)|_{\partial X}
                    \cap\Gamma=\varnothing
 \quad\text{on the chosen base patch}.
 \tag{SC6}
\]
The operator has output support in that base patch and
normal collar. If a sequence of interior wave-front points
of \(B_bu\) approached its compact boundary output set,
compactness of the cosphere would give a boundary
wave-front limit, contradicting (SC6). After shrinking
the normal cutoff once, the new operator is multiplication
of the old \(B_b\) on the left by a smooth \(t\)-cutoff
equal to one at zero; it remains tangential and elliptic at
\(q\). Its output has empty compressed
wave-front set throughout its support; outside that
support it is zero. The finite cover (GW7) therefore gives
\(B_bu\in\mathcal A\). As \(B_bu\in\mathcal A'\)
by (GT6), (SP15) gives \(B_bu\in C^\infty(X)\).
We have proved the exact equivalence
\[
 q\notin\operatorname{WF}_b(u)
 \quad\Longleftrightarrow\quad
 \exists\,B_b=b(x,D')\text{ proper, elliptic at }q,
             \ B_bu\in C^\infty(X).
 \tag{SC7}
\]
The existence assertion is coordinate independent: (GL10)
preserves the embedded tangential hyperplane, (GW6) is
intrinsic, and in either boundary chart the explicit
cutoff construction above supplies a tester. No assertion
of all-interior pseudolocality at pure normal covectors
is needed; the explicit exception in GT15 remains intact.

## 10. Examples

**Example 10.1 (a changed defining function).** Keep the tangential coordinates and set \(\bar x_n=e^{f(x')}x_n\), where \(f\) is any smooth real function. The exact compressed coordinate law (GL10) reads
\[
 \xi'=\bar\xi'+df\,\bar\rho,
 \qquad \rho=\bar\rho.
\]
Here \(df\) is the ordinary coordinate differential, and the hyperplane \(\rho=0\) remains the embedded \(T^*\partial X\). The tangential component does change when \(\bar\rho\ne0\); discarding that term would break the coordinate law.

**Example 10.2 (the conormal-only Hardy deficit).** In one normal dimension \(u(t)=\chi(t)t^{-3/4}H(t)\) has conormal order zero. Its first derivative has size \(t^{-7/4}\), so \(t u'(t)\) still has size \(t^{-3/4}\) and is not square-integrable. The exact delta test (HS6) shows why this function is outside \(\mathcal A'\). Thus the failure of the direct Hardy iteration has an explicit function behind it.

**Example 10.3 (the raw and corrected normal equations).** Take \(P=D_n+a_0\), with constant scalar \(a_0\), and \(u(t)=e^{-t}\) on \(t>0\). Then \(f=(i+a_0)e^{-t}\). The supported extension is \(U=H(t)e^{-t}\). Its *raw* distributional derivative is \(D_nU=iH(t)e^{-t}-i\delta(t)\), so
\[
 PU-Hf=-i\delta(t),\qquad t(PU-Hf)=0.
\]
The corrected derivative (GD9) is \(\nabla_n^{\mathrm{int}}U=iH(t)e^{-t}\), which has the prescribed interior restriction. The boundary delta is essential to the weighted equation and to its uniqueness calculation.

## 11. Exercises and complete solutions

**Exercise 11.1.** Let \(\bar x'=x'\) and \(\bar x_n=e^{f(x')}x_n\). Prove the inverse compressed covector law and identify which part is an ordinary cotangent covector on the boundary.

**Solution.** Example 10.1 gives \(\rho=\bar\rho\) and \(\xi'=\bar\xi'+df\,\bar\rho\). Solving without suppressing the mixed term gives \(\bar\rho=\rho\), \(\bar\xi'=\xi'-df\,\rho\). At \(x_n=0\), the embedded ordinary boundary cotangent vectors are exactly \(\rho=0\), so their tangential coordinate is unchanged. For a compressed covector with \(\rho\ne0\), the tangential coordinate changes by the precise term \(-df\,\rho\). These two formulas are inverses on the entire compressed fibre, not only on the boundary hyperplane.

**Exercise 11.2.** In one normal dimension replace the exponent \(3/4\) in Example 10.2 by any \(\beta\) with \(1/2<\beta<1\). Find its conormal order, its safe weighted \(L^2\) threshold, and use a boundary delta test to decide membership in \(\mathcal A'\).

**Solution.** The normal Fourier amplitude of \(\chi(t)t^{-\beta}H(t)\) has order \(\mu=\beta-1\). Formula (HS1) with \(n=1\) has \(\mu=m-1/4\), so the exact conormal order is \(m=\beta-3/4\). Formula (HS2) for the undifferentiated function requires \(\nu>m+1/4=\beta-1/2\) for \(t^{\nu}u\in L^2\). Its unweighted squared integral diverges because \(2\beta>1\). For any nonnegative unit-mass \(q\in C_c^\infty((1,2))\), the test \(q_\varepsilon=Q_\varepsilon\delta\) converges to \(\delta\) in every conormal order above \(1/4\), while the actual pairing is \(\varepsilon^{-\beta}\int_1^2s^{-\beta}q(s)ds\to+\infty\). Continuity on those conormal tests is required of \(\mathcal A'\), so the function is not in that dual class.

**Exercise 11.3.** For the constant-coefficient normal equation in Example 10.3, let \(V=U+c\delta\) be another supported extension, with scalar \(c\). Compute \(t(PV-Hf)\) exactly and verify that the weighted equation forces \(c=0\).

**Solution.** The original extension has \(t(PU-Hf)=0\). For the added term, \(P(c\delta)=cD_n\delta+a_0c\delta=-ic\delta'+a_0c\delta\). The exact distribution identities are \(t\delta=0\) and \(t\delta'=-\delta\). Thus \(t(PV-Hf)=ic\delta\). It vanishes only when \(c=0\). This is the \(m=1,\mu=0\) leading-coefficient case of (GE19)–(GE20); the lower coefficient \(a_0\) remains in the calculation but is killed by the *actual* factor \(t\delta=0\).

## 12. References and onward use

The local formulas used as prerequisites are written in the linked lessons at the start of this chapter. The compressed geometry, full kernel factors, boundary trace, normal extension and wave-front tests in Sections 1–11 form one continuous argument; later boundary-value lessons use these exact supported and restricted objects.