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, Totally characteristic operators on the half space, From symbol estimates to operators on every Sobolev scale, and Detecting regularity without choosing coordinates. 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.

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.