When finite defects force one-sided ellipticity

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

A closed range with only finitely many null solutions forces a pseudodifferential symbol to be injective at high frequency. A range with only finitely many missing targets forces the separate surjective condition. The two need not coincide for a rectangular system. This lesson proves both directions, then carries every source and target weight through a mixed-order system. It ends with an example showing exactly why an adapted graph domain changes the conclusion.

The required entry results are Finite defects under perturbation, From symbol estimates to operators on every Sobolev scale, and Symbols, finite defects, and the index on a closed manifold. Their compactness, complete symbol composition, and order-changing maps are recalled at the points where they enter. We use $D=-i\partial$, retain every matrix factor in its source-to-target order, and never identify distinct bundle fibers without the stated Hermitian metric.

1. Objects and the four one-sided assertions

Let $X$ be a compact smooth manifold without boundary. Let $E,F$ be finite-rank complex Hermitian bundles, with their half-density factors retained in the Sobolev spaces. Fix $m,s\in\mathbb R$ and $$P\in\Psi^m(X;E\otimes\Omega_X^{1/2},F\otimes\Omega_X^{1/2}), \qquad P_s:H^s(X;E\otimes\Omega_X^{1/2}) \longrightarrow H^{s-m}(X;F\otimes\Omega_X^{1/2}). \tag{OM1}$$ Here $\Psi^m$ has the $S^m_{1,0}$ full-symbol bounds in local bundle frames, and $p\in S^m/S^{m-1}$ is its original principal class. A representative $p$ is used when writing a pointwise high-frequency inequality. Changing representatives changes it by $S^{m-1}$, so the inequalities below are independent of that choice after increasing the lower frequency cutoff. The positive function $\langle\xi\rangle$ is defined by one fixed Riemannian cotangent metric; the original $p$, its bundle maps and all orders remain explicit.

The left condition is $$\|p(x,\xi)w\|_F\ge c\langle\xi\rangle^m\|w\|_E \quad (|\xi|\ge R,\ w\in E_x). \tag{OM2}$$ The right condition is the corresponding lower bound for the actual Hermitian adjoint $p(x,\xi)^*:F_x\to E_x$: $$\|p(x,\xi)^*v\|_E\ge c'\langle\xi\rangle^m\|v\|_F \quad (|\xi|\ge R',\ v\in F_x). \tag{OM3}$$ These conditions allow unequal ranks. The left one says injectivity with a uniform high-frequency bound, and the right one says surjectivity with a uniform high-frequency bound. Neither implies the other for a rectangular bundle map.

2. Finite kernel and closed range give a compact-error estimate

Assume $P_s$ has finite-dimensional kernel $N$ and closed range. The restriction to the orthogonal complement $N^\perp\subset H^s(E)$ is a bounded bijection onto $\operatorname{ran}P_s$. Both are Hilbert spaces, so the bounded inverse theorem gives $$\|u\|_{H^s} \le C\bigl(\|P_su\|_{H^{s-m}}+\|\Pi_Nu\|_{H^s}\bigr). \tag{OM4}$$ The projection $\Pi_N$ is finite rank and compact; no smoothness of $N$ has been assumed.

The same hypotheses give the lower-order estimate required in the original Sobolev spaces: $$\|u\|_{H^s} \le C_s\bigl(\|P_su\|_{H^{s-m}}+\|u\|_{H^{s-1}}\bigr). \tag{OM5}$$ If this failed, choose $u_j$ with $\|u_j\|_{H^s}=1$ and both terms on the right tending to zero. Equation (OM4) shows $\operatorname{dist}_{H^s}(u_j,N)\to0$. Choose $n_j\in N$ with $\|u_j-n_j\|_{H^s}\to0$. The continuous inclusion $H^s\hookrightarrow H^{s-1}$ gives $\|n_j\|_{H^{s-1}}\to0$. Its restriction to the finite-dimensional $N$ is injective, so its $H^{s-1}$ and $H^s$ norms on $N$ are equivalent. Thus $\|n_j\|_{H^s}\to0$, contradicting $\|u_j\|_{H^s}=1$. This argument does not silently make the unknown kernel smooth.

3. A local wave packet tests the complete original principal map

We record the quantitative packet fact used to derive (OM2). Let $A\in\Psi^0(X;E,F)$, with complete local symbol $a(x,\eta)$, and choose a smaller chart whose closure lies in one bundle trivialization. Let $\lambda=|\xi_\lambda|\to\infty$, with $x_\lambda$ in that smaller chart, and let $w_\lambda$ be a unit vector in its local input frame. Choose $\chi\in C_c^\infty(\mathbb R^n)$ with $\|\chi\|_2=1$, and set $$v_\lambda(x) =\lambda^{n/4}\chi\!\left(\lambda^{1/2}(x-x_\lambda)\right) e^{\,i\langle x-x_\lambda,\xi_\lambda\rangle}w_\lambda . \tag{OM6}$$ The coefficient in (OM6) is written in the coordinate half-density frame $|dx|^{1/2}$, using the stated input and output bundle frames. The unit condition means $w_\lambda^\dagger H_E(x_\lambda)w_\lambda=1$ in the original input metric at the center. The norm of the constant output matrix in (OM7) is taken in the specified output frame. The complete calculation below retains the fixed cotangent metric and its chart comparison constants; it includes the frequency derivatives needed for the operator estimate, not only the base derivatives.

The half-density and frame metrics vary smoothly on the shrinking support, so $$\|v_\lambda\|_{L^2}=1+O(\lambda^{-1/2}),\qquad v_\lambda\rightharpoonup0\text{ in }L^2,\qquad \|Av_\lambda-a(x_\lambda,\xi_\lambda)v_\lambda\|_{L^2} =O(\lambda^{-1/2}). \tag{OM7}$$ The weak limit follows by Cauchy–Schwarz on a shrinking ball for each fixed $L^2$ test, first for bounded smooth tests and then by density.

For completeness, the last bound has two exact parts. On the support of $v_\lambda$, the mean-value formula and the uniform $x$-derivatives of $a$ give $\|(a(x,\xi_\lambda)-a(x_\lambda,\xi_\lambda))v_\lambda\|_2 =O(\lambda^{-1/2})$. The Fourier transform of the packet is centered at $\xi_\lambda$, with width $\lambda^{1/2}$. Cut it to $|\eta-\xi_\lambda|\le\lambda/2$; the complementary Schwartz tail is $O(\lambda^{-N})$ in every fixed Sobolev norm, for every $N$. On the cut region the exact integral identity $$a(x,\eta)-a(x,\xi_\lambda) =\sum_{r=1}^n(\eta_r-\xi_{\lambda,r}) \int_0^1\partial_{\eta_r}a (x,\xi_\lambda+\theta(\eta-\xi_\lambda))\,d\theta \tag{OM8}$$ has coefficient symbols with every fixed $x$-derivative bounded by $C\lambda^{-1}$. The finite-derivative $L^2$ symbol estimate from the linked symbol-calculus lesson, applied after the frequency cutoff, and $\|(D-\xi_\lambda)v_\lambda\|_2=O(\lambda^{1/2})$ give $O(\lambda^{-1/2})$. Operators cut off away from the coordinate diagonal have smooth kernels; repeated integration by parts in the packet’s oscillation bounds their output by $O(\lambda^{-N})$ for each $N$. This proves (OM7), uniformly in the selected chart. A finite chart cover provides uniformity on $X$.

The complete coordinate packet proof

The operator bound used here is the full finite-derivative symbol estimate, (E23), proved there in (E24)–(E27) and (EC12)–(EC14). Its derivative count can be taken as $L_n=4N_0$, with an integer $N_0>n/2$. The Fourier conventions are (E10), and the exact kernel reconstruction is (E12).

1. Exact objects, metrics, density and local realization

Let $X$ be the original compact smooth manifold without boundary, and let $E,F$ be the original complex Hermitian bundles. Keep their half-density factors. Let $$A\in\Psi^0_{1,0} (X;E\otimes\Omega_X^{1/2},F\otimes\Omega_X^{1/2}). \tag{PW1}$$ We give the proof on a component of dimension $n\geq1$. A zero-dimensional component has no covectors tending to infinity and consequently contributes no packet sequence to the necessity argument.

Choose one coordinate chart $x=(x_1,\ldots,x_n)$, input and output bundle frames, and nested relatively compact open subsets $$K\subset U_0,\qquad \overline {U_0}\subset U_1,\qquad \overline {U_1}\subset U_2,\qquad \overline {U_2}\subset U , \tag{PW2}$$ where $K$ is compact and every packet center $x_\lambda$ belongs to $K$. Choose fixed smooth cutoffs $\psi,\zeta$, with $\psi=1$ near $\overline {U_0}$, $\operatorname{supp}\psi\subset U_1$, $\zeta=1$ near $\overline {U_1}$, and $\operatorname{supp}\zeta\subset U_2$. Use the coordinate half-density frame $|dx|^{1/2}$. Write $H_E(x),H_F(x)$ for the positive Hermitian metric matrices in the two specified frames. Thus, with the Hermitian pairing linear in its first variable, $$h_E(u,v)=v^\dagger H_E(x)u,\qquad \|u(x)|dx|^{1/2}\|_{L^2(E)}^2 =\int u(x)^\dagger H_E(x)u(x)\,dx . \tag{PW3}$$ There is no missing density multiplier in PW3: the square of the chosen half-density is exactly $|dx|$. Under another coordinate system $y=\Phi(x)$, the coefficient transforms with $|\det D\Phi^{-1}(y)|^{1/2}$; squaring that factor and changing variables preserves precisely PW3. Bundle frame changes transform the matrices and coefficients together, preserving the same integral.

On the compact coordinate region let $$0<h_E^-\leq\lambda_{\min}(H_E(x)),\qquad 0<h_F^-\leq\lambda_{\min}(H_F(x)),\qquad \lambda_{\max}(H_E(x))\leq h_E^+,\quad \lambda_{\max}(H_F(x))\leq h_F^+ . \tag{PW4}$$ All metric derivatives used below have finite suprema there. Write $|\eta|_{\rm c}$ for the Euclidean coordinate covector norm and keep $|\eta|_{g,x}$ for the fixed Riemannian cotangent norm already chosen in OM1. There are fixed constants $$0<\kappa_-\leq\kappa_+,\qquad \kappa_-|\eta|_{\rm c}\leq|\eta|_{g,x} \leq\kappa_+|\eta|_{\rm c}. \tag{PW5}$$ For example their squares can be taken as the minimum and maximum of the eigenvalues of the coordinate cotangent metric matrix on the compact region. These are exact inequalities between the two original norms.

Let $a(x,\eta)$ be a complete local left symbol for $A$, with all the original $S^0_{1,0}$ bounds $$\|\partial_x^\alpha\partial_\eta^\beta a(x,\eta)\| \leq M_{\alpha,\beta}\langle\eta\rangle_{\rm c}^{-|\beta|}, \qquad \langle\eta\rangle_{\rm c}=(1+|\eta|_{\rm c}^2)^{1/2}. \tag{PW6}$$ Its source and target coordinate dimensions are kept distinct. The local realization of the pseudodifferential class means $\zeta A\psi=\zeta\operatorname{Op}(a)\psi+R$, where $R$ has a smooth localized kernel. Here left quantization is exactly $$\widehat u(\eta)=\int e^{-ix\cdot\eta}u(x)\,dx,\qquad \operatorname{Op}(a)u(x)=(2\pi)^{-n} \int e^{ix\cdot\eta}a(x,\eta)\widehat u(\eta)\,d\eta , \qquad D_r=-i\partial_{x_r}. \tag{PW7}$$ These are E10’s conventions. Put $\widetilde a=\zeta a$, extended by zero outside the chart. It is a global Euclidean $S^0_{1,0}$ symbol. The exact derivative formula is $$\partial_x^\alpha\partial_\eta^\beta\widetilde a =\sum_{\nu\leq\alpha}{\alpha\choose\nu} (\partial_x^\nu\zeta) (\partial_x^{\alpha-\nu}\partial_\eta^\beta a). \tag{PW8}$$ This retains every output-cutoff contribution. In particular PW6 holds for $\widetilde a$, with new fixed constants $\widetilde M_{\alpha,\beta}$ bounded by the finite sum in PW8.

2. The packet and its exact norm and Fourier transform

Keep the original packet profile $\chi\in C_c^\infty(\mathbb R^n)$ with $\int|\chi(z)|^2\,dz=1$, and put $R_\chi=\sup\{|z|_{\rm c}:z\in\operatorname{supp}\chi\}$. For the original fixed metric define $$\lambda=|\xi_\lambda|_{g,x_\lambda}\longrightarrow\infty, \qquad w_\lambda^\dagger H_E(x_\lambda)w_\lambda=1. \tag{PW9}$$ The second equality is the meaning of unit vector needed in OM6. It does not require an orthonormal coordinate frame. It gives $|w_\lambda|_{\rm c}^2\leq(h_E^-)^{-1}$. For all sufficiently large $\lambda$, the packet support is contained in $U_0$, uniformly over $x_\lambda\in K$. The actual section, with the local frame identifications made explicit, is $$v_\lambda(x)= \lambda^{n/4}\chi\!\left(\sqrt\lambda(x-x_\lambda)\right) e^{\,i(x-x_\lambda)\cdot\xi_\lambda} w_\lambda\,|dx|^{1/2}. \tag{PW10}$$ The local bundle frame multiplying $w_\lambda$ is understood in PW10, and the section is extended by zero. Its exact norm is $$\|v_\lambda\|_{L^2(E)}^2 =\int|\chi(z)|^2 w_\lambda^\dagger H_E(x_\lambda+\lambda^{-1/2}z)w_\lambda\,dz . \tag{PW11}$$ This follows from the exact change $dx=\lambda^{-n/2}dz$; that factor cancels the $\lambda^{n/2}$ from the squared packet amplitude. The full metric difference is $$H_E(x_\lambda+\lambda^{-1/2}z)-H_E(x_\lambda) =\lambda^{-1/2}\sum_{j=1}^n z_j \int_0^1\partial_{x_j}H_E (x_\lambda+t\lambda^{-1/2}z)\,dt . \tag{PW12}$$ The segments lie inside the fixed coordinate region for large $\lambda$. PW4, PW9 and PW12 give a uniform $O(\lambda^{-1/2})$ error in PW11 and therefore $$\|v_\lambda\|_{L^2(E)}^2=1+O(\lambda^{-1/2}), \qquad \|v_\lambda\|_{L^2(E)}=1+O(\lambda^{-1/2}). \tag{PW13}$$

For clarity about another possible half-density presentation, if one writes PW10 relative to $\rho(x)=r(x)|dx|^{1/2}$, $r>0$, then its norm is instead $$\int|\chi(z)|^2r(x_\lambda+\lambda^{-1/2}z)^2 w_\lambda^\dagger H_E(x_\lambda+\lambda^{-1/2}z)w_\lambda\,dz . \tag{PW14}$$ Under PW9 this tends to $r(x_\lambda)^2$, with the same uniform metric-and-density error. Thus PW13 as written requires the coordinate half-density frame, or the explicit center condition $r(x_\lambda)^2 w_\lambda^\dagger H_E(x_\lambda)w_\lambda=1$. If a general frame is used, its $r$ factor also stays in the Fourier integral; it cannot be silently dropped. The following formulas use the stated coordinate frame PW3.

Writing $v_\lambda$ for its coordinate coefficient in PW7, its forward Fourier transform is exactly $$\widehat v_\lambda(\eta) =\lambda^{-n/4}e^{-ix_\lambda\cdot\eta} \widehat\chi\!\left( \frac{\eta-\xi_\lambda}{\sqrt\lambda}\right)w_\lambda . \tag{PW15}$$ Indeed $x=x_\lambda+z/\sqrt\lambda$ gives $$-x\cdot\eta+(x-x_\lambda)\cdot\xi_\lambda =-x_\lambda\cdot\eta -z\cdot(\eta-\xi_\lambda)/\sqrt\lambda , \quad \lambda^{n/4}dx=\lambda^{-n/4}dz .$$ In particular the constant phase is $e^{-ix_\lambda\cdot\eta}$. Replacing it by $e^{-ix_\lambda\cdot(\eta-\xi_\lambda)}$ would change this original packet by the extra factor $e^{ix_\lambda\cdot\xi_\lambda}$. Inverse transformation retains the complete constant: $$v_\lambda(x)= (2\pi)^{-n}\lambda^{n/4} e^{i(x-x_\lambda)\cdot\xi_\lambda} \int e^{i\sqrt\lambda(x-x_\lambda)\cdot z} \widehat\chi(z)\,dz\,w_\lambda . \tag{PW16}$$ Plancherel in these conventions is $\|u\|_{L^2(dx)}^2=(2\pi)^{-n}\int|\widehat u(\eta)|^2d\eta$. For every multiindex $\beta$, PW15 has the full derivative $$\partial_\eta^\beta\widehat v_\lambda(\eta) =\lambda^{-n/4}e^{-ix_\lambda\cdot\eta} \sum_{\mu\leq\beta}{\beta\choose\mu} (-ix_\lambda)^{\beta-\mu}\lambda^{-|\mu|/2} (\partial^\mu\widehat\chi) ((\eta-\xi_\lambda)/\sqrt\lambda)w_\lambda . \tag{PW17}$$ Every derivative of $\widehat\chi$ decreases faster than every power: differentiate its defining compact integral and integrate by parts in each original coordinate. No Fourier constant is introduced by that forward transform.

The exact centered differentiation identity is $$(D_r-\xi_{\lambda,r})v_\lambda =-i\lambda^{n/4+1/2} (\partial_r\chi)(\sqrt\lambda(x-x_\lambda)) e^{i(x-x_\lambda)\cdot\xi_\lambda}w_\lambda . \tag{PW18}$$ Consequently its Euclidean norm squared is $\lambda|w_\lambda|_{\rm c}^2\int|\partial_r\chi|^2$, while its geometric norm squared retains the exact integrand $$\lambda\int|\partial_r\chi(z)|^2 w_\lambda^\dagger H_E(x_\lambda+\lambda^{-1/2}z) w_\lambda\,dz . \tag{PW19}$$ Both yield a uniform $O(\lambda^{1/2})$ norm.

3. The frequency cutoff, all of its derivatives and the tail

Set $c_g=\max(1,\kappa_+)$. Choose a fixed $\vartheta\in C_c^\infty(\mathbb R^n)$, equal to one when $|q|_{\rm c}\leq1/2$, supported where $|q|_{\rm c}\leq1$, and with $0\leq\vartheta\leq1$. Define $$q_\lambda(\eta)=\frac{2c_g}{\lambda}(\eta-\xi_\lambda), \quad \vartheta_\lambda(\eta)=\vartheta(q_\lambda(\eta)), \quad t_\lambda=(1-\vartheta_\lambda)(D)v_\lambda . \tag{PW20}$$ This keeps $\lambda$ as the original metric norm. The cutoff support has $|\eta-\xi_\lambda|_{\rm c}\leq\lambda/(2c_g)$. When $\lambda$ is a coordinate norm and $c_g=1$, this is the radius $\lambda/2$ stated in the original OM8 discussion. For a general metric PW20 records the needed chart constant. On that support, for every $0\leq\theta\leq1$, $$|\xi_\lambda+\theta(\eta-\xi_\lambda)|_{\rm c} \geq \lambda/\kappa_+-\lambda/(2c_g) \geq \lambda/(2c_g),\qquad |\eta|_{\rm c}\leq\lambda/\kappa_-+\lambda/(2c_g). \tag{PW21}$$ Every derivative of the cutoff is exactly $$\partial_\eta^\gamma\vartheta_\lambda =(2c_g)^{|\gamma|}\lambda^{-|\gamma|} (\partial^\gamma\vartheta)(q_\lambda). \tag{PW22}$$ It is supported in the same fixed-radius rescaled ball. Every positive-order derivative of $1-\vartheta_\lambda$ is the negative of PW22; its zeroth derivative is $1-\vartheta_\lambda$. Thus every tail-transform derivative is exactly the product sum $$\partial_\eta^\beta\widehat t_\lambda =\sum_{\gamma\leq\beta}{\beta\choose\gamma} \partial_\eta^\gamma(1-\vartheta_\lambda) \partial_\eta^{\beta-\gamma}\widehat v_\lambda , \tag{PW23}$$ with PW17 and PW22 supplying all factors and signs.

For every fixed real $s$, the tail’s exact Euclidean Sobolev norm is $$\begin{split} \|t_\lambda\|_{H^s(\mathbb R^n)}^2 &=(2\pi)^{-n}|w_\lambda|_{\rm c}^2 \int \langle\xi_\lambda+\sqrt\lambda z\rangle_{\rm c}^{2s} |1-\vartheta(2c_g z/\sqrt\lambda)|^2 |\widehat\chi(z)|^2\,dz . \end{split} \tag{PW24}$$ The two powers $\lambda^{-n/2}$ and $\lambda^{n/2}$ from PW15 and $d\eta$ cancel exactly. The nonzero integrand has $|z|_{\rm c}\geq\sqrt\lambda/(4c_g)$. Put $s_+=\max(s,0)$. For $\lambda\geq1$, $$\langle\xi_\lambda+\sqrt\lambda z\rangle_{\rm c} \leq \sqrt2(1+\kappa_-^{-1})\lambda \langle z\rangle_{\rm c}. \tag{PW25}$$ Use $|\widehat\chi(z)|\leq C_L\langle z\rangle_{\rm c}^{-L}$. For $\lambda\geq(4c_g)^2$ and $2(L-s_+)>n$, integration outside the displayed ball gives $$\|t_\lambda\|_{H^s}^2 \leq C_{s,L}\lambda^{3s_++n/2-L}. \tag{PW26}$$ For example this follows by comparison with $\int_R^\infty r^{n-1-2(L-s_+)}dr =R^{n-2(L-s_+)}/(2(L-s_+)-n)$, multiplied by the sphere area; here $R=\sqrt\lambda/(4c_g)$. All constants from PW24–PW25 remain in $C_{s,L}$. For each fixed $s$ and $N$, choosing $L>2N+3s_++n/2$ proves $$\|t_\lambda\|_{H^s}=O(\lambda^{-N}). \tag{PW27}$$ This is uniform in the centers, directions and metric-unit vectors. It also keeps all centered differentiation factors: for a fixed multiindex $\delta$, replace PW24’s integrand by $\lambda^{|\delta|}|z^\delta|^2$ times that integrand to obtain $$\|(D-\xi_\lambda)^\delta t_\lambda\|_{H^s}^2 \leq C_{s,L,\delta} \lambda^{3s_++2|\delta|+n/2-L}. \tag{PW28}$$ Thus these tail terms decrease faster than every prescribed power as well.

4. Every derivative of the mean-value coefficient

The exact identity OM8, now for the cutoff-extended symbol, is $$\widetilde a(x,\eta)-\widetilde a(x,\xi_\lambda) =\sum_{r=1}^n(\eta_r-\xi_{\lambda,r}) \int_0^1\partial_{\eta_r}\widetilde a (x,\xi_\lambda+\theta(\eta-\xi_\lambda))\,d\theta . \tag{PW29}$$ Define the actual cutoff coefficients $$b_{\lambda,r}(x,\eta)=\vartheta_\lambda(\eta) \int_0^1\partial_{\eta_r}\widetilde a (x,\xi_\lambda+\theta(\eta-\xi_\lambda))\,d\theta . \tag{PW30}$$ For all multiindices $\alpha,\beta$, their complete derivative formula is $$\begin{split} \partial_x^\alpha\partial_\eta^\beta b_{\lambda,r} &=\sum_{\gamma\leq\beta}{\beta\choose\gamma} (2c_g)^{|\gamma|}\lambda^{-|\gamma|} (\partial^\gamma\vartheta)(q_\lambda)\\ &\quad{}\times\int_0^1\theta^{|\beta-\gamma|} (\partial_x^\alpha \partial_\eta^{\beta-\gamma+e_r}\widetilde a) (x,\xi_\lambda+\theta(\eta-\xi_\lambda))\,d\theta . \end{split} \tag{PW31}$$ If desired, substitute PW8 into the last line to retain each individual output-cutoff derivative as well. Each factor $\theta^{|\beta-\gamma|}$ comes from differentiating the affine frequency path; there is no derivative of either center parameter, because these are fixed parameters when differentiating $x,\eta$. The binomial coefficients include every way derivatives can hit the frequency cutoff. Differentiation under the integral is justified on the compact interval by the smooth integrand and its uniform bounds PW6 and PW21.

Every nonzero term in PW31 is supported where PW21 holds. Consequently $$\begin{split} \|\partial_x^\alpha\partial_\eta^\beta b_{\lambda,r}\|_\infty &\leq (2c_g)^{1+|\beta|} \lambda^{-1-|\beta|} \sum_{\gamma\leq\beta}{\beta\choose\gamma} \|\partial^\gamma\vartheta\|_\infty \frac{\widetilde M_{\alpha,\beta-\gamma+e_r}} {1+|\beta-\gamma|}. \end{split} \tag{PW32}$$ Here the denominator is the exact integral $\int_0^1\theta^{|\beta-\gamma|}d\theta$. The factor $(2c_g)^{1+|\beta|}$ is the product of the cutoff factor in PW31 and the lower-frequency comparison in PW21. In particular $b_{\lambda,r}$ is a uniformly bounded family in $S^{-1}_{1,0}$, and $\lambda b_{\lambda,r}$ is a uniformly bounded family in $S^0_{1,0}$: on its support $\langle\eta\rangle_{\rm c}\leq (1+\kappa_-^{-1}+(2c_g)^{-1})\lambda$, and outside that support all derivatives vanish. More directly, E23 and PW32 give $$\|\operatorname{Op}(b_{\lambda,r})u\|_{L^2(dx)} \leq C_n \max_{|\alpha|+|\beta|\leq L_n} \|\partial_x^\alpha\partial_\eta^\beta b_{\lambda,r}\|_\infty \|u\|_{L^2(dx)} \leq C_r\lambda^{-1}\|u\|_{L^2(dx)} . \tag{PW33}$$ This uses a fixed derivative count and finitely many original symbol seminorms. It is therefore uniform, rather than a separate boundedness assertion for each $\lambda$.

5. Ordered decomposition and the spatial term

Let $M_{\widetilde a(\cdot,\xi_\lambda)}$ denote multiplication by the actual $x$-dependent matrix. PW29 gives the exact operator identity on the packet $$\begin{split} \operatorname{Op}(\widetilde a)v_\lambda -M_{\widetilde a(\cdot,\xi_\lambda)}v_\lambda &=\sum_{r=1}^n \operatorname{Op}(b_{\lambda,r}) (D_r-\xi_{\lambda,r})v_\lambda\\ &\quad{}+\operatorname{Op}(\widetilde a)t_\lambda -M_{\widetilde a(\cdot,\xi_\lambda)}t_\lambda . \end{split} \tag{PW34}$$ To check every product in this identity, transform the rightmost $(D_r-\xi_{\lambda,r})v_\lambda$. Its transform is exactly $(\eta_r-\xi_{\lambda,r})\widehat v_\lambda$. Thus the first line of PW34 has left symbol $(\widetilde a(x,\eta)-\widetilde a(x,\xi_\lambda)) \vartheta_\lambda(\eta)$. The second line has the same difference times $1-\vartheta_\lambda(\eta)$. Their sum is the original left-symbol difference. The derivative operator stands on the right; no matrix factor is commuted, and no composition correction is discarded.

E23 also bounds the fixed operator $\operatorname{Op}(\widetilde a)$. Multiplication by $\widetilde a(x,\xi_\lambda)$ is bounded uniformly by PW6 and PW8. Hence PW18, PW27 and PW33 prove $$\|\operatorname{Op}(\widetilde a)v_\lambda -M_{\widetilde a(\cdot,\xi_\lambda)}v_\lambda\|_{L^2(dx)} \leq C\lambda^{-1/2}. \tag{PW35}$$ The output is supported in the fixed support of $\zeta$; the factor $H_F$ in PW3 converts this to the geometric norm with the factor $(h_F^+)^{1/2}$. The input norms in PW33 and PW18 likewise retain PW4’s uniform coordinate comparison.

Let the frozen output section be $$f_\lambda(x)= \lambda^{n/4}\chi(\sqrt\lambda(x-x_\lambda)) e^{i(x-x_\lambda)\cdot\xi_\lambda} a(x_\lambda,\xi_\lambda)w_\lambda\,|dx|^{1/2}, \tag{PW36}$$ using the specified output frame. This is the precise meaning of $a(x_\lambda,\xi_\lambda)v_\lambda$ in OM7: the constant coordinate matrix sends the input coordinates to the output coordinates. It does not identify $E_x$ with $F_x$. On the support of $v_\lambda$, $\zeta=1$ and $$a(x,\xi_\lambda)-a(x_\lambda,\xi_\lambda) =\sum_{j=1}^n(x_j-x_{\lambda,j}) \int_0^1\partial_{x_j}a (x_\lambda+t(x-x_\lambda),\xi_\lambda)\,dt . \tag{PW37}$$ PW6, PW9, PW11 and the compact support of $\chi$ give $$\|M_{\widetilde a(\cdot,\xi_\lambda)}v_\lambda -f_\lambda\|_{L^2(F)} \leq C\lambda^{-1/2} \left(\int |z|_{\rm c}^2|\chi(z)|^2dz\right)^{1/2}. \tag{PW38}$$ All the spatial factors and the target metric remain in this estimate. Writing $z_\lambda=a(x_\lambda,\xi_\lambda)w_\lambda$, its frozen output norm is exactly $$\|f_\lambda\|_{L^2(F)}^2 =\int|\chi(z)|^2 z_\lambda^\dagger H_F(x_\lambda+\lambda^{-1/2}z) z_\lambda\,dz . \tag{PW39}$$ The target version of PW12 and PW4 give $$\left|\|f_\lambda\|_{L^2(F)}^2 -\|a(x_\lambda,\xi_\lambda)w_\lambda\|_{F_{x_\lambda}}^2\right| \leq C\lambda^{-1/2} \|a(x_\lambda,\xi_\lambda)w_\lambda\|_{F_{x_\lambda}}^2 . \tag{PW40}$$ This relative estimate includes the case $z_\lambda=0$, when both sides vanish exactly.

6. The off-diagonal kernel and every smoothing contribution

For large $\lambda$, $\psi v_\lambda=v_\lambda$. The exact localized decomposition is $$Av_\lambda =\operatorname{Op}(\widetilde a)v_\lambda +Rv_\lambda+(1-\zeta)A\psi v_\lambda . \tag{PW41}$$ The $R$ term has a smooth localized kernel by its definition. The last term has a smooth kernel because the supports of $1-\zeta$ and $\psi$ are separated. This also follows directly from the full local Fourier kernel, with all signs retained. For $z=x-y\ne0$, E12 and distributional frequency integration by parts give $$\begin{split} \partial_x^\alpha\partial_y^\beta K_a(x,y) &=(2\pi)^{-n}|x-y|_{\rm c}^{-2M} \sum_{\gamma\leq\alpha}{\alpha\choose\gamma} \int e^{i(x-y)\cdot\eta}\\ &\quad{}\times(-\Delta_\eta)^M \left[(i\eta)^{\alpha-\gamma}(-i\eta)^\beta \partial_x^\gamma a(x,\eta)\right]\,d\eta . \end{split} \tag{PW42}$$ Choose $2M>n+|\alpha|+|\beta|$. The full differentiated amplitude is $$\begin{split} (-\Delta_\eta)^M [P_{\alpha,\beta,\gamma}(\eta)\partial_x^\gamma a] &=(-1)^M\sum_{|\nu|=M}\frac{M!}{\nu!} \sum_{\delta\leq2\nu}{2\nu\choose\delta} (\partial_\eta^{2\nu-\delta} P_{\alpha,\beta,\gamma})\\ &\qquad{}\times (\partial_\eta^\delta\partial_x^\gamma a), \quad P_{\alpha,\beta,\gamma} =(i\eta)^{\alpha-\gamma}(-i\eta)^\beta . \end{split} \tag{PW43}$$ Its nonzero terms have frequency bound $C\langle\eta\rangle_{\rm c}^{|\alpha-\gamma|+|\beta|-2M}$, which is integrable. For a monomial $\partial_\eta^\tau\eta^\omega$ equals $\omega!\eta^{\omega-\tau}/(\omega-\tau)!$ when $\tau\leq\omega$, and is zero otherwise; thus PW43 keeps the full coefficient and every zero. The original inverse factor $(2\pi)^{-n}$ and both differentiation signs stay in PW42. The distributional identities E12 and EC15 justify PW42 before any assertion of absolute convergence. Multiplication by $|x-y|_{\rm c}^{-2M}$ is smooth away from $z=0$; the integrable amplitude then identifies the resulting restriction. Here is also the full compact-cutoff passage. Choose a fixed smooth $\omega$, equal to one on $|\eta|_{\rm c}\leq c_0$ and zero on $|\eta|_{\rm c}\geq c_1$, with $0<c_0<c_1$. Put $B=P_{\alpha,\beta,\gamma}\partial_x^\gamma a$ and $d=|\alpha-\gamma|+|\beta|$. For every $\nu$ in PW43, $$\partial_\eta^{2\nu}[B(x,\eta)\omega(\eta/R)] =\sum_{\sigma\leq2\nu}{2\nu\choose\sigma} (\partial_\eta^{2\nu-\sigma}B)(x,\eta) R^{-|\sigma|}(\partial^\sigma\omega)(\eta/R). \tag{PW43a}$$ The derivatives of $B$ have their full product-rule sums as in PW43, with the corresponding total derivative order. The $\sigma=0$ term tends to $\partial_\eta^{2\nu}B$ in absolute integral by its integrable bound. Every $\sigma>0$ term has the original annular support $c_0R\leq|\eta|_{\rm c}\leq c_1R$, and its absolute integral has the bound $${2\nu\choose\sigma}R^{-|\sigma|} \|\partial^\sigma\omega\|_\infty \int_{c_0R\leq|\eta|_{\rm c}\leq c_1R} C_{\nu,\sigma}\langle\eta\rangle_{\rm c}^{d-2M+|\sigma|}d\eta \leq C'_{\nu,\sigma}R^{n+d-2M}\longrightarrow0. \tag{PW43b}$$ The constants include the annulus volume $\omega_nc_1^nR^n$, $c_0,c_1$ and all polynomial coefficients. The exponent is negative by the stated choice of $M$. Thus every frequency-cutoff derivative vanishes separately, and the original oscillatory identity passes to PW42 with all boundary terms accounted for. For every pair $\alpha,\beta$, this is a continuous derivative. Consequently the kernel is smooth away from the coordinate diagonal. For output cutoff $c(x)$ and input cutoff $d(y)$, the exact remaining derivative sum is $$\partial_x^\alpha\partial_y^\beta[c(x)K_a(x,y)d(y)] =\sum_{\rho\leq\alpha}\sum_{\sigma\leq\beta} {\alpha\choose\rho}{\beta\choose\sigma} (\partial_x^\rho c)(x) (\partial_x^{\alpha-\rho}\partial_y^{\beta-\sigma}K_a)(x,y) (\partial_y^\sigma d)(y). \tag{PW43c}$$ For the actual off-diagonal term, $c=1-\zeta$, $d=\psi$: the zeroth output derivative is $1-\zeta$, and each positive output derivative is $-\partial_x^\rho\zeta$. All such contributions retain the smooth-kernel bounds just proved. In other charts the same argument applies. Compactness and the positive support separation give uniform bounds for the resulting global smooth kernel and all its input and output derivatives.

Here is the exact packet bound for any such smooth matrix kernel $S(x,y)$. Its local action on the actual coordinate half-density coefficient is $$Sv_\lambda(x)=\lambda^{n/4} \int S(x,y)\chi(\sqrt\lambda(y-x_\lambda)) e^{i(y-x_\lambda)\cdot\xi_\lambda}w_\lambda\,dy . \tag{PW44}$$ The input integration measure is precisely $dy$; the kernel’s half-density factors produce the output half-density as in PW3. Set $$L_\lambda=\frac{\xi_\lambda\cdot\partial_y} {i|\xi_\lambda|_{\rm c}^2}; \qquad L_\lambda e^{i(y-x_\lambda)\cdot\xi_\lambda} =e^{i(y-x_\lambda)\cdot\xi_\lambda}. \tag{PW45}$$ Compact support of the packet envelope removes all input boundary terms. Repeated integration by parts gives the exact finite sum $$\begin{split} Sv_\lambda(x) &=\frac{\lambda^{n/4}(-1)^M} {(i|\xi_\lambda|_{\rm c}^2)^M} \sum_{|\gamma|=M}\frac{M!}{\gamma!}\xi_\lambda^\gamma \sum_{\delta\leq\gamma}{\gamma\choose\delta} \lambda^{|\delta|/2}\\ &\quad{}\times\int e^{i(y-x_\lambda)\cdot\xi_\lambda} (\partial_y^{\gamma-\delta}S)(x,y) (\partial^\delta\chi)(\sqrt\lambda(y-x_\lambda)) w_\lambda\,dy . \end{split} \tag{PW46}$$ The sign $(-1)^M$, denominator $i^M$, multinomial coefficients and all envelope derivatives remain visible. Every fixed output derivative has the same formula with the corresponding derivative of $S$. PW5 gives $$\frac{|\xi_\lambda^\gamma|} {|\xi_\lambda|_{\rm c}^{2M}} \leq|\xi_\lambda|_{\rm c}^{-M} \leq\kappa_+^M\lambda^{-M}.$$ The envelope’s exact $L^1$ integral is $\lambda^{-n/2}\|\partial^\delta\chi\|_1$. Using PW9 in PW46 therefore yields, for every fixed output derivative order, $$\|\partial_x^\alpha Sv_\lambda\|_\infty \leq C_{\alpha,M}\lambda^{-M/2-n/4}. \tag{PW47}$$ Finitely many output charts, the retained smooth metrics and the finite volume of $X$ convert this to geometric $L^2$. For each requested power $N$, choose $M\geq2N$ to obtain $$\|Rv_\lambda\|_{L^2(F)} +\|(1-\zeta)A\psi v_\lambda\|_{L^2(F)} =O(\lambda^{-N}). \tag{PW48}$$ The proof applies separately to every smooth remainder; none is absorbed into the symbol error without an estimate.

PW35, PW38, PW41 and PW48 prove the required full estimate $$\boxed{\ \|Av_\lambda-f_\lambda\|_{L^2(F)} \leq C\lambda^{-1/2}\ } . \tag{PW49}$$ The constant depends only on the fixed chart, cutoffs, metric bounds, profile and finitely many symbol seminorms. It is independent of the moving center, covector direction, frequency and metric-unit vector.

7. Weak convergence, order-minus-one errors and uniform necessity

The support is contained in the coordinate ball $B(x_\lambda,R_\chi/\sqrt\lambda)$, whose coordinate volume is $\omega_nR_\chi^n\lambda^{-n/2}$. For any fixed $g\in L^2(E)$, geometric Cauchy–Schwarz gives $$|\langle v_\lambda,g\rangle_{L^2(E)}| \leq \|v_\lambda\|_{L^2(E)} \|\mathbf1_{\operatorname{supp}v_\lambda}g\|_{L^2(E)} \longrightarrow0. \tag{PW50}$$ Absolute continuity of the integral of $g^\dagger H_Eg$ proves the convergence for these moving sets, uniformly with respect to their centers. PW13 bounds the first factor. Thus the original packets are weakly null.

For completeness let $\mathcal R\in\Psi^{-1}_{1,0}(X;E,F)$, let $r\in S^{-1}_{1,0}$ be its complete local symbol, and retain its localized smooth remainder and off-diagonal term. In the following Euclidean formulas use $r=\zeta r_{\rm loc}$, extended by zero, exactly as in PW8. Thus each base derivative contains the full sum of derivatives of $\zeta$ and of the original complete symbol $r_{\rm loc}$; at the packet center the two symbols coincide. Its cutoff symbol has every derivative $$\partial_x^\alpha\partial_\eta^\beta [r(x,\eta)\vartheta_\lambda(\eta)] =\sum_{\gamma\leq\beta}{\beta\choose\gamma} (\partial_x^\alpha\partial_\eta^{\beta-\gamma}r)(x,\eta) (2c_g)^{|\gamma|}\lambda^{-|\gamma|} (\partial^\gamma\vartheta)(q_\lambda). \tag{PW51}$$ On this support PW21 gives $\langle\eta\rangle_{\rm c}\geq\lambda/(2c_g)$. Hence each full derivative in PW51 is $O(\lambda^{-1-|\beta|})$, with its finite binomial sum retained. E23 then bounds $\operatorname{Op}(r\vartheta_\lambda)$ by $C\lambda^{-1}$. The exact tail decomposition $$\operatorname{Op}(r)v_\lambda =\operatorname{Op}(r\vartheta_\lambda)v_\lambda +\operatorname{Op}(r)t_\lambda \tag{PW52}$$ and PW27, together with PW48 for the smoothing term, prove $$\|\mathcal Rv_\lambda\|_{L^2(F)}=O(\lambda^{-1}) \quad\text{for each fixed }\mathcal R\in\Psi^{-1}_{1,0}(X;E,F). \tag{PW53}$$ Here $\mathcal R$ is distinct from the smooth local remainder $R$ of PW41.

Every point of $X$ lies in one member of a finite cover by smaller coordinate regions $K$ of the kind used above. Taking the maximum of their finitely many constants proves uniformity on $X$. For the contradiction argument one can instead select a subsequence whose centers all lie in one such compact smaller region. The original metric frequency $\lambda$ is retained on each chart through PW5 and PW20.

Apply this to the exact operator in OM9, $$A_s=J_E^{-s}:L^2(E)\longrightarrow H^s(E),\qquad B_{s-m}=J_F^{s-m}:H^{s-m}(F)\longrightarrow L^2(F),\qquad T=B_{s-m}P_sA_s . \tag{PW54}$$ These are the specified bounded isomorphisms, with their original source and target spaces, metrics and half-densities. Retain the ordered original principal factors $$\widetilde p(x,\xi) =\langle\xi\rangle_{g,x}^{\,s-m} p(x,\xi)\langle\xi\rangle_{g,x}^{-s}. \tag{PW55}$$ The full local symbol of $T$ equals this representative plus $r\in S^{-1}_{1,0}$, including all the lower-order composition terms. PW53 estimates that complete error; it is not presumed zero. The scalar factors in PW55 are positive and the displayed ordered product has the exact value $\langle\xi\rangle_{g,x}^{-m}p(x,\xi)$. This equality is an explicit comparison with the original $p$; it does not remove $p$, its order or either operator factor.

If OM2 fails, choose for each integer $j$ a covector $(x_j,\xi_j)$ with $|\xi_j|_{g,x_j}\geq j$ and a vector $\|w_j\|_{E_{x_j}}=1$ such that $$\|\widetilde p(x_j,\xi_j)w_j\|_{F_{x_j}}<1/j . \tag{PW56}$$ The failure of a uniform positive lower bound gives exactly these choices. Passing to one smaller chart preserves PW9, with $\lambda_j=|\xi_j|_{g,x_j}\to\infty$. PW49, PW40 and PW53 give $\|Tv_{\lambda_j}\|_2\to0$, while PW13 gives $\|v_{\lambda_j}\|_2\to1$. If $N=\ker T$ is finite dimensional, its orthogonal projection sends PW50’s weakly null packets to zero in norm: for an actual orthonormal basis $e_1,\ldots,e_d$ of $N$, $$\|\Pi_Nv_{\lambda_j}\|_2^2 =\sum_{\ell=1}^d |\langle v_{\lambda_j},e_\ell\rangle|^2\longrightarrow0. \tag{PW57}$$ No smoothness of $N$ is required. Closed range and finite kernel of $P_s$ pass through PW54 to $T$, so the already-proved compact estimate OM4 gives $$\|v_{\lambda_j}\|_2 \leq C\bigl(\|Tv_{\lambda_j}\|_2+ \|\Pi_Nv_{\lambda_j}\|_2\bigr)\longrightarrow0. \tag{PW58}$$ This contradicts PW13 and proves OM2 for the original symbol $p$ with its original weight $\langle\xi\rangle_{g,x}^{m}$. The proof uses the complete $S^0_{1,0}$ high-frequency symbol and does not require a homogeneous cosphere representative.