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\begin{document}

\section{Symbols, operators and Sobolev
scales}\label{symbols-operators-and-sobolev-scales}

Frequency differentiation improves a symbol, while differentiation in
the base variable may worsen it. The balance between those two effects
controls an expansion. Boundedness requires a different argument: a
decomposition into frequency bands continues to work when the two
effects exactly balance. We develop these mechanisms separately, then
combine them to construct inverses and act on Sobolev and Besov spaces.

We use the Fourier conventions proved in
\href{prerequisite-bridges.pdf}{Fourier transforms, finite spectra and
convex separation} and applied in
\href{gauss-transform-estimates.pdf}{Quadratic Fourier multipliers at a
moving scale}: the forward kernel is \(e^{-ix\cdot\xi}\), inverse
transformation has factor \((2\pi)^{-n}\), and \(D=-i\partial\). Fourier
inversion and Plancherel, including their extension to \(L^2\), are
explicit prerequisites. The additional entry contracts are completeness
of \(L^2\), density of Schwartz functions in \(L^2\), the usual
integration and dominated-convergence theorems for Lebesgue measure,
Taylor's formula and the multivariable product rule. Distributional
Fourier transformation, invertible linear pullback on tempered
distributions, and the tempered Schwartz kernel theorem are the same
explicit contracts as in Section 4 of \href{weyl-metric-products.pdf}{Two
measuring scales, one Weyl product}. In particular the kernel theorem
means that every continuous linear map
\(\mathcal S(\mathbb R^n)\to\mathcal S'(\mathbb R^n)\) has exactly one
kernel in \(\mathcal S'(\mathbb R^{2n})\), and conversely. Its full
proof is
\href{weyl-metric-products.pdf\#section*.27}{Sections4.1--4.5 of
the Weyl-product lesson}, including the exact global kernel and
strong-dual maps.
\href{prerequisite-bridges.pdf\#section*.31}{Sections8.1--8.4 of
the Fourier lesson} prove partial transforms and the full original
distributional and L2 extensions.
\href{banach-foundation-bridges.pdf\#section*.94}{The Banach and
measure lesson} supplies completeness and density;
\href{metric-foundation-bridges.pdf\#section*.106}{the metric and
calculus lesson} supplies the complete Taylor and product maps.

The multiplier results in Sections 7--8 of
\href{gauss-transform-estimates.pdf}{Quadratic Fourier multipliers at a
moving scale} and the classical metric verification in Section 8 of
\href{weyl-metric-products.pdf}{Two measuring scales, one Weyl product}
are available here. We use those results with specified metrics below.
All statements are for a fixed finite dimension. Scalar symbols may be
replaced by matrices between fixed finite-dimensional complex Hermitian
spaces: norms replace absolute values, products keep their displayed
order, and adjoints are conjugate transposes. The inverse theorem uses
square matrices and a lower bound for their smallest singular value. No
infinite-dimensional operator-valued extension is implicit.

\subsection{1. Symbol classes and the order
budget}\label{symbol-classes-and-the-order-budget}

Fix \[
0<\rho\leq1,\qquad 0\leq\delta<1,\qquad m\in\mathbb R,
\qquad \langle\xi\rangle=(1+|\xi|^2)^{1/2}.
\tag{E1}
\] The space \(S^m_{\rho,\delta}\) consists of smooth functions on
\(\mathbb R_x^n\times\mathbb R_\xi^n\) for which every seminorm \[
p_{m,L}(a)=\max_{|\alpha|+|\beta|\leq L}
\sup_{x,\xi}\langle\xi\rangle^{-m+\rho|\alpha|-\delta|\beta|}
\|\partial_\xi^\alpha\partial_x^\beta a(x,\xi)\|
\tag{E2}
\] is finite. Bounds hold for all base points with the same constants.
Estimates only on compact sets in \(x\) define a different, local
topology. We write \(S^m=S^m_{1,0}\),
\(S^{-\infty}=\bigcap_m S^m_{\rho,\delta}\), and
\(S^\infty_{\rho,\delta}=\bigcup_mS^m_{\rho,\delta}\). The intersection
is independent of the parameters in (E1): it consists precisely of
functions all of whose derivatives decrease faster than every power of
\(\langle\xi\rangle\), uniformly in \(x\).

These are Fréchet spaces. Indeed, a Cauchy sequence in all seminorms has
uniformly Cauchy derivatives on every compact set. Successive
integration along coordinate segments identifies the limits as
derivatives of a smooth function. Passing to the limit in each weighted
uniform bound proves convergence in that seminorm and completeness. The
seminorm family is countable and separates points.

Differentiation is continuous with its exact order change, \[
\partial_\xi^\alpha\partial_x^\beta:
S^m_{\rho,\delta}\longrightarrow
S^{m-\rho|\alpha|+\delta|\beta|}_{\rho,\delta}.
\tag{E3}
\] Multiplication is continuous bilinear from orders \(m,m'\) to order
\(m+m'\). To see the finite-seminorm assertion, expand a derivative of
\(ab\): each term is \(\binom\alpha\gamma\binom\beta\nu
(\partial_\xi^\gamma\partial_x^\nu a)
(\partial_\xi^{\alpha-\gamma}\partial_x^{\beta-\nu}b)\). The powers
contributed by the two factors add to
\(m+m'-\rho|\alpha|+\delta|\beta|\). Summing the finitely many
coefficients gives \(p_{m+m',L}(ab)\leq C_Lp_{m,L}(a)p_{m',L}(b)\). This
calculation also explains why exchanging two matrix factors is not
allowed.

We will need one elementary stability property. If
\(a_1,\ldots,a_k\in S^0_{\rho,\delta}\) and \(F\) is a smooth
real-variable function on \(\mathbb C^k\), then
\(F(a_1,\ldots,a_k)\in S^0_{\rho,\delta}\). Their ranges have compact
closure in a bounded subset of \(\mathbb C^k\), so every derivative of
\(F\) needed at a given stage is bounded there. An iterated chain-rule
term is a derivative of \(F\) times a product of positive-order
derivatives of the \(a_j\); their total frequency and base derivative
counts are \(|\alpha|\) and \(|\beta|\). Equation (E2) follows. The same
proof works when \(F\) is defined on an open neighborhood of that
compact closure. In particular inversion of a matrix-valued order-zero
symbol whose inverse is uniformly bounded preserves order zero:
differentiate \(a^{-1}a=I\), obtaining
\(\partial a^{-1}=-a^{-1}(\partial a)a^{-1}\), and iterate with the
original factor order.

\subsection{2. Summing symbol expansions and preserving
support}\label{summing-symbol-expansions-and-preserving-support}

\textbf{Frequency rescaling.} If \(a\in S^0\), then
\(a_\varepsilon(x,\xi)=a(x,\varepsilon\xi)\), \(0\leq\varepsilon\leq1\),
is a bounded family in \(S^0\), and \[
a_\varepsilon\longrightarrow a(x,0)\quad\hbox{in }S^t
\quad(t>0).
\tag{E4}
\] For \(0<t\leq1\), each fixed \(S^t\) seminorm of the difference is at
most \(C\varepsilon^t\), with a finite number of seminorms of \(a\) in
\(C\). For derivatives with \(\alpha=0\), boundedness and the mean-value
formula give a bound
\(C\min(1,\varepsilon|\xi|)\leq C(\varepsilon|\xi|)^t\). For
\(q=|\alpha|\geq1\), derivatives of the limit vanish and \[
\varepsilon^q\langle\xi\rangle^{q-t}
\langle\varepsilon\xi\rangle^{-q}\leq\varepsilon^t,
\tag{E5}
\] because
\(\varepsilon\langle\xi\rangle\leq\langle\varepsilon\xi\rangle\) and
\(q\geq t\). The estimates also prove boundedness at order zero. Values
\(t>1\) follow by the inclusion \(S^1\subset S^t\). Convergence in
\(S^0\) need not hold: \(a(\xi)=\xi_1/\langle\xi\rangle\) has
\(a(x,0)=0\) and \(\sup_\xi|a(\varepsilon\xi)|=1\) for every positive
\(\varepsilon\).

\textbf{Summation theorem.} Let \(a_j\in S^{m_j}_{\rho,\delta}\),
\(j\geq0\), with \(m_j\to-\infty\), without assuming monotonicity. Put
\(M_k=\max_{j\geq k}m_j\). There exists \(a\in S^{M_0}_{\rho,\delta}\)
such that \[
\operatorname{supp}a\subset\bigcup_j\operatorname{supp}a_j,
\qquad a-\sum_{j<k}a_j\in S^{M_k}_{\rho,\delta}\quad(k\geq0).
\tag{E6}
\] Any two such sums differ by \(S^{-\infty}\). Permuting the terms does
not change their sum modulo that space.

\textbf{Proof.} Choose a smooth function \(\chi\) equal to one near zero
and with compact support. For each \(j\) choose \(R_j\to\infty\),
strictly increasing if necessary, so that \[
\|(1-\chi(\xi/R_j))a_j\|_{m_j+1,L}\leq2^{-j}
\quad(0\leq L\leq j).
\tag{E7}
\] Here \(\|\cdot\|_{m,L}\) means (E2). This choice is possible: on the
support of the high-frequency cutoff, \(\langle\xi\rangle^{-1}\to0\)
uniformly as \(R_j\to\infty\). A frequency derivative of order
\(\gamma\) hitting the cutoff gives \(R_j^{-|\gamma|}\), supported where
\(|\xi|\asymp R_j\); since \(\rho\leq1\), it is bounded by a constant
times \(\langle\xi\rangle^{-\rho|\gamma|}\). The remaining factors have
their stipulated symbol bounds. For a fixed finite set of derivatives
these estimates prove (E7).

Let \(A_j=(1-\chi(\xi/R_j))a_j\) and \(a=\sum_jA_j\). The sum is locally
finite in frequency, hence smooth. For fixed \(k,L\), choose
\(J\geq\max(k,L)\) so large that \(m_j+1\leq M_k\) for all \(j\geq J\).
The tail \(\sum_{j\geq J}A_j\) converges in the order-\(M_k\),
derivative-\(L\) seminorm by (E7). The remaining terms in
\(a-\sum_{j<k}a_j\) are finitely many symbols of order at most \(M_k\),
together with finitely many \(-\chi(\xi/R_j)a_j\). The latter are in
\(S^{-\infty}\), since their frequency support is bounded and all base
derivatives are uniformly bounded there. This proves every
differentiated estimate in (E6).

Support is also preserved at points outside the union. Near such a point
only finitely many \(A_j\) can be nonzero, and each of those vanishes in
some neighborhood of the point; intersect those neighborhoods.
Uniqueness follows by taking \(k\to\infty\). For a permutation and a
specified finite initial segment of it, let \(M\) be the largest order
among the omitted terms. Choose an original initial segment containing
the specified terms and all terms of order greater than \(M\).
Subtracting its expansion expresses the new remainder as finitely many
symbols of order at most \(M\), plus a remainder of that order. This
proves the permuted version of (E6). \(\square\)

\textbf{Recognition from values.} Suppose \(a\) is smooth, every one of
its derivatives has some polynomial bound in \(\langle\xi\rangle\),
uniformly in \(x\), and \[
\left\|a-\sum_{j<k}a_j\right\|\leq C_k\langle\xi\rangle^{\mu_k},
\qquad \mu_k\to-\infty.
\tag{E8}
\] Then \(a\) has the full symbol expansion (E6). In particular no
differentiated remainder assumptions are needed in (E8).

To prove this, subtract a sum \(a_0\) constructed above. The difference
\(f=a-a_0\) decreases faster than every frequency power: for a
prescribed power choose \(k\) for which both \(\mu_k\) and \(M_k\) are
sufficiently negative. Every derivative of \(f\) still has a polynomial
bound. If \(\partial_j^2 f\) has growth exponent \(q\), the
one-dimensional Taylor formula in either a base or a frequency
coordinate gives, for \(0<h\leq1\), \[
|\partial_j f(x,\xi)|
\leq h^{-1}|f((x,\xi)+he_j)-f(x,\xi)|
+C h\langle\xi\rangle^q.
\tag{E9}
\] The weight at the shifted point is comparable to the original weight.
Given an arbitrary desired decay power \(N\), take
\(h=\langle\xi\rangle^{-L}\) with \(L>q+N\), then use the already known
decay of \(f\) of degree greater than \(L+N\). Both terms in (E9) are
\(O(\langle\xi\rangle^{-N})\). Thus every first derivative decreases
rapidly. Reapply the same argument to each derivative already treated;
the necessary second derivative has a polynomial bound by hypothesis.
Induction proves rapid decrease of every derivative, so
\(f\in S^{-\infty}\). This proof and the summation construction work
throughout (E1), including \(\delta>\rho\).

\textbf{Complex homogeneous expansions.} For \(z\in\mathbb C\) and
\(h=1/q\), with \(q\) a positive integer, a polyhomogeneous symbol of
degree \(z\) and step \(h\) is a symbol \(a\in S^{\operatorname{Re}z}\)
satisfying \(a\sim\sum_{j\geq0}a_j\) in the sense of (E6), where \(a_j\)
is homogeneous of degree \(z-jh\) outside the unit frequency ball. The
convention is \(t^z=e^{z\log t}\) for \(t>0\). Smooth angular
coefficients whose base and angular derivatives are uniformly bounded
have the required symbol estimates: polar differentiation lowers the
radial degree by one for each frequency derivative. Multiplying each
homogeneous term by a fixed cutoff near zero makes it globally smooth;
such choices differ in \(S^{-\infty}\). The summation theorem then
constructs the symbol. Without the uniform angular/base bounds, a
homogeneous formula alone would give only a local-in-base symbol.
Integer frequency differentiations change the degree by an integer,
which is an integral multiple of the stipulated step \(h\).

\subsection{3. Quantization, kernels and
signs}\label{quantization-kernels-and-signs}

For every symbol in (E1), left quantization is \[
\operatorname{Op}(a)u(x)=(2\pi)^{-n}
\int e^{ix\cdot\xi}a(x,\xi)\widehat u(\xi)\,d\xi,
\qquad u\in\mathcal S.
\tag{E10}
\] This integral and all its differentiated versions converge
absolutely. The map \((a,u)\mapsto\operatorname{Op}(a)u\) is continuous
bilinear from \(S^m_{\rho,\delta}\times\mathcal S\) into \(\mathcal S\),
throughout (E1). Indeed, output derivatives create finitely many powers
of \(\xi\) and base derivatives of \(a\). To estimate
\(x^\gamma\partial_x^\beta\operatorname{Op}(a)u\), move \(|\gamma|\)
frequency derivatives off the exponential by integration by parts. Each
resulting integrand is a derivative of \(a\), a polynomial, and a
derivative of \(\widehat u\). Its absolute integral is bounded by one
sufficiently large Schwartz seminorm of \(u\) times finitely many
seminorms (E2), after choosing the remaining weight less than \(-n\).
Boundary terms vanish by the same rapid decrease. This gives every
Schwartz seminorm and the asserted joint continuity.

Differentiation and one integration by parts give \[
[\operatorname{Op}(a),D_j]=i\operatorname{Op}(\partial_{x_j}a),
\qquad
[\operatorname{Op}(a),x_j]=-i\operatorname{Op}(\partial_{\xi_j}a).
\tag{E11}
\] If \(a(x,\xi)=\sum_\alpha a_\alpha(x)\xi^\alpha\), then
\(\operatorname{Op}(a)=\sum_\alpha a_\alpha(x)D^\alpha\): coefficients
stand on the left. Such a polynomial belongs to a global symbol class
only when its coefficients have the relevant uniform derivative bounds.

There is also an exact definition for every
\(a\in\mathcal S'(\mathbb R^{2n})\). Take the partial inverse Fourier
transform in \(\xi\), call its second variable \(z\), and use the
invertible linear substitution \(z=x-y\). The result is a tempered
kernel \(K_a\). Conversely, \[
K_a(x,y)=(2\pi)^{-n}\int e^{i(x-y)\cdot\xi}a(x,\xi)\,d\xi,
\quad
a(x,\xi)=\int e^{-iz\cdot\xi}K_a(x,x-z)\,dz
\tag{E12}
\] are inverse operations on tempered distributions. The displayed
integrals mean Fourier transforms there, not potentially divergent
ordinary integrals. Each step is a continuous linear isomorphism, so the
symbol--kernel correspondence is bijective and continuous in both
directions. Combined with the explicit tempered kernel theorem contract,
it identifies arbitrary continuous \(\mathcal S\to\mathcal S'\) maps
with tempered symbol-distributions. For smooth symbols (E12) agrees with
(E10), by testing against two Schwartz functions and Fourier inversion.

Use the sesquilinear pairing \((u,v)=\int u\overline v\), linear in
\(u\). Transposing and conjugating the kernel sends \(K_a(x,y)\) to
\(K_a(y,x)^*\). Its left symbol is \[
a^\dagger=Q_1(a^*),\qquad
Q_t=\exp\bigl(it\langle D_x,D_\xi\rangle\bigr).
\tag{E13}
\] Here \(Q_t\) is the Fourier multiplier \(e^{itp\cdot q}\) in the
variables dual to \((x,\xi)\). It preserves \(\mathcal S\) and
\(\mathcal S'\) because all derivatives of that multiplier grow at most
polynomially. To check the sign directly, apply (E12) first to
\(a(x,\xi)=e^{i(p\cdot x+q\cdot\xi)}\). Its operator is multiplication
by \(e^{ip\cdot x}\) followed by translation \(u(x)\mapsto u(x+q)\). Its
adjoint is therefore \(v(x)\mapsto e^{-ip\cdot(x-q)}v(x-q)\), whose
symbol is \(e^{ip\cdot q}e^{-i(p\cdot x+q\cdot\xi)}\). This is exactly
(E13). Integrating this identity against Schwartz Fourier transforms
proves it for Schwartz symbols, and continuity of the kernel and Fourier
operations proves it for all tempered symbols. Thus
\((\operatorname{Op}(a)u,v)=(u,\operatorname{Op}(a^\dagger)v)\) whenever
\(u,v\in\mathcal S\), with distributional pairing when needed.

\subsection{4. Gaussian estimates for the
product}\label{gaussian-estimates-for-the-product}

Assume now \(\delta\leq\rho\). Set \(\kappa=\rho-\delta\). Section 8 of
\href{weyl-metric-products.pdf}{Two measuring scales, one Weyl product}
applies to \[
g_{(x,\xi)}(y,\eta)=\langle\xi\rangle^{2\delta}|y|^2
+\langle\xi\rangle^{-2\rho}|\eta|^2,
\qquad m_a=\langle\xi\rangle^m.
\tag{E14}
\] Its slow variation and temperateness hold uniformly, and its
symplectic parameter is \(\langle\xi\rangle^{-\kappa}\). Consequently,
for every fixed real \(t\), \[
Q_t:S^m_{\rho,\delta}\longrightarrow S^m_{\rho,\delta},
\quad
Q_ta-\sum_{|\alpha|<N}\frac{t^{|\alpha|}}{\alpha!}
\partial_\xi^\alpha D_x^\alpha a
\in S^{m-N\kappa}_{\rho,\delta}.
\tag{E15}
\] For each output seminorm only finitely many input seminorms are used.
These maps preserve convergence in the local smooth topology on bounded
source sets. The phase convention matters: for \(A(p,q)=t p\cdot q\),
the symmetric representative is \(t/2\) times the swap map, so the Gauss
parameter is \(|t|\langle\xi\rangle^{-\kappa}/2\). The finite-bound
theorem applies to the original metric itself; the complete receiving
calculation below retains every parameter-dependent factor. This proves
(E15) directly from Section 8 of
\href{gauss-transform-estimates.pdf}{Quadratic Fourier multipliers at a
moving scale} with precisely the hypotheses checked in Section 8 of
\href{weyl-metric-products.pdf}{Two measuring scales, one Weyl product}.

\textbf{The unchanged metric and its full remainder constants.}

Retain E14, E1's parameter range and the original weight
\(m_a(X)=\langle\xi\rangle^m\). Assume \(\delta\leq\rho\) and set
\(\kappa=\rho-\delta\geq0\), \(X=(x,\xi)\). The original metric is

\[
 g_X(y,\eta)=\langle\xi\rangle^{2\delta}|y|^2+
            \langle\xi\rangle^{-2\rho}|\eta|^2,\qquad
 m_a(X)=\langle\xi\rangle^m.
 \tag{EC5}
\]

For \(n\geq1\) and \(t\ne0\), the exact phase \(A_t(p,q)=tp\cdot q\) has
the symmetric map

\[
 B_t=\frac t2\begin{pmatrix}0&I_n\\I_n&0\end{pmatrix},\qquad
 g_X(B_t(p,q))=\frac{t^2}{4}
 \bigl(\langle\xi\rangle^{2\delta}|q|^2+
       \langle\xi\rangle^{-2\rho}|p|^2\bigr).
 \tag{EC6}
\]

Taking the dual of this full positive quadratic form gives, on every
original direction,

\[
 g_X^{A_t}(y,\eta)=\frac4{t^2}
 \bigl(\langle\xi\rangle^{2\rho}|y|^2+
       \langle\xi\rangle^{-2\delta}|\eta|^2\bigr)
 =\frac4{t^2}g_X^\sigma(y,\eta),\qquad
 h_{g,A_t}(X)=\frac{|t|}{2}\langle\xi\rangle^{-\kappa}
 \leq h_*=\frac{|t|}{2}.
 \tag{EC7}
\]

Indeed the ratio of each original metric coefficient to its phase-dual
coefficient is \(t^2\langle\xi\rangle^{-2\kappa}/4\). Its positive
square root is the exact supremum defining \(h\). No direction is lost.
The phase-dual form is finite on the entire original space, since
\(B_t\) is invertible.

For the original temperateness and weight inequalities, retain the exact
comparison

\[
 1+g_Y^\sigma(X-Y)
 =1+\frac{t^2}{4}g_Y^{A_t}(X-Y)
 \leq\max(1,t^2/4)\bigl(1+g_Y^{A_t}(X-Y)\bigr).
 \tag{EC8}
\]

Thus every occurrence of an original metric constant \(C_g\) with
distance exponent \(L_g\) becomes \(C_g\max(1,t^2/4)^{L_g}\), and each
original weight constant \(C_m\) with its own exponent \(L_m\) becomes
\(C_m\max(1,t^2/4)^{L_m}\). Local continuity and slow variation remain
those of the unchanged \(g\) and unchanged weight. The
\href{gauss-transform-estimates.pdf\#section*.46}{original Gauss
finite-bound theorem G24--G26} applies with \(h_*=|t|/2\). Its packing
step is in dimension \(2n\) and retains the full factor
\((1+|t|/2)^{2n}\), in addition to the original structural constants and
their displayed changes. That factor alone does not describe all
dependence on \(t\) of the theorem's constant.

For \(N,l\geq0\) and \(T_j=(y_j,\eta_j)\), its exact receiving estimate
is

\[
 \begin{split}
 |\partial_{T_1}\cdots\partial_{T_l}R_{N,t}a(X)|
 &\leq C_{N,l,t}\langle\xi\rangle^m
       \left(\frac{|t|}{2}\langle\xi\rangle^{-\kappa}\right)^N\\
 &\quad{}\times
       \prod_{j=1}^{l}
       \left(\langle\xi\rangle^{2\delta}|y_j|^2+
             \langle\xi\rangle^{-2\rho}|\eta_j|^2\right)^{1/2}
       p_{\leq J_{N,l,t}}(a;m_a,g),\\
 R_{N,t}a&=Q_ta-\sum_{j<N}\frac{(iA_t(D))^j}{j!}a .
 \end{split}
 \tag{EC9}
\]

The constants and finite derivative budget carry the actual unchanged
local metric constants, the full EC8 temperateness/weight constants and
the finite \(h_*\) counting constants. For coordinate derivatives
\(\alpha\) in \(\xi\) and \(\beta\) in \(x\), choose the corresponding
coordinate directions in EC9. Their product is
\(\langle\xi\rangle^{-\rho|\alpha|+\delta|\beta|}\). Hence

\[
 |\partial_\xi^\alpha\partial_x^\beta R_{N,t}a(X)|
 \leq C_{N,\alpha,\beta,t}
       (|t|/2)^N
       \langle\xi\rangle^{m-N\kappa-\rho|\alpha|+\delta|\beta|}
       p_{\leq J_{N,\alpha,\beta,t}}(a;m_a,g).
 \tag{EC10}
\]

The source metric seminorms are exactly controlled by finitely many E2
coordinate seminorms: expand each directional derivative into its \(2n\)
coordinate components and use
\(|y_k|\leq\langle\xi\rangle^{-\delta}g_X(T)^{1/2}\) and
\(|\eta_k|\leq\langle\xi\rangle^\rho g_X(T)^{1/2}\). Each ordered
product expansion has finitely many terms, at most \((2n)^l\) at order
\(l\), and all factors remain. Conversely insert the coordinate
directions to bound each coordinate seminorm by its metric seminorm.
Thus EC10 proves the original E15 class inclusion with finite E2
source-seminorm control, retaining the full \((|t|/2)^N\) rather than
replacing its value by one.

The Taylor coefficients are the original ones:

\[
 \begin{split}
 \sum_{j<N}\frac{(iA_t(D))^j}{j!}a
 &=\sum_{j<N}\sum_{|\alpha|=j}
       \frac{i^j t^j}{\alpha!}D_x^\alpha D_\xi^\alpha a\\
 &=\sum_{j<N}\sum_{|\alpha|=j}
       \frac{i^j(-i)^jt^j}{\alpha!}
       \partial_\xi^\alpha D_x^\alpha a
 =\sum_{|\alpha|<N}\frac{t^{|\alpha|}}{\alpha!}
       \partial_\xi^\alpha D_x^\alpha a .
 \end{split}
 \tag{EC11}
\]

For \(N=0\) the sum is empty and \(R_{0,t}=Q_t\). The finite-bound
theorem also gives bounded-set local smooth convergence, continuity and
uniqueness on the original symbol class. When \(t=0\), \(Q_0=I\): for
\(N\geq1\) the term of order zero is \(a\) and all positive-order terms
vanish, so the remainder is exactly zero; for \(N=0\) the remainder is
\(a\). These cases use the identity directly, without assigning a
phase-dual metric or evaluating a zero denominator. If \(n=0\) all
symbols are functions on a one-point space and the phase is zero for
every \(t\); the same identity argument applies. For \(\kappa=0\) the
estimate has no improving frequency order. None of these arguments
rescale or replace the original \(g\).

For composition we need a parameter version that retains more
information before restricting to a diagonal. Let \[
c(x,\xi,y,\eta)=a(x,\eta)b(y,\xi),\qquad
\mathcal B=\exp(i\langle D_y,D_\eta\rangle)c.
\tag{E16}
\] At fixed \((x,\xi)\), let \(s=\langle\xi\rangle\) and equip the
\((y,\eta)\) variables with \[
G_{(y,\eta)}(v,w)=s^{2\delta}|v|^2+
\langle\eta\rangle^{-2\rho}|w|^2.
\tag{E17}
\] The evaluation points needed for composition have \(\eta=\xi\). At
these points the Gauss parameter for the phase \(p_y\cdot p_\eta\) is
\(s^{-\kappa}/2\). We verify the uniform global hypotheses for that
evaluation instead of assuming uncertainty at every \(\eta\).

Write \(r=\langle\eta\rangle\), \(d=|\eta-\xi|\). The phase-dual
distance based at \((y,\eta)\) includes \(4s^{-2\delta}d^2\). Both
ratios \(r/s\) and \(s/r\) are bounded by a fixed power of
\(1+s^{-2\delta}d^2\), with constants depending only on \(\delta<1\). If
\(r\) and \(s\) are within a factor two this is immediate. If \(r>2s\),
then \(d>r/2\) and \(s^{-2\delta}d^2\geq (r/s)^2/4\), since
\(s^{2-2\delta}\geq1\). If \(s>2r\), then \(d>s/2\), so the same
expression is at least \(s^{2(1-\delta)}/4\); choosing a sufficiently
large power bounds \(s/r\leq s\). These inequalities control the ratio
of the two frequency coefficients of (E17), every real power weight
\(r^q\), and their reciprocals. The base coefficient is constant. Slow
variation follows from \(\rho\leq1\) and the Lipschitz inequality for
\(\langle\eta\rangle\); the same comparison proves local continuity of
\(r^q\). Thus every condition of the observation-point form of Sections
7--8 of \href{gauss-transform-estimates.pdf}{Quadratic Fourier
multipliers at a moving scale} holds, with constants independent of
\(x,\xi,y\).

The tensor \(c\) has weight \(r^{m_1}s^{m_2}\) in (E17): each
\(y\)-derivative costs at most \(s^\delta\), and each
\(\eta\)-derivative costs at most \(r^{-\rho}\). Applying the Gauss
theorem and then restricting \(y=x,\eta=\xi\) gives a remainder of order
\(m_1+m_2-N\kappa\). To obtain every derivative estimate, differentiate
(E16) first. Derivatives commute with its constant-coefficient
multiplier. For example \(\partial_\xi^\alpha\partial_x^\beta
\partial_\eta^{\alpha'}\partial_y^{\beta'}c\) has weight
\(r^{m_1-\rho|\alpha'|+\delta|\beta|}
s^{m_2-\rho|\alpha|+\delta|\beta'|}\). Apply the same theorem with those
modified real exponents. Finally, differentiation of the diagonal
restriction is the sum of the corresponding derivatives in the two sets
of variables. Each term has the required total order in (E2). The
product rule bounds every input seminorm by a finite sum of products of
seminorms of \(a\) and \(b\).

For the classical case \((\rho,\delta)=(1,0)\), this argument holds at
every \((y,\eta)\), not just \(\eta=\xi\), because (E17) no longer
depends on \(s\). It gives the stronger, pre-diagonal estimate \[
\begin{split}
&\left\|\partial_\xi^\alpha\partial_x^\beta
\partial_\eta^{\alpha'}\partial_y^{\beta'}
\left(\mathcal B-
\sum_{|\gamma|<N}\frac{\partial_\eta^\gamma a(x,\eta)
D_y^\gamma b(y,\xi)}{\gamma!}\right)\right\|\\
&\hspace{12mm}\leq C
\langle\eta\rangle^{m_1-N-|\alpha'|}
\langle\xi\rangle^{m_2-|\alpha|}.
\end{split}
\tag{E18}
\] In particular the loss is attached to the transformed frequency
variable before evaluation. The constants in (E18) also involve only
finitely many source seminorms.

All these Gauss limits agree with the distributional multipliers
wherever both are defined. Multiply the symbols by cutoffs in both
variables that tend to one. They stay in bounded symbol sets and
converge locally smoothly; their polynomial growth gives convergence in
\(\mathcal S'\). The distributional multiplier is continuous there,
while the Gauss estimate gives the same locally smooth limit on its
evaluation set. This also justifies differentiating with respect to the
fixed parameters in (E16): differentiated approximants have the
estimates just proved, converge locally, and their limits are successive
derivatives by the fundamental theorem of calculus along parameter
segments.

\subsection{5. Adjoints and composition}\label{adjoints-and-composition}

\textbf{Theorem.} For \(\delta\leq\rho\), the adjoint symbol (E13)
belongs to \(S^m_{\rho,\delta}\), and left quantization extends
continuously to \(\mathcal S'\) by transposition of its adjoint's
Schwartz action. If \(a\in S^{m_1}_{\rho,\delta}\) and
\(b\in S^{m_2}_{\rho,\delta}\), then \[
\operatorname{Op}(a)\operatorname{Op}(b)=\operatorname{Op}(a\circ b),
\quad a\circ b=\mathcal B(x,\xi,x,\xi)
\in S^{m_1+m_2}_{\rho,\delta}.
\tag{E19}
\] The identity holds on both \(\mathcal S\) and \(\mathcal S'\). The
product is continuous bilinear. For every \(N\geq0\), \[
a^\dagger-\sum_{|\alpha|<N}
\frac{\partial_\xi^\alpha D_x^\alpha(a^*)}{\alpha!}
\in S^{m-N\kappa}_{\rho,\delta},
\tag{E20}
\] \[
a\circ b-\sum_{|\alpha|<N}
\frac{(\partial_\xi^\alpha a)(D_x^\alpha b)}{\alpha!}
\in S^{m_1+m_2-N\kappa}_{\rho,\delta}.
\tag{E21}
\] Each remainder map has finite-seminorm control as described above.
Adjoint transformation sends bounded sequences converging locally
smoothly to sequences with the same property; the product has the
analogous bounded-set continuity in both factors.

\textbf{Proof.} The symbol assertions and their estimates follow from
(E15)--(E18), including \(N=0\). Formula (E13) and the Schwartz action
prove extension to tempered distributions. For Schwartz symbols and
\(u\in\mathcal S\), Fubini's theorem applied to two instances of (E10)
gives \[
(a\circ b)(x,\xi)=(2\pi)^{-n}
\iint e^{i(x-y)\cdot(\eta-\xi)}a(x,\eta)b(y,\xi)\,dy\,d\eta,
\tag{E22}
\] which equals (E16) restricted to the diagonal. The sign can also be
checked by taking \(a=a(\xi)\), \(b=e^{iv\cdot x}b_0(\xi)\): (E22)
becomes \(e^{iv\cdot x}a(\xi+v)b_0(\xi)\), agreeing with (E21).

Choose compactly supported smooth approximants \(a_j,b_j\) as at the end
of Section 4. Their symbols and products remain bounded in their
respective orders and converge locally smoothly. For any fixed
\(u\in\mathcal S\), (E10) and dominated convergence give local smooth
convergence of the outputs. The Schwartz estimates of Section 3 bound
all output Schwartz seminorms uniformly. Local convergence plus those
stronger uniform seminorms gives convergence in every Schwartz seminorm:
outside a large base ball use one extra power of \(\langle x\rangle\),
and inside use local convergence. Thus
\(\operatorname{Op}(a_j)u\to\operatorname{Op}(a)u\) in \(\mathcal S\),
and similarly for \(b_j\) and their products. Boundedness of the
Schwartz operator seminorms allows passage through the double
composition. This proves (E19) on \(\mathcal S\).

Taking adjoints there identifies the transpose of the product with the
reverse product of the transposes. Apply their continuous extensions to
\(\mathcal S'\) and test against \(\mathcal S\) to obtain (E19) on
\(\mathcal S'\). Equivalently, the identities of pairings determine the
extension uniquely. Kernel injectivity proves associativity of \(\circ\)
and that the constant symbol \(I\) is its unit. \(\square\)

When \(\delta<\rho\), (E20)--(E21) are asymptotic expansions in
decreasing orders. When \(\delta=\rho<1\), closure and every finite
remainder estimate survive, but all their target orders are the original
order. Taking more terms then supplies no smoothing assertion. This
distinction will be used in the inverse construction.

\subsection{\texorpdfstring{6. A finite-derivative \(L^2\)
estimate}{6. A finite-derivative L\^{}2 estimate}}\label{a-finite-derivative-l2-estimate}

We first prove an estimate that involves no symbol order. Suppose
\(a(x,\xi)\) is smooth and all derivatives up to a sufficiently large
fixed order are bounded. Then \[
\|\operatorname{Op}(a)u\|_2\leq C_n M\|u\|_2,
\qquad
M=\max_{|\alpha|+|\beta|\leq L_n}
\|\partial_\xi^\alpha\partial_x^\beta a\|_\infty.
\tag{E23}
\] The number \(L_n\) is finite; the proof permits \(L_n=4N\) for any
integer \(N>n/2\). Only this estimate, not an optimal derivative count,
is needed.

We record the integral estimate used in its proof. If a measurable
kernel \(K(s,t)\) on any two copies of a measure space satisfies
\(\sup_t\int|K(s,t)|\,ds\leq A\) and \(\sup_s\int|K(s,t)|\,dt\leq B\),
then its canonical pairing operator has \(L^2\) norm at most
\(\sqrt{AB}\). The raw integral agrees globally under semifinite output
measure, as proved below. Pointwise Cauchy--Schwarz gives \[
\left|\int K(s,t)u(t)\,dt\right|^2
\leq\left(\int|K(s,t)|\,dt\right)
\left(\int|K(s,t)||u(t)|^2\,dt\right).
\tag{E24}
\] On semifinite output measures, integration in \(s\) on finite pieces
by Tonelli and the global recovery below give \(AB\|u\|_2^2\). The raw
integral therefore supplies the bounded extension there. This includes
continuous kernels with equal marginal bound \(A=B=C\) and norm at most
\(C\).

\textbf{The exact measure-space interpretation.} For arbitrary measure
spaces, the operator in the preceding statement is the canonical pairing
operator, not an unrestricted raw-integral operator. Write \(\mu\) for
the output measure in \(s\) and \(\nu\) for the input measure in \(t\).
Their original constants are column \(A\) and row \(B\); this is
\(A_{\rm SC}=B\), \(B_{\rm SC}=A\) in
\href{totally-characteristic-operators.pdf\#section*.6}{the
complete Schur proof, Section1.1}. Joint product measurability is
required.

For finite-valued representatives \(u\in L^2(\nu)\) and
\(v\in L^2(\mu)\), their actual supports are sigma-finite, since \[
 S_u=\bigcup_{k\geq1}\{|u|>1/k\},\qquad
 \nu\{|u|>1/k\}\leq k^2\|u\|_2^2,
\] and the same bound holds for \(v\). On the original restricted
product \(S_v\times S_u\), Tonelli and Cauchy--Schwarz therefore prove
\[
 \begin{split}
 b_K(u,v)&=\int_{S_v}\int_{S_u}
               K(s,t)u(t)\overline{v(s)}\,d\nu(t)\,d\mu(s),\\
 \int_{S_v\times S_u}|K|\,|u|^2\,d(\mu\otimes\nu)&\leq A\|u\|_2^2,\\
 \int_{S_v\times S_u}|K|\,|v|^2\,d(\mu\otimes\nu)&\leq B\|v\|_2^2,\\
 |b_K(u,v)|&\leq\sqrt{AB}\|u\|_2\|v\|_2.
 \end{split}
 \tag{E24a}
\] Absolute convergence also proves invariance under changing the
original null representatives and sesquilinearity. The complete Hilbert
representation argument SC10--SC13 gives the unique operator \(T_K\)
with \(\langle T_Ku,v\rangle=b_K(u,v)\) and \(\|T_K\|\leq\sqrt{AB}\),
including both zero-constant cases.

SC14--SC23 prove the raw integral's measurability and existence on every
finite output piece, the exact finite-piece comparison with \(T_Ku\),
and the onto isometry \(L^2(\mu)\to F_\mu/N_\mu\). The raw
representative has class \(i_\mu(T_Ku)\). Under semifinite output
measure SC24 proves that the divergent set is globally null and that
this raw representative equals \(T_Ku\) in the original global
\(L^2(\mu)\). The explicit SC2--SC6 counterexample shows why truncation
alone cannot establish this global assertion on arbitrary output
measures.

In every application below the measure is Lebesgue measure, or its
original multiple \(d\mu(q,p)=(2\pi)^{-n}dq\,dp\), hence is sigma-finite
and semifinite. Indeed the boxes \([-k,k]^{2n}\) have their exact finite
phase-space measures \((2\pi)^{-n}(2k)^{2n}\) and exhaust the space. A
measurable set with positive measure then has a positive finite-measure
intersection with one of these boxes. The same proof applies to the
ordinary position and frequency measures. Thus the raw integral and all
bounds below have the proved meaning. The finite matrix case keeps the
original operator norm and both vector dimensions, exactly as SC26
proves.

\textbf{Proof of (E23).} Fix \(\phi\in\mathcal S(\mathbb R^n)\) with
\(\|\phi\|_2=1\), and put \(\phi_{q,p}(x)=e^{ip\cdot x}\phi(x-q)\). Use
phase-space measure \(d\mu(q,p)=(2\pi)^{-n}dq\,dp\). The packet
transform \(Vu(q,p)=(u,\phi_{q,p})\) is an isometry from \(L^2\) into
\(L^2(d\mu)\): Plancherel in \(p\), followed by integration in \(q\),
gives \[
\int|Vu(q,p)|^2d\mu(q,p)
=\iint |u(x)|^2|\phi(x-q)|^2\,dx\,dq=\|u\|_2^2.
\tag{E25}
\] The synthesis map \(V^*f=\int f(Q)\phi_Q\,d\mu(Q)\), initially for
bounded compactly supported \(f\), has norm at most one: pairing with an
\(L^2\) function, applying Cauchy--Schwarz and (E25), and taking the
supremum over unit vectors proves this bound. Completeness extends
synthesis to every \(f\in L^2(d\mu)\). Polarization of (E25) gives
\(V^*V=I\). For Schwartz \(u\), \(Vu\) decreases rapidly in both packet
variables; integration by parts in \(x\) controls the momentum variable
and the product of two Schwartz functions controls the position
variable. Thus the reconstruction \(u=\int Vu(Q)\phi_Q\,d\mu(Q)\)
converges in \(\mathcal S\).

Consider the matrix of \(A=\operatorname{Op}(a)\) between two packets,
with output packet \((q,p)\) and input packet \((q',p')\). Substitute
\(x=q+s\), \(\xi=p'+t\) in (E10). The integral below is the amplitude of
this matrix; its full scalar phase is computed immediately afterward: \[
(2\pi)^{-n}\iint
e^{i[s\cdot(p'-p)+t\cdot(q-q')]}
e^{is\cdot t}a(q+s,p'+t)\widehat\phi(t)\overline{\phi(s)},ds\,dt.
\tag{E26}
\]

Keep the original \(\phi_{q,p}(x)=e^{ip\cdot x}\phi(x-q)\), its \(L^2\)
norm one, \(d\mu(q,p)=(2\pi)^{-n}dq\,dp\), and the linear-first pairing.
Directly changing variables in the original forward transform gives

\[
 \widehat{\phi_{q',p'}}(\xi)
 =e^{-iq'\cdot(\xi-p')}\widehat\phi(\xi-p').
 \tag{EC12}
\]

Insert this expression into E10 and pair with \(\phi_{q,p}\). With
\(x=q+s\) and \(\xi=p'+t\), the full phase before any grouping is

\[
 (q+s)\cdot(p'+t)-q'\cdot t-p\cdot(q+s)
 =q\cdot(p'-p)+s\cdot(p'-p)
  +t\cdot(q-q')+s\cdot t.
 \tag{EC13}
\]

Consequently the actual original packet matrix is

\[
 \begin{split}
 (\operatorname{Op}(a)\phi_{q',p'},\phi_{q,p})
 &=e^{iq\cdot(p'-p)}(2\pi)^{-n}
   \iint e^{i[s\cdot(p'-p)+t\cdot(q-q')]}\\
 &\quad{}\times e^{is\cdot t}
       a(q+s,p'+t)\widehat\phi(t)\overline{\phi(s)}\,ds\,dt .
 \end{split}
 \tag{EC14}
\]

The integration measure is \(ds\,dt\): the comma before \(ds\) in the
previous E26 is typographical punctuation, not an amplitude factor. All
scalar phase factors, the original inverse Fourier multiplier and the
matrix order remain. The omitted factor has absolute value one for these
real packet parameters, so E27's absolute-value estimate and both Schur
marginals remain valid. It is independent of \(s,t\) and therefore
survives every displayed integration by parts unchanged.

Apply \((1-\Delta_s)^N(1-\Delta_t)^N\) to the last three factors by
integration by parts. Derivatives of \(e^{is\cdot t}\) create only
polynomials in \(s,t\); derivatives of the two fixed Schwartz functions
absorb those polynomials. Consequently the absolute integral of all
differentiated amplitudes is bounded by \(C_{n,N,\phi}M\), uniformly in
all four packet parameters. The derivative order of \(a\) is at most
\(4N\). We obtain \[
|(A\phi_{q',p'},\phi_{q,p})|
\leq CM\langle q-q'\rangle^{-2N}
\langle p-p'\rangle^{-2N}.
\tag{E27}
\] Since \(2N>n\), both phase-space marginals are integrable and
uniformly bounded by \(CM\). Equation (E24) bounds the corresponding
operator \(\mathcal M\) on \(L^2(d\mu)\). The zeroth bound on \(a\)
already makes (E10) a continuous map \(\mathcal S\to\mathcal S'\), since
its outputs are bounded functions. For Schwartz \(u,v\), their packet
reconstructions and (E27) give \((Au,v)=(\mathcal M Vu,Vv)\). All
integrals converge by the decay just proved. Hence \(A=V^*\mathcal M V\)
on \(\mathcal S\) as distributions, which establishes both membership of
\(Au\) in \(L^2\) and the norm bound. No \(L^2\) boundedness of \(A\)
was assumed in forming its packet matrix. This proves (E23), and density
gives the extension. \(\square\)

The calculation works for fixed-size matrices by estimating the matrix
norm in (E26) and applying the scalar majorant to the norm of the vector
input. It therefore has no positivity or scalar-commutativity
assumption.

Two consequences also isolate the lower-order mechanisms in the
classical calculus. If \(a\in S^q_{\rho,\delta}\) with \(q<-n\), its
kernel is an ordinary continuous function. Every
\((x-y)^\alpha K_a(x,y)\) is the inverse-frequency transform multiplied
by the exact power of \(i\) specified below, applied to
\(\partial_\xi^\alpha a\). That derivative has integrable absolute value
uniformly in \(x\). Dominated convergence on compact base sets proves
continuity. The needed integration by parts uses a cutoff first and then
lets it tend to one.

For the original inverse-frequency kernel E12, differentiating \(a\) in
\(\xi\) and integrating by parts gives

\[
 K_{\partial_\xi^\alpha a}(x,y)
 =(-i)^{|\alpha|}(x-y)^\alpha K_a(x,y),\qquad
 (x-y)^\alpha K_a(x,y)
 =i^{|\alpha|}K_{\partial_\xi^\alpha a}(x,y).
 \tag{EC15}
\]

These are also exact distributional identities: frequency
differentiation is multiplication by \(-i(x-y)\) under the inverse
transform, proved by testing the defining tempered transforms. When
\(q<-n\), every displayed frequency derivative is integrable uniformly
in \(x\). For an ordinary integration proof insert a smooth compact
cutoff \(\chi(\xi/R)\). Each boundary term contains a positive cutoff
derivative, at least one factor \(R^{-1}\), and lies where \(|\xi|\) is
of order \(R\). The earlier intermediate estimate claimed an absolute
bound by a constant times \(R^{n+q-|\alpha|}\), or a smaller power; this
exponent is valid when \(\rho=1\) but is excessive in the general range
\(0<\rho<1\). The following correction keeps the entire product rule and
proves the actual vanishing bound.

The convergence still follows from \(q<-n\), with every original
derivative factor retained. Choose the original cutoff to equal one on
\(|\xi|\leq c_0\) and vanish on \(|\xi|\geq c_1\), with \(0<c_0<c_1\).
Take \(R\geq\max(1,c_0^{-1})\). The complete derivative is \[
 \partial_\xi^\alpha\bigl(a(x,\xi)\chi(\xi/R)\bigr)
 =\sum_{\beta\leq\alpha}{\alpha\choose\beta}
   (\partial_\xi^{\alpha-\beta}a)(x,\xi)
   R^{-|\beta|}(\partial^\beta\chi)(\xi/R).
 \tag{EC16}
\] For each \(0<\beta\leq\alpha\), put
\(s_\beta=q-\rho|\alpha-\beta|<0\). Its original annular support lies in
\(c_0R\leq|\xi|\leq c_1R\). On this set
\(\langle\xi\rangle^{s_\beta}\leq c_0^{s_\beta}R^{s_\beta}\). Equation
E2 and the original coordinate volume therefore give, with \(\omega_n\)
the volume of the Euclidean unit ball, \[
 \begin{split}
 &\int\left\|{\alpha\choose\beta}
       (\partial_\xi^{\alpha-\beta}a)(x,\xi)
       R^{-|\beta|}(\partial^\beta\chi)(\xi/R)\right\|\,d\xi\\
 &\qquad\leq{\alpha\choose\beta}\,
       p_{q,|\alpha-\beta|}(a)\,
       \|\partial^\beta\chi\|_\infty\,
       \omega_nc_1^n c_0^{q-\rho|\alpha-\beta|}
       R^{n+q-\rho|\alpha-\beta|-|\beta|}.
 \end{split}
 \tag{EC17}
\] Every factor is retained, and the bound is uniform in the original
base point. Because \(\rho>0\) and \(|\beta|\geq1\), every exponent is
at most \(n+q-1<0\). Thus every boundary term tends to zero. The term
\(\beta=0\) converges in absolute integral by dominated convergence
against the integrable original \(\partial_\xi^\alpha a\). The
integration-by-parts phase has absolute value one, so these are
sufficient absolute bounds for that passage. This proves EC15 as an
ordinary integral with its exact power of \(i\). When \(|\alpha|=0\),
there are no boundary terms; in dimension zero there is no positive
frequency multiindex. The resulting absolute kernel and norm conclusions
remain unchanged.

Here is an exact counterexample to the excessive earlier absolute
exponent. Take \(n=1\), \(0<\rho<1\), \(\delta=0\), and \(q<-1\). Choose
smooth \(\theta\) equal to zero for \(\xi\leq1\) and one for
\(\xi\geq2\). Define \(a(\xi)=0\) for \(\xi\leq1\) and
\(a(\xi)=\theta(\xi)\xi^q e^{i\xi^{1-\rho}}\) for \(\xi>1\). It is
smooth at the join, and for \(\xi\geq2\) every derivative has the full
finite form \[
 a^{(j)}(\xi)=e^{i\xi^{1-\rho}}
       \sum_{k=0}^j C_{j,k}\xi^{q-j+k(1-\rho)},
 \qquad C_{0,0}=1,
 \tag{EC18}
\] where coefficients outside \(0\leq k\leq j\) are zero and direct
differentiation gives \[
 C_{j+1,k}=\bigl(q-j+k(1-\rho)\bigr)C_{j,k}
                  +i(1-\rho)C_{j,k-1}.
 \tag{EC19}
\] Each exponent is at most \(q-\rho j\). On the compact transition
interval all terms of the product with \(\theta\) are bounded with the
original binomial coefficients; there are no position derivatives. Hence
\(a\in S^q_{\rho,0}\) with the original E2 seminorms. On the positive
tail, \[
 |a'(\xi)|=\left|q\xi^{q-1}+i(1-\rho)\xi^{q-\rho}\right|
                 \geq(1-\rho)\xi^{q-\rho}.
 \tag{EC20}
\] A smooth compact cutoff which equals one near zero must have a
nonzero derivative on some positive interval; continuity supplies a
closed interval \(J\subset(0,\infty)\) on which \(|\chi'|\) is bounded
below by a positive constant. For \(R\) large enough that
\(RJ\subset[2,\infty)\), the original \(\alpha=2,\beta=1\) boundary term
satisfies \[
 \int_{RJ}\left|2a'(\xi)R^{-1}\chi'(\xi/R)\right|\,d\xi
 \geq2(1-\rho)R^{q-\rho}
            \int_J t^{q-\rho}|\chi'(t)|\,dt>0.
 \tag{EC21}
\] The last integral is a fixed positive number. Division by the earlier
proposed power \(R^{q-1}\) gives growth by \(R^{1-\rho}\), so that
stronger absolute bound fails. This is a lower bound for the absolute
integral of this particular full product-rule term; it does not assert a
lower bound for its oscillatory integral or for a sum with cancellation.
The corrected EC17 bound still tends to zero and proves the unchanged
original identity and its receiving estimates.

The full kernel estimate is
\(|K_a(x,y)|\leq C_N\langle x-y\rangle^{-N}\) for every \(N\). Taking
\(N>n\) verifies both marginals in (E24). For classical symbols this
proves boundedness at all sufficiently negative orders without the
packet estimate. If boundedness is known at order \(2q\), the identity
\(\|\operatorname{Op}(a)u\|_2^2=(\operatorname{Op}(a^\dagger\circ a)u,u)\)
proves it at order \(q\), since the composed symbol has order \(2q\). A
finite number of doublings brings any fixed \(q<0\) below \(-n\). This
proves boundedness at every negative order, including order minus one.

For scalar classical \(a\in S^0\), choose \(M>\sup|a|^2\) with a fixed
positive gap and set \(c=(M-|a|^2)^{1/2}\in S^0\), using Section 1.
Equations (E20)--(E21) give \(a^\dagger\circ a+c^\dagger\circ c=M+r\),
\(r\in S^{-1}\). Taking the quadratic form on Schwartz functions and
discarding \(\|\operatorname{Op}(c)u\|_2^2\geq0\) proves order-zero
boundedness from the negative-order result. Thus the classical
square-root argument is justified in full, while the packet and band
argument below supplies the additional equality endpoint where a
decreasing remainder is unavailable.

\subsection{\texorpdfstring{7. \(L^2\) boundedness at equal derivative
costs}{7. L\^{}2 boundedness at equal derivative costs}}\label{l2-boundedness-at-equal-derivative-costs}

\textbf{Theorem.} If \(\delta\leq\rho\) in (E1), then \[
a\in S^0_{\rho,\delta}
\quad\Longrightarrow\quad
\|\operatorname{Op}(a)\|_{L^2\to L^2}
\leq C p_{0,L}(a)
\tag{E28}
\] for some finite \(L\), with \(C,L\) depending only on the fixed
parameters and dimension. In particular (E28) includes
\(0<\rho=\delta<1\).

\textbf{Proof.} Take a smooth dyadic partition
\(1=\sum_{j\geq0}\chi_j(\xi)\), with \(\chi_0\) supported in a fixed
ball and, for \(j\geq1\), \(\chi_j\) supported where
\(2^{j-1}\leq|\xi|\leq2^{j+1}\). All frequency derivatives obey the
corresponding \(2^{-j|\alpha|}\) bounds. Set \(a_j=a\chi_j\). Under the
unitary dilation \(u(x)\mapsto 2^{-jn\rho/2}u(2^{-j\rho}x)\), the symbol
of the conjugated operator is \[
\widetilde a_j(X,\Xi)=a_j(2^{-j\rho}X,2^{j\rho}\Xi).
\tag{E29}
\] Its derivatives are uniformly bounded: a base derivative gives
\(2^{-j\rho}2^{j\delta}\leq1\), and a frequency derivative gives
\(2^{j\rho}2^{-j\rho}=1\). Cutoff derivatives satisfy the same bounds
because \(\rho\leq1\). Hence (E23) gives
\(\|\operatorname{Op}(a_j)\|\leq C p_{0,L_n}(a)\), uniformly in \(j\).
This alone would not justify summing their norms.

Separate each \(a_j\) into a symbol with small base Fourier support and
an error. Choose \(\psi\in\mathcal S\) with
\(\widehat\psi\in C_c^\infty\), equal to one near zero, and with its
Fourier support in the unit ball. With the Fourier normalization fixed
above, \(\int\psi=1\), and every nonconstant polynomial moment of
\(\psi\) is zero. For \(j\geq1\), put \(\lambda_j=2^{j-4}\) and \[
\ell_j(x,\xi)=\int\psi(z)a_j(x-z/\lambda_j,\xi)\,dz,
\qquad r_j=a_j-\ell_j.
\tag{E30}
\] Convolution preserves the uniform derivative bounds used in (E29), up
to \(\|\psi\|_1\). Taylor expansion in \(x\) to order \(K-1\), together
with the vanished moments, gives \[
\|\partial_x^\beta\partial_\xi^\alpha r_j\|_\infty
\leq C_K\lambda_j^{-K}
\max_{|\nu|=K}\|\partial_x^{\beta+\nu}
\partial_\xi^\alpha a_j\|_\infty
\leq C p_{0,L}(a)
2^{-jK(1-\delta)}2^{j\delta|\beta|-j\rho|\alpha|}.
\tag{E31}
\] The integral Taylor remainder is bounded using
\(\int|z|^K|\psi(z)|dz<\infty\). Thus after the dilation (E29), (E23)
implies \[
\|\operatorname{Op}(r_j)\|_{2\to2}
\leq C p_{0,L}(a)2^{-jK(1-\delta)}.
\tag{E32}
\] Choose for example \(K=1\). Since \(\delta<1\), these norms are
summable, with finite bound depending on \(1-2^{-(1-\delta)}\). Larger
\(K\) is available when needed.

The symbol \(\ell_j\) has base Fourier support \(|\theta|\leq2^{j-4}\),
in the distributional sense; multiplication by
\(\widehat\psi(\theta/\lambda_j)\) proves this even though the base
variable is not integrable. Its frequency support remains that of
\(\chi_j\). In the Fourier representation of (E10), output frequency
equals input frequency plus base Fourier frequency. Hence
\(\operatorname{Op}(\ell_j)\) kills inputs outside the \(j\)-th
frequency annulus and its outputs are supported in a slightly enlarged
annulus, for example \(2^{j-2}\leq|\eta|\leq2^{j+2}\). One can verify
the support assertion by testing against compactly supported frequency
test functions with disjoint sum supports, using the distributional
Fourier formula; it requires no pointwise Fourier transform of
\(\ell_j\).

Let \(P_j\) be the sharp \(L^2\) projection onto the support annulus of
\(\chi_j\). Then
\(\operatorname{Op}(\ell_j)=\operatorname{Op}(\ell_j)P_j\), and
\(\sum_j\|P_ju\|_2^2\leq C\|u\|_2^2\) because the annuli overlap only
finitely often. The enlarged output annuli have the same finite-overlap
property. Pointwise Cauchy--Schwarz in Fourier space and Plancherel
therefore give, for every finite sum, \[
\left\|\sum_j\operatorname{Op}(\ell_j)u\right\|_2^2
\leq C\sum_j\|\operatorname{Op}(\ell_j)P_ju\|_2^2
\leq C p_{0,L}(a)^2\|u\|_2^2.
\tag{E33}
\] The same estimate for tails shows convergence for each \(u\in L^2\),
since the tail of \(\sum\|P_ju\|^2\) tends to zero. The single
low-frequency term \(a_0\) is bounded by (E23). Adding (E32) proves a
bounded operator with estimate (E28). On Schwartz inputs the dyadic
symbol sum converges in (E10), with all Schwartz output seminorms, by
the rapid decrease of \(\widehat u\) and the already established
seminorm estimates. The bounded operator is therefore exactly
\(\operatorname{Op}(a)\). \(\square\)

This proof uses \(\delta\leq\rho\) in the rescaling and \(\delta<1\) in
the summable error. It never treats the zero gain \(\rho-\delta\) as
positive.

There is a useful complementary result with no frequency
differentiability. Suppose \(a(x,\xi)\) is measurable and, for each
\(\xi\), is \(C^{n+1}\) in \(x\), with \[
\sup_\xi\sum_{|\alpha|\leq n+1}
\int\|D_x^\alpha a(x,\xi)\|\,dx\leq M.
\tag{E34}
\] Then \(\operatorname{Op}(a)\) is bounded on \(L^2\), with norm at
most \(C_nM\). Define it initially when \(\widehat u\in C_c^\infty\).
Let \(A(\theta,\xi)=(2\pi)^{-n}\mathcal F_xa(\theta,\xi)\). Integration
by parts in the sense of integrable weak derivatives gives \[
\|A(\theta,\xi)\|\leq C_nM\langle\theta\rangle^{-n-1}.
\tag{E35}
\] Indeed the Fourier transforms of all derivatives of order at most
\(n+1\) are bounded by their \(L^1\) norms, and the corresponding
monomials dominate \(\langle\theta\rangle^{n+1}\). Cutoffs in \(x\),
followed by their limit in these \(L^1\) derivative norms, justify the
integration by parts. Fubini gives
\(\widehat{\operatorname{Op}(a)u}(\eta)=\int A(\eta-\xi,\xi)\widehat u(\xi)d\xi\).
Both kernel marginals are bounded by \(C_nM\) using (E35). Apply (E24)
and Plancherel, then density. Joint measurability and the derivative
hypotheses suffice; no hidden \(\xi\)-smoothness is used.

\subsection{8. Sobolev and Besov
mapping}\label{sobolev-and-besov-mapping}

For real \(s\), let \(H^s\) be the tempered distributions whose Fourier
transforms are locally square integrable and satisfy \[
\|u\|_{H^s}^2=(2\pi)^{-n}
\int\langle\xi\rangle^{2s}|\widehat u(\xi)|^2\,d\xi<\infty.
\tag{E36}
\] The multiplier \(J^s=\langle D\rangle^s\) is an isometric isomorphism
\(H^s\to L^2\), by the Fourier \(L^2\) contract. Its symbol belongs to
\(S^s_{\rho,\delta}\) in (E1): actual frequency derivatives lose one
order, which is at least the required loss \(\rho\), and all nonzero
base derivatives vanish.

For \(a\in S^m_{\rho,\delta}\) with \(\delta\leq\rho\), the composition
theorem gives
\(J^{s-m}\operatorname{Op}(a)J^{-s}\in\operatorname{Op}(S^0_{\rho,\delta})\).
The finite-seminorm bounds in (E21) and (E28) therefore prove \[
\operatorname{Op}(a):H^s\longrightarrow H^{s-m}
\quad\hbox{continuously for every real }s.
\tag{E37}
\] This is the restriction of the already defined tempered-distribution
operator: it agrees first on Schwartz functions and then by continuity
and density in \(H^s\). For integer nonnegative \(s\), (E36) is
equivalent to the sum of squared \(L^2\) norms of the derivatives
through order \(s\), by expanding \((1+|\xi|^2)^s\); negative and
noninteger orders retain the Fourier definition.

The relevant Besov scale uses an \(\ell^p\) index, not an \(L^p\) index
in the base variable. Its domain consists of tempered distributions with
locally square-integrable Fourier transforms. Set \(A_0=\{|\xi|<1\}\),
\(A_j=\{2^{j-1}\leq|\xi|<2^j\}\) for \(j\geq1\), and let \(\Pi_j\) be
their sharp Fourier projections. For \(1\leq p\leq\infty\), define \[
\|u\|_{B^s_{2,p}}=
\left\|(2^{js}\|\Pi_ju\|_2)_{j\geq0}\right\|_{\ell^p},
\tag{E38}
\] with the supremum when \(p=\infty\). A distribution in this space has
a locally \(L^2\) Fourier transform. The sharp projections are
consequently well defined; conversely a sequence of \(L^2\) functions on
these annuli with at most polynomial growth defines a tempered
distribution by summation. Cauchy--Schwarz on each annulus and rapid
decay of a Schwartz test function make that series convergent. Thus
(E38) is a Banach norm, by completeness of the weighted \(\ell^p\) sum
of the Hilbert spaces \(L^2(A_j)\). It is equivalent to the norm using
\(\|\Pi_ju\|_{H^s}\), since \(\langle\xi\rangle\asymp2^j\) on each
annulus. The latter is the convention often denoted \({}^pH_{(s)}\). In
particular \(B^s_{2,2}=H^s\) with equivalent norms. Smooth dyadic
cutoffs give the same spaces: each smooth piece meets finitely many
sharp annuli and vice versa, so the two finite-overlap triangle
estimates give norm equivalence.

Here is a proof of the exact interpolation implication needed for the
calculus. Suppose a linear operator \(T\) is consistent and bounded from
\(H^{s_-}\) to \(H^{s_--m}\) and from \(H^{s_+}\) to \(H^{s_+-m}\), with
\(s_-<s<s_+\). On \(\Pi_ju\), the two estimates imply \[
2^{k(s-m)}\|\Pi_kT\Pi_ju\|_2
\leq C\min\{2^{(k-j)(s-s_-)},
2^{(k-j)(s-s_+)}\}\,
2^{js}\|\Pi_ju\|_2.
\tag{E39}
\] For example insert \(2^{-k(s_--m)}\|T\Pi_ju\|_{H^{s_--m}}\), then use
\(\|\Pi_ju\|_{H^{s_-}}\leq C2^{js_-}\|\Pi_ju\|_2\); the other endpoint
is identical. The sequence \(c_l=\min(2^{l(s-s_-)},2^{l(s-s_+)})\) is
summable over \(l\in\mathbb Z\). For a nonnegative sequence \(v\),
\(\|c*v\|_{\ell^p}\leq\|c\|_{\ell^1}\|v\|_{\ell^p}\): for finite sums
this is the triangle inequality for translations; monotone passage to
the sum proves the assertion, also for \(p=1,\infty\). Equation (E39)
thus bounds the output norm (E38).

For a general input of finite (E38) norm, the annular sum converges in
\(H^{s_-}\), because its weighted \(\ell^2\) norm there is bounded by an
exponentially decreasing sequence times a bounded sequence. Hence \(Tu\)
is already defined by the lower endpoint. Passing to each fixed output
projection and using (E39) proves the asserted Besov bound, even for
\(p=\infty\), where finite annular sums need not converge in the Besov
norm. For \(p<\infty\) those sums do converge in that norm by the tail
property of \(\ell^p\).

Applying this implication to (E37), with any \(s_-<s<s_+\), proves \[
\operatorname{Op}(a):B^s_{2,p}\longrightarrow B^{s-m}_{2,p}
\quad(1\leq p\leq\infty,\ s\in\mathbb R).
\tag{E40}
\] All constants are controlled by finitely many symbol seminorms.
Multiplication by any smooth function with all derivatives bounded is
the order-zero special case; in particular it is continuous on every
space in (E38). For a distribution \(u\) on an open set, its product
with \(\chi\in C_c^\infty\) of that set extends by zero to a compactly
supported tempered distribution. The local space is therefore defined by
the seminorms \(\|\chi u\|_{B^s_{2,p}}\) of these extensions. Equivalent
cutoff families follow by partitions of unity and this multiplication
estimate. Coordinate invariance and the later microlocal assertions
require their own geometric arguments; (E40) supplies their exact global
continuity input.

\textbf{Editorial extension: the same original annuli for every positive
sequence exponent.} The restriction \(p\ge1\) in (E38)--(E40) is
unnecessary for the mapping theorem. The smaller exponents give complete
metric vector spaces, with the following exact proofs. Retain every
original annulus \(A_j\), sharp projection \(\Pi_j\), Fourier
convention, and weight \(2^{js}\). For \(0<p<1\), put \[
N_{s,p}(u)=\left(\sum_{j\ge0}(2^{js}\|\Pi_ju\|_2)^p\right)^{1/p},
\qquad d_{s,p}(u,v)=N_{s,p}(u-v)^p.
\tag{BQ1}
\] The domain is the same locally square-integrable Fourier domain as in
(E38), with finite displayed sum. For \(a,b\ge0\),
\((a+b)^p\le a^p+b^p\): after treating \(a=0\) separately, differentiate
\((1+t)^p-1-t^p\) for \(t>0\). Its derivative is
\(p((1+t)^{p-1}-t^{p-1})<0\), and its limit at zero is zero. The
original \(L^2\) triangle inequality, followed by this inequality and
summation, proves \[
N_{s,p}(u+v)^p\le N_{s,p}(u)^p+N_{s,p}(v)^p,
\qquad N_{s,p}(zu)=|z|N_{s,p}(u).
\tag{BQ2}
\] If the sum vanishes, every original Fourier piece vanishes, hence the
distribution vanishes. Thus \(d_{s,p}\) is a translation-invariant
metric; scalar multiplication is continuous because of homogeneity and
(BQ2). No norm triangle inequality is required for this assertion.

Here is the exact annular reconstruction and completeness argument.
Choose arbitrary \(F_j\in L^2(A_j)\), extended by zero, with
\(v_j=2^{js}(2\pi)^{-n/2}\|F_j\|_2\) in \(\ell^p\). Then
\(v_j\le\|v\|_{\ell^p}\). Define \(F=\sum_jF_j\) on the disjoint
original annuli, and define its inverse Fourier distribution by \[
\langle u,\varphi\rangle=(2\pi)^{-n}\sum_{j\ge0}
       \int_{A_j}F_j(\xi)\widehat\varphi(-\xi)\,d\xi.
\tag{BQ3}
\] For \(j\ge1\), retain the actual volume bound
\(|A_j|\le\omega_n2^{jn}\). If
\(C_N=\sup_\xi\langle\xi\rangle^N|\widehat\varphi(-\xi)|\),
Cauchy--Schwarz bounds the absolute value of the \(j\)-th summand,
including its Fourier factor, by
\((2\pi)^{-n/2}\omega_n^{1/2}2^NC_N2^{j(n/2-s-N)}v_j\). Choose an
integer \(N>n/2-s\). The complete geometric tail converges. The \(j=0\)
term is at most \((2\pi)^{-n/2}v_0\|\widehat\varphi\|_{L^2(A_0)}\).
These estimates also prove continuity in a finite Schwartz seminorm and
continuous passage from the displayed sequence metric to
\(\mathcal S'\). The actual Fourier transform is \(F\); every compact
frequency set meets finitely many original annuli, so it is locally
\(L^2\). Plancherel gives \(\|\Pi_ju\|_2=(2\pi)^{-n/2}\|F_j\|_2\), with
the original factor retained. This is an onto isometry for (BQ1) between
its domain and the weighted sequence of original Hilbert spaces.

For a Cauchy sequence \(u^{(r)}\) in \(d_{s,p}\), each weighted Fourier
piece is Cauchy in its actual Hilbert norm. Let \(F_j\) be its limit.
For every fixed \(r\), finite partial sums and their increasing limit
give \[
\sum_j\left(2^{js}(2\pi)^{-n/2}
       \|\widehat{\Pi_ju^{(r)}}-F_j\|_2\right)^p
\le\liminf_{t\to\infty}N_{s,p}(u^{(r)}-u^{(t)})^p.
\tag{BQ4}
\] One fixed \(r\) and (BQ2) show that the limiting sequence has finite
sum. Equation (BQ3) constructs its distribution \(u\). The Cauchy bound
in (BQ4) then gives \(d_{s,p}(u^{(r)},u)\to0\). Thus the metric is
complete, with the full original Fourier domain, rather than a
completion containing unidentified objects. Finite annular sums converge
in this metric by the tail of the actual \(\ell^p\) sum.

All earlier comparisons retain their original constants. On each
\(A_j\), \(2^{-j}\langle\xi\rangle\) lies between \(1/2\) and
\(\sqrt2\), also when \(j=0\). For every real \(t\), set \[
a_t=\min(2^{-t},2^{t/2}),\qquad b_t=\max(2^{-t},2^{t/2}).
\] Then
\(a_t2^{jt}\|\Pi_ju\|_2\le\|\Pi_ju\|_{H^t}\le b_t2^{jt}\|\Pi_ju\|_2\).
In particular, for \(\varepsilon=s-s_->0\), disjointness of the original
annuli gives \[
\|u\|_{H^{s_-}}^2
 =\sum_j\|\Pi_ju\|_{H^{s_-}}^2
 \le {b_{s_-}^2\over1-2^{-2\varepsilon}}N_{s,p}(u)^2.
\tag{BQ5}
\] The same bound for tails proves convergence of the annular sum in the
actual lower-endpoint Sobolev space.

For completeness, take smooth dyadic multipliers \(\phi_k\) with
\(\sum_k\phi_k=1\), \(\sup_k\|\phi_k\|_\infty\le M\), and
\(\phi_k|_{A_j}=0\) unless \(|k-j|\le L\), including the low piece. The
original Fourier multiplier and the \(L^2\) triangle inequality give \[
\begin{aligned}
\sum_k(2^{ks}\|\phi_k(D)u\|_2)^p
 &\le M^p2^{p|s|L}(2L+1)\sum_j(2^{js}\|\Pi_ju\|_2)^p,\\
\sum_j(2^{js}\|\Pi_ju\|_2)^p
 &\le 2^{p|s|L}(2L+1)\sum_k(2^{ks}\|\phi_k(D)u\|_2)^p.
\end{aligned}
\tag{BQ6}
\] The first uses \(\phi_k(D)u=\sum_{|j-k|\le L}\phi_k(D)\Pi_ju\); the
second uses \(\Pi_ju=\sum_{|k-j|\le L}\Pi_j\phi_k(D)u\). Each sum is
finite. Apply (BQ2) to each sum, then count at most \(2L+1\) appearances
of each input. Thus the same smooth-cutoff space is obtained for these
exponents as well.

Retain precisely the two consistent endpoint maps in (E39), with
operator norms \(M_-\) and \(M_+\). They give that formula with, for
example, \(C=\max(M_-b_{s_-}/a_{s_--m},M_+b_{s_+}/a_{s_+-m})\). For
finite annular input sums put \(v_j=2^{js}\|\Pi_ju\|_2\) and
\(y_k=2^{k(s-m)}\|\Pi_kTu\|_2\). Equation (E39) gives
\(y_k\le C\sum_jc_{k-j}v_j\), with its unchanged sequence \(c_l\). For
\(0<p<1\), (BQ2), now for nonnegative scalars, gives the exact
replacement for the convex triangle step: \[
\begin{aligned}
\sum_{k\ge0}y_k^p
 &\le C^p\sum_{k,j\ge0}c_{k-j}^pv_j^p
 \le C^p\left(\sum_{l\in\mathbb Z}c_l^p\right)\sum_{j\ge0}v_j^p,\\
\sum_{l\in\mathbb Z}c_l^p
 &=1+{2^{-p(s-s_-)}\over1-2^{-p(s-s_-)}}
       +{2^{-p(s_+-s)}\over1-2^{-p(s_+-s)}}.
\end{aligned}
\tag{BQ7}
\] Both denominators are positive. For a general input, (BQ5) defines
\(Tu\) by the original lower endpoint. Annular input partial sums
converge there; each fixed output projection converges in \(L^2\), since
its Sobolev weight has a positive lower bound on its original annulus.
Finite output sums, then their increasing limit, pass (BQ7) to this
actual \(Tu\). Consequently its target metric is finite, and the map is
continuous, with norm bound \(C(\sum_lc_l^p)^{1/p}\). This proves the
implication without supposing a norm triangle inequality for \(p<1\),
and without redefining \(T\).

Apply the proved implication to the actual operator (E37). Combining it
with the original proof for \(p\ge1\) yields \[
\operatorname{Op}(a):B^s_{2,p}\longrightarrow B^{s-m}_{2,p}
\quad(0<p\le\infty,\ s\in\mathbb R),
\tag{BQ8}
\] under the original hypotheses (E1), \(\delta\le\rho\), and
\(a\in S^m_{\rho,\delta}\). For \(p<1\) the displayed Besov space means
(BQ1); for the earlier exponents it means (E38). All bounds still use
finitely many original symbol seminorms. Finite-dimensional vector norms
follow by the same Hilbert and triangle arguments; every matrix product
in the original composition remains ordered.

The multiplication special case and local definition after (E40) now
hold for every positive \(p\). To check the cutoff-family assertion
explicitly, let a finite partition \(\sum_i\psi_i=1\) on a neighborhood
of \(\operatorname{supp}\chi\) be subordinate to regions on which test
cutoffs \(\chi_i\) are one. Then
\(\chi u=\sum_i(\chi\psi_i)(\chi_i u)\). Each multiplier is smooth with
bounded derivatives. Apply (BQ8) to each term and (BQ2) to the finite
sum for \(p<1\); use the original triangle estimate for the other
exponents. A locally finite family reduces to such a finite family on
the actual compact support. This proves the equivalence of the
corresponding local membership tests. It makes no coordinate-invariance
claim.

The added exponents do change the space. For \(n\ge1\), choose \(F_j\)
supported in the original \(A_j\), with
\((2\pi)^{-n/2}\|F_j\|_2=(1+j)^{-a}\), \(j\ge1\), and \(F_0=0\). The
annular reconstruction is (BQ3). Its \(H^0\) norm squared is exactly
\(\sum_{j\ge1}(1+j)^{-2a}\), whereas its \(B^0_{2,p}\) metric to zero is
exactly \(\sum_{j\ge1}(1+j)^{-ap}\). Comparing each decreasing power
with its integral on consecutive unit intervals proves convergence
precisely when the exponent is greater than one. Thus \(1/2<a\le1/p\)
gives an actual \(H^0\) element outside \(B^0_{2,p}\). This is a proved
editorial consequence of the original annular construction, without a
novelty claim.

\subsection{9. Elliptic inversion}\label{elliptic-inversion}

The parameters satisfy \(0<\rho\leq1\), \(0\leq\delta<\rho\),
\(m\in\mathbb R\), and \(\kappa=\rho-\delta>0\). Every ellipticity bound
below has constants \(c>0\) and \(R\geq0\), uniform in the displayed
original base variable. The matrix norm is the induced norm of its
original vector norm.

Assume \(\delta<\rho\), so \(\kappa>0\), and let
\(a\in S^m_{\rho,\delta}\) be scalar. The following global ellipticity
condition is equivalent to the existence of a symbol
\(b\in S^{-m}_{\rho,\delta}\) with both products inverse modulo
smoothing: \[
|a(x,\xi)|\geq c\langle\xi\rangle^m
\quad\hbox{for all }x\hbox{ and }|\xi|\geq R,
\tag{E41}
\] \[
a\circ b-I\in S^{-\infty},\qquad
b\circ a-I\in S^{-\infty}.
\tag{E42}
\] Moreover either of the two relations in (E42), for a given \(b\),
implies the other and determines \(b\) uniquely modulo \(S^{-\infty}\).
Either implies \[
ab-I\in S^{-\kappa}_{\rho,\delta}.
\tag{E43}
\] In the classical case the gain in (E43) is one. For square matrices
replace (E41) by \(\|a(x,\xi)v\|\geq c\langle\xi\rangle^m\|v\|\); all
assertions remain true.

\textbf{Proof.} On the original elliptic region the entire original
matrix \(a(x,\xi)\) is invertible and its inverse satisfies
\(\|a(x,\xi)^{-1}\|\leq c^{-1}\langle\xi\rangle^{-m}\). The full ordered
derivative recurrence and every resulting constant are calculated in
Section9.1 below, giving all original \(S^{-m}_{\rho,\delta}\) inverse
estimates. Choose the actual frequency cutoff \(\theta\), supported in
that invertibility region and equal to one outside a larger ball. Define
\(b_0=\theta a^{-1}\), extended by zero across the interior neighborhood
where \(\theta=0\). The complete cutoff product in Section9.1 proves
\(b_0\in S^{-m}_{\rho,\delta}\) and retains \(ab_0=b_0a=\theta I\) at
every frequency. The full composition remainder (E21), with its actual
bounded-frequency term \((1-\theta)I\), therefore gives
\(r=I-a\circ b_0\in S^{-\kappa}\) and
\(\ell=I-b_0\circ a\in S^{-\kappa}\).

For a right inverse take the formal series
\(\sum_{j\geq0}b_0\circ r^{\circ j}\), whose terms have orders
\(-m-j\kappa\). Use (E6) to obtain a sum \(b_R\). Its first \(N\) terms
satisfy the exact telescoping identity \[
a\circ\sum_{j<N}b_0\circ r^{\circ j}=I-r^{\circ N}.
\tag{E44}
\] Continuity of composition and the remainder order in (E6) show
\(a\circ b_R-I\in S^{-N\kappa}\) for every \(N\), hence it is smoothing.
Similarly the series \(\sum_{j\geq0}\ell^{\circ j}\circ b_0\) gives
\(b_L\) with \(b_L\circ a-I\) smoothing. The class \(S^{-\infty}\) is a
two-sided ideal for composition with a finite-order symbol: apply (E19)
at arbitrarily negative source orders. Therefore \[
b_L-b_R=b_L\circ(I-a\circ b_R)
+(b_L\circ a-I)\circ b_R\in S^{-\infty}.
\tag{E45}
\] Either choice is a two-sided inverse.

Conversely, a right relation in (E42) and (E21) with \(N=1\) give (E43).
In fact (E43) alone implies (E41). Since \(\kappa>0\), \(ab\) is
uniformly within \(1/2\) of \(I\) at large frequency. In the scalar case
\(1/2\leq|a||b|\leq C|a|\langle\xi\rangle^{-m}\), proving (E41). In the
square matrix case \(ab\) is invertible there by the convergent
geometric series in operator norm; a square matrix with a right inverse
is invertible by finite-dimensional linear algebra, and
\(\|a^{-1}\|=\|b(ab)^{-1}\|\leq C\langle\xi\rangle^{-m}\). Starting
instead with a left relation, (E21) gives
\(ba-I\in S^{-\kappa}_{\rho,\delta}\). At sufficiently large \(|\xi|\),
\(ba\) is invertible, hence so is the square matrix \(a\), with
\(a^{-1}=(ba)^{-1}b\) and \(\|a^{-1}\|\le C\langle\xi\rangle^{-m}\).
Differentiating \(a^{-1}a=I\) gives its \(S^{-m}_{\rho,\delta}\) bounds
there. The exact ordered identity \(ab-I=a(ba-I)a^{-1}\) places \(ab-I\)
in \(S^{-\kappa}_{\rho,\delta}\) at large frequency. On the remaining
compact frequency region, the smooth symbol also belongs to that class.
Thus (E43) follows from either side. The construction already gives a
two-sided inverse \(c\), and if \(a\circ b-I\) is smoothing then
\(b-c=c\circ(a\circ b-I)-(c\circ a-I)\circ b\) is smoothing. The other
side follows. This also proves uniqueness and covers a given one-sided
inverse. \(\square\)

The geometric-series argument here is formal in decreasing symbol
orders, followed by (E6); no convergence of the operator Neumann series
is assumed. At \(\delta=\rho\) the orders \(-m-j\kappa\) do not tend to
\(-\infty\). The construction therefore makes no elliptic-parametrix
claim at that endpoint. A positive lower bound for the order-zero symbol
is a different fact from a positive symbolic remainder gain.

\subsubsection{9.1. The full inverse of the original
symbol}\label{the-full-inverse-of-the-original-symbol}

The inverse theorem requires derivative estimates before its finite
composition errors can be corrected. The original symbol \(a(x,\xi)\),
all of its entries and every derivative remain the working object. The
original symbol estimates, composition map and support-controlled
asymptotic sum are those proved in Euclidean Sections1/2/4/5. Their
original constants and full remainders are retained.

Let \(a\) be scalar or a square finite matrix in \(S^m_{\rho,\delta}\),
where \(0<\rho\leq1\), \(0\leq\delta<\rho\), \(m\in\mathbb R\), and
\(\kappa=\rho-\delta>0\). Retain the entire original ellipticity
assumption:

\[
\|a(x,\xi)v\|\geq c\langle\xi\rangle^m\|v\|
\quad\hbox{for every original }x,\ |\xi|\geq R,\ v,
\qquad\langle\xi\rangle=(1+|\xi|^2)^{1/2}.
\tag{OI1}
\]

For a scalar, take the ordinary complex modulus. The matrix norm is the
induced operator norm of the displayed original vector norm. No change
of basis or discarded coefficient is implicit. This lower bound makes
\(a(x,\xi)\) injective, hence invertible because its domain and codomain
have the same finite dimension. Substituting \(v=a^{-1}w\) into OI1
gives \(\|a^{-1}(x,\xi)\|\leq c^{-1}\langle\xi\rangle^{-m}\). The
inverse is smooth on this actual invertibility region, by the original
finite determinant/cofactor formula and its nonzero determinant.

The full original symbol estimate for every base/frequency multi-index
is

\[
\|\partial_x^\beta\partial_\xi^\alpha a(x,\xi)\|
\leq C_{\alpha\beta}
\langle\xi\rangle^{m-\rho|\alpha|+\delta|\beta|}.
\tag{OI2}
\]

Set \(h=a^{-1}\) on that region. Differentiating the original identity
\(ah=I\) by the complete joint product rule gives, for
\(|\alpha|+|\beta|>0\),

\[
\partial_x^\beta\partial_\xi^\alpha h
=-h\!
\sum_{\substack{0<(\nu_x,\nu_\xi)\leq(\beta,\alpha)}}
\binom{\beta}{\nu_x}\binom{\alpha}{\nu_\xi}
(\partial_x^{\nu_x}\partial_\xi^{\nu_\xi}a)
(\partial_x^{\beta-\nu_x}
     \partial_\xi^{\alpha-\nu_\xi}h).
\tag{OI3}
\]

There is no permutation of the three ordered matrix factors in any
summand. Each final inverse derivative has strictly smaller total order.
Induction starts with OI1 and then uses OI2. The exact three powers in
each summand multiply to

\[
\begin{split}
&\langle\xi\rangle^{-m}
\langle\xi\rangle^{m-\rho|\nu_\xi|+\delta|\nu_x|}
\langle\xi\rangle^{-m-\rho|\alpha-\nu_\xi|
                              +\delta|\beta-\nu_x|}\\
&\hspace{15mm}
=\langle\xi\rangle^{-m-\rho|\alpha|+\delta|\beta|}.
\end{split}
\tag{OI4}
\]

The equality uses only the componentwise index decompositions in the
original sum. If \(H_{\alpha\beta}\) bounds the inverse derivative, the
induction supplies the explicit permissible constant

\[
H_{\alpha\beta}
=c^{-1}\!
\sum_{\substack{0<(\nu_x,\nu_\xi)\leq(\beta,\alpha)}}
\binom{\beta}{\nu_x}\binom{\alpha}{\nu_\xi}
C_{\nu_\xi,\nu_x}
H_{\alpha-\nu_\xi,\beta-\nu_x},
\qquad H_{00}=c^{-1}.
\tag{OI5}
\]

Every term is present. An increased constant remains permissible, but no
contribution is absorbed into a substitute symbol. This proves the
entire original inverse derivative estimate on \(|\xi|\geq R\).

Choose a smooth original frequency cutoff \(\theta(\xi)\), supported in
\(|\xi|>R\), and equal to one for \(|\xi|\geq R_1\), with \(R_1>R\).
Such a cutoff is supplied by the finite calculus bump construction;
retain its actual derivatives. Define \(b_0=\theta h\) on the
invertibility region and zero elsewhere. Since \(\theta\) vanishes on a
neighborhood of its inner support boundary, the extension is smooth. The
full derivative formula is

\[
\partial_x^\beta\partial_\xi^\alpha b_0
=\sum_{\lambda\leq\alpha}\binom{\alpha}{\lambda}
(\partial_\xi^\lambda\theta)
(\partial_x^\beta\partial_\xi^{\alpha-\lambda}h).
\tag{OI6}
\]

The \(\lambda=0\) term has the order proved in OI4. For every
\(\lambda\ne0\), the cutoff derivative has support in the compact
annulus \(R<|\xi|<R_1\). There all positive and negative powers of the
full \(\langle\xi\rangle\) are bounded above and away from zero, with
constants determined by both radii and the displayed real exponents;
OI3--OI5 remain uniform in every original \(x\). Thus every term of OI6
satisfies the target global \(S^{-m}_{\rho,\delta}\) estimate. In
particular \(ab_0=b_0a=\theta I\), retaining the entire cutoff factor,
rather than silently asserting identity at bounded frequency.

Use the original full composition, denoted \(\circ\), whose oscillatory
formula is

\[
(a\circ b)(x,\xi)=(2\pi)^{-n}
\iint e^{i(x-y)\cdot(\eta-\xi)}
           a(x,\eta)b(y,\xi)\,dy\,d\eta .
\tag{OI7}
\]

For symbols its value is the distributional/regularized limit proved in
E19--E22, rather than an unjustified absolute integral. The exact finite
expansion and its full remainder are

\[
a\circ b-\sum_{|\gamma|<N}
\frac{(\partial_\xi^\gamma a)(D_x^\gamma b)}{\gamma!}
\in S^{m_a+m_b-N\kappa}_{\rho,\delta},
\qquad D_x=-i\partial_x .
\tag{OI8}
\]

With \(N=1\), \(m_a=m\), \(m_b=-m\), and the retained bounded-frequency
term \(I-\theta I\), this proves both actual defects \[
r=I-a\circ b_0\in S^{-\kappa}_{\rho,\delta},\qquad
\ell=I-b_0\circ a\in S^{-\kappa}_{\rho,\delta}.
\tag{OI9}
\] The term \((1-\theta)I\) is smoothing in this global symbol class:
its base derivatives of positive order are zero and every frequency
derivative is supported in a bounded set. The other defect terms are
exactly the composition remainders from OI8.

Retain the ordered finite sums \[
b_{R,N}=\sum_{j=0}^{N-1}b_0\circ r^{\circ j},
\qquad
b_{L,N}=\sum_{j=0}^{N-1}\ell^{\circ j}\circ b_0.
\tag{OI10}
\] Their terms have orders \(-m-j\kappa\) by the proved composition
bound; the original zeroth powers are \(I\). Associativity and the unit
\(I\) are established by Fourier/kernel injectivity in Euclidean
Section5. They give the entire exact telescoping identities \[
a\circ b_{R,N}=I-r^{\circ N},\qquad
b_{L,N}\circ a=I-\ell^{\circ N}.
\tag{OI11}
\] For example each right summand contributes
\((I-r)\circ r^{\circ j}=r^{\circ j}-r^{\circ(j+1)}\); summing all \(j\)
leaves both stated endpoints. The left computation keeps the reverse
matrix order and has the same complete telescoping endpoints.

Since \(\kappa>0\), the support-controlled sum theorem in Euclidean
Section2 applies to those actual sequences: it supplies
\(b_R,b_L\in S^{-m}_{\rho,\delta}\) with
\(b_R-b_{R,N},b_L-b_{L,N}\in S^{-m-N\kappa}_{\rho,\delta}\). Every tail
is retained as an actual symbol. Composing each such tail with \(a\) and
using OI11 yields \[
a\circ b_R-I\in S^{-N\kappa}_{\rho,\delta},\qquad
b_L\circ a-I\in S^{-N\kappa}_{\rho,\delta}
\quad\hbox{for every }N.
\tag{OI12}
\] The actual positive \(\kappa\) makes those orders tend to minus
infinity, so the two errors are smoothing. The smoothing class is a
two-sided ideal, by applying the proved composition estimate with
arbitrary negative order for its smoothing factor. The full ordered
identity \[
b_L-b_R
=b_L\circ(I-a\circ b_R)
 +(b_L\circ a-I)\circ b_R
\in S^{-\infty}
\tag{OI13}
\] then makes either one a two-sided inverse. No convergence of the
operator geometric series is assumed.

For completeness retain both one-sided converses from the original
theorem. A smoothing right error for an arbitrary
\(b\in S^{-m}_{\rho,\delta}\), together with OI8 at \(N=1\), gives
\(ab-I\in S^{-\kappa}\). Its original norm is at most \(1/2\) above a
uniform frequency radius. The finite matrix geometric series makes
\(ab\) invertible there; hence the square \(a\) is invertible and
\(a^{-1}=b(ab)^{-1}\), with norm at most \(2C_b\langle\xi\rangle^{-m}\).
Taking reciprocals of this inverse norm proves OI1 with a corresponding
positive constant. A left smoothing error similarly gives
\(ba-I\in S^{-\kappa}\), the exact inverse \(a^{-1}=(ba)^{-1}b\), and
the same original lower bound. OI3 proves all inverse derivatives in
that region. The ordered pointwise identity \[
ab-I=a(ba-I)a^{-1}
\tag{OI14}
\] and the full product derivative rule then give
\(ab-I\in S^{-\kappa}\) at large frequency. On the complementary
bounded-frequency set the original smooth symbols have the same
estimate, including every base derivative.

The construction above already supplies a two-sided inverse \(c\). For a
given right inverse \(b\), retain \[
b-c=c\circ(a\circ b-I)-(c\circ a-I)\circ b .
\tag{OI15}
\] Both terms are smoothing, so \(b-c\) is smoothing and the other side
follows by the ideal property. For a given left inverse the exact
companion identity \[
b-c=(b\circ a-I)\circ c-b\circ(a\circ c-I)
\tag{OI16}
\] gives the same conclusion. This proves uniqueness modulo the actual
smoothing class, preserving the original matrix order in each case. All
assertions of E41--E45 are thereby reconstructed from the entire
original symbol and its full derivative formulas. At \(\delta=\rho\),
the displayed tail orders no longer decrease; this calculation does not
claim an inverse at that endpoint.

The preceding edition used the auxiliary expression
\(a_0(x,\xi)=q(\xi)a(x,\xi)\), where the full positive factor is
\(q(\xi)=\langle\xi\rangle^{-m}=(1+|\xi|^2)^{-m/2}\). The new inverse
proof above calculates directly with the entire original \(a(x,\xi)\).
Here we retain the exact relation to the preceding expression, including
all scalar and differentiated factors.

For every real \(\xi\) and real \(m\), \(q(\xi)>0\) and its full inverse
is \(q(\xi)^{-1}=\langle\xi\rangle^m=(1+|\xi|^2)^{m/2}\). Thus
\(a=q^{-1}a_0\), and \(a_0\) is invertible exactly where \(a\) is. If
their original finite matrix rank is \(r\), then \[
\det a_0=q^r\det a,\qquad
\ker a_0=\ker a,\qquad
a_0^{-1}=a^{-1}q^{-1}.
\tag{OC1}
\] The determinant equality follows from multilinearity in every one of
the \(r\) columns. The kernel equality follows from multiplication by
the nonzero scalar. The inverse identity follows by multiplying on both
sides, retaining the displayed order. At \(\xi=0\) the factor is one; it
introduces no excluded frequency point or additional zero. In rank zero
every determinant is one, and the unique empty linear maps satisfy the
same relations.

The complete derivatives of the scalar factor can be retained without
hiding their contributions. Write \(g(\xi)=1+\sum_j\xi_j^2\) and
\(\nu=-m/2\). For a list \(J\) of labeled derivative positions whose
coordinate counts are \(\alpha\), the full chain formula is \[
\partial_\xi^\alpha q
=\sum_{\substack{\pi\text{ a partition of }J\\
                  |B|\leq2\ (B\in\pi)}}
 \left(\prod_{j=0}^{|\pi|-1}(\nu-j)\right)
 g^{\nu-|\pi|}
 \prod_{B\in\pi}\partial_{\xi_B}g,
\quad
\partial_{\xi_j}g=2\xi_j,\quad
\partial_{\xi_j}\partial_{\xi_k}g=2\delta_{jk}.
\tag{OC2}
\] The empty partition gives the unchanged zeroth derivative. All larger
blocks have zero derivative and therefore make zero contributions to the
original full chain formula. The finite chain rule proved in the
calculus chapter gives this equality by induction: differentiating a
factor either adds the new label to one existing block or creates its
singleton block, which generates each next labeled partition exactly
once.

If a term has \(s\) singleton blocks and \(t\) two-label blocks, then
\(|\alpha|=s+2t\), \(|\pi|=s+t\). Its full \(g\)-power times its
singleton coordinate factors has an absolute bound of order
\(\langle\xi\rangle^{-m-2(s+t)+s}
=\langle\xi\rangle^{-m-|\alpha|}\). The actual falling-factorial
product, each factor \(2\), each Kronecker delta and every
repeated-coordinate partition remain in OC2; their finite sum supplies
its bound constant. The same proof with \(\nu=m/2\) gives all
derivatives of the full \(q^{-1}\). The original product formula for the
earlier auxiliary expression is \[
\partial_x^\beta\partial_\xi^\alpha a_0
=\sum_{\lambda\leq\alpha}\binom{\alpha}{\lambda}
 (\partial_\xi^\lambda q)
 (\partial_x^\beta\partial_\xi^{\alpha-\lambda}a).
\tag{OC3}
\] Each summand has its complete power \[
\langle\xi\rangle^{-m-|\lambda|}
\langle\xi\rangle^{m-\rho|\alpha-\lambda|+\delta|\beta|}
=\langle\xi\rangle^{-\rho|\alpha|-(1-\rho)|\lambda|+\delta|\beta|}.
\tag{OC4}
\] Since \(0<\rho\leq1\), these actual exponents prove the earlier
\(S^0_{\rho,\delta}\) statement, with all its derivative terms retained.
They are a comparison of the two expressions; OI1--OI16 continue to use
the original full symbol.

On the original invertibility region and with the same cutoff
\(\theta\), the preceding and current preliminary inverse expressions
obey the entire factor identity \[
b_{\mathrm{preceding}}
=\theta q a_0^{-1}
=\theta q a^{-1}q^{-1}
=\theta a^{-1}=b_0 .
\tag{OC5}
\] Both expressions are extended by zero across the region where the
cutoff vanishes, so this equality holds as a global smooth symbol even
at frequencies where \(a\) is singular. No inverse is evaluated at those
singular frequencies.

To compare every derivative, put \(h=a^{-1}\) on the invertibility
region and use a joint base/frequency multi-index \(\gamma\). The
complete four-factor derivative is \[
\partial^\gamma(\theta q h q^{-1})
=\sum_{\gamma_1+\gamma_2+\gamma_3+\gamma_4=\gamma}
\frac{\gamma!}{\gamma_1!\gamma_2!\gamma_3!\gamma_4!}
(\partial^{\gamma_1}\theta)(\partial^{\gamma_2}q)
(\partial^{\gamma_3}h)(\partial^{\gamma_4}q^{-1}).
\tag{OC6}
\] All scalar derivatives commute with the actual matrix derivative of
\(h\). Group the terms with \(\lambda=\gamma_2+\gamma_4\). Their inner
sum is the entire product derivative \[
\sum_{\gamma_2+\gamma_4=\lambda}
\frac{\lambda!}{\gamma_2!\gamma_4!}
(\partial^{\gamma_2}q)(\partial^{\gamma_4}q^{-1})
=\partial^\lambda(qq^{-1})
=\begin{cases}1,&\lambda=0,\\0,&|\lambda|>0.\end{cases}
\tag{OC7}
\] This proves the cancellations of all scalar derivative contributions
rather than omitting them. Substitution in OC6 gives exactly OI6's
remaining original cutoff and inverse derivative sum. At the
zero-extension boundary the symbols are smooth and coincide, so all
derivatives coincide there too. Their actual quantizations are equal by
the full original Fourier formula, hence their left and right
composition defects and finite ordered inverse powers coincide. The new
proof supplies each original inverse derivative and every subsequent
remainder directly, while this comparison preserves the exact provenance
of the preceding expression.

\subsection{10. Hypoelliptic polynomials and the symbol
calculus}\label{hypoelliptic-polynomials-and-the-symbol-calculus}

The origin of more general symbol orders includes reciprocals of
hypoelliptic polynomials. The quantitative hypoelliptic polynomial
theorem, proved in this section, states: if a nonzero complex polynomial
\(P\) defines a hypoelliptic constant-coefficient operator, meaning that
\(P(D)u\) smooth implies \(u\) smooth on every open set for every
distribution \(u\), then \(P(\xi)\ne0\) for all sufficiently large real
\(|\xi|\), and there are constants \(c>0,C,R\) such that \[
\left|\frac{\partial^\alpha P(\xi)}{P(\xi)}\right|
\leq C|\xi|^{-c|\alpha|}
\quad (\alpha\ne0,\ \xi\in\mathbb R^n,\ |\xi|>R).
\tag{E48}
\] Only finitely many nonzero derivatives occur, so a common \(C\) is
possible. We prove every step below, including the finite real
polynomial projection theorem, its power-growth consequence, the
original complex-zero separation and every reciprocal derivative. The
algebraic and spectral proofs are in
\href{stable-prerequisite-bridges.pdf\#section*.24}{Stable
prerequisites} and
\href{prerequisite-bridges.pdf\#section*.4}{Fourier
prerequisites}. The compactness and calculus proofs are in
\href{metric-foundation-bridges.pdf\#section*.42}{Metric
foundations}, the Hilbert and closed-graph proofs in
\href{banach-foundation-bridges.pdf\#section*.54}{Banach
foundations}, and the original-circle coefficient formula in
\href{conormal-transmission.pdf\#section*.84}{Conormal
transmission}.

\subsubsection{10.1. Polynomial sign counts through their original
multiplication
matrices}\label{polynomial-sign-counts-through-their-original-multiplication-matrices}

Let \[
F(t)=\sum_{j=0}^{d}a_jt^j,\qquad a_j\in\mathbb R,\quad d\geq1,\quad a_d\ne0.
\tag{HP1}
\] In \(E_F=\mathbb R[t]/(F)\), polynomial division proves that the
classes \(1,t,\ldots,t^{d-1}\) form a basis. Multiplication by the
original class of \(t\) has matrix \(C_F\): column \(j<d-1\) is the
coordinate vector of \(t^{j+1}\), and its last column has entries
\(-a_j/a_d\), \(0\leq j<d\). Indeed \[
a_dt^d+\sum_{j=0}^{d-1}a_jt^j=0\quad\hbox{in }E_F,
\qquad F(C_F)=0,\qquad F(\lambda)=a_d\det(\lambda I_d-C_F).
\tag{HP2}
\] For the determinant identity, expand along the last column of
\(\lambda I_d-C_F\). Its row-\(j\) cofactor is \(\lambda^j\): the upper
shift part contributes \(j\) diagonal factors \(\lambda\), the lower
part contributes \(d-1-j\) factors \(-1\), and its cofactor sign is
\((-1)^{j+d-1}\), making the total sign positive. The last-column
entries are \(a_j/a_d\) for \(j<d-1\) and \(\lambda+a_{d-1}/a_d\) in the
last row. Their sum is \(\lambda^d+\sum_{j=0}^{d-1}(a_j/a_d)\lambda^j\).
Multiplying by the actual \(a_d\) proves HP2, including \(d=1\). Its
leading factor has not been suppressed.

For a real polynomial \(h\), define the symmetric trace matrix \[
H_{F,h}=\left[\operatorname{Tr}\big(h(C_F)C_F^{j+k}\big)\right]_{0\leq j,k<d}.
\tag{HP3}
\] It represents the original bilinear form \[
\mathcal H_{F,h}(u,v)=\operatorname{Tr}_{E_F}(m_{huv}),\qquad
u=\sum_{j=0}^{d-1}u_jt^j,\quad v=\sum_{k=0}^{d-1}v_kt^k,
\tag{HP4}
\] where \(m_w\) is multiplication by the class of \(w\). Evaluating
each basis pair proves HP3. Its entries are rational functions of all
the coefficients, with denominators powers of the actual nonzero
\(a_d\).

The complex-root theorem and conjugation give the full original
factorization \[
F(t)=a_d
\prod_{\ell=1}^{r}(t-r_\ell)^{m_\ell}
\prod_{\ell=1}^{s}\big((t-b_\ell)^2+c_\ell^2\big)^{n_\ell},
\quad c_\ell>0,\quad
d=\sum_\ell m_\ell+2\sum_\ell n_\ell.
\tag{HP5}
\] The real roots and the nonreal conjugate pairs are distinct. All
multiplicities are retained. Bézout for these coprime powered factors
gives the isomorphism from \(E_F\) to the product of their quotient
algebras, sending each polynomial to all of its remainder classes.
Injectivity follows because divisibility by all coprime factors implies
divisibility by their product; multiplication by \(a_d^{-1}\) identifies
that ideal with \((F)\). For surjectivity, write \(G_\ell\) for a
powered factor and \(B_\ell=\prod_{j\ne\ell}G_j\). Bézout gives
\(u_\ell B_\ell+v_\ell G_\ell=1\), so the classes
\(e_\ell=u_\ell B_\ell\) are one modulo their own factor and zero modulo
the others; their sum is one, and \(\sum_\ell e_\ell w_\ell\) receives
any prescribed tuple of remainder classes.

At a real root the remainder map is \[
[u]\longmapsto\sum_{j=0}^{m_\ell-1}
\frac{u^{(j)}(r_\ell)}{j!}\epsilon^j
\quad\hbox{in }\mathbb R[\epsilon]/(\epsilon^{m_\ell}).
\tag{HP6}
\] For a nonreal pair \(z_\ell=b_\ell+ic_\ell,\bar z_\ell\),
complexifying and using the same Bézout maps gives both Taylor
remainders. The conjugation-fixed pairs are
\((w(\epsilon),\overline{w(\epsilon)})\). Thus the original real
quotient is isomorphic as a real algebra to
\(\mathbb C[\epsilon]/(\epsilon^{n_\ell})\), by the Taylor remainder at
\(z_\ell\), with all derivatives through \(n_\ell-1\). This proves its
real dimension \(2n_\ell\), while preserving both conjugate factors.

Multiplication by \(huv\) is triangular in the real basis
\(1,\epsilon,\ldots,\epsilon^{m_\ell-1}\), with every diagonal entry
\(h(r_\ell)u(r_\ell)v(r_\ell)\). Its trace is \[
m_\ell h(r_\ell)u(r_\ell)v(r_\ell).
\tag{HP7}
\] Its form has the scalar block \(m_\ell h(r_\ell)\) and \(m_\ell-1\)
zero directions; if \(h(r_\ell)=0\), the whole block vanishes. On the
complex local factor its real trace is
\(2n_\ell\operatorname{Re}(h(z_\ell)u(z_\ell)v(z_\ell))\). In constant
coordinates \(1,i\) its block is \[
2n_\ell\begin{pmatrix}
\operatorname{Re}h(z_\ell)&-\operatorname{Im}h(z_\ell)\\
-\operatorname{Im}h(z_\ell)&-\operatorname{Re}h(z_\ell)
\end{pmatrix},
\qquad \det=-(2n_\ell)^2|h(z_\ell)|^2,
\tag{HP8}
\] with \(2n_\ell-2\) further zero directions. If \(h(z_\ell)\ne0\), its
eigenvalues are \(2n_\ell|h(z_\ell)|,-2n_\ell|h(z_\ell)|\); otherwise
both vanish.

The product remainder isomorphism has an invertible real coordinate
matrix \(S\). The earlier expression \(S^TH_{F,h}S\) used this
coefficient-to-remainder map in the wrong direction; the exact corrected
map is as follows.

Keep the polynomial \(F\), its actual nonzero leading coefficient
\(a_d\), the coefficient basis \(1,t,\ldots,t^{d-1}\), every real-root
multiplicity and both members of each complex conjugate pair in
HP1--HP8. Let \(\psi\) be the proved real-algebra isomorphism from
\(E_F\) to the product of the local remainder algebras. Let \(S\) be its
matrix, taking an original coefficient column \(u\) to the local
remainder column \(Su\). Invertibility follows from the exact
injectivity and surjectivity argument in the preceding CRT and
Taylor-remainder arguments HP5--HP6.

Write \(B\) for the direct sum of the full real and complex trace blocks
in HP7--HP8, including every zero direction. For multiplication by any
original \(w\), the matrices of multiplication before and after \(\psi\)
are conjugate: \(\psi m_w\psi^{-1}=m_{\psi(w)}\). The trace is invariant
under this conjugation because, for arbitrary square matrices \(X,Y\),
direct summation gives
\(\operatorname{Tr}(XY)=\sum_{j,k}X_{jk}Y_{kj}=\operatorname{Tr}(YX)\).
Since \(\psi\) is an algebra map, the full original trace form
consequently satisfies

\[
 u^T H_{F,h}v=(Su)^T B(Sv)=u^T S^TBSv
 \quad\hbox{for every }u,v\in\mathbb R^d.
 \tag{EC1}
\]

Taking every pair of coordinate vectors proves the exact matrix
equalities

\[
 H_{F,h}=S^TBS,\qquad
 B=S^{-T}H_{F,h}S^{-1}.
 \tag{EC2}
\]

Thus \(S\) itself is the coefficient-to-remainder map, while the map
from local remainder coordinates to coefficient coordinates is
\(S^{-1}\). The earlier \(S^TH_{F,h}S\) applied that map in the wrong
direction.

The inertia conclusion survives with the correct morphism. A subspace
\(U\) of original coefficient columns is positive for \(H\) if and only
if \(SU\) is positive for \(B\), by EC1; \(S\) preserves dimension and
has an inverse. Thus the maximum dimensions of positive subspaces agree.
The same proof applies to negative subspaces. Also \(Hu=0\) if and only
if \(B(Su)=0\) by EC2, so \(S\) identifies the nullspaces. This proves
preservation of positive, negative and zero counts, with no omission of
nilpotent directions. Combining the original HP7--HP8 blocks therefore
gives precisely the original HP9 signature formula, including roots at
which \(h\) vanishes.

For the exact counterexample \(F=(t-2)(t-3)\), \(h=1\), all original
coefficients are \(a_2=1,a_1=-5,a_0=6\). The coefficient-basis companion
matrix and the remainder map are

\[
 C_F=\begin{pmatrix}0&-6\\1&5\end{pmatrix},\qquad
 H_{F,1}=\begin{pmatrix}2&5\\5&13\end{pmatrix},\qquad
 S=\begin{pmatrix}1&2\\1&3\end{pmatrix},\qquad
 S^{-1}=\begin{pmatrix}3&-2\\-1&1\end{pmatrix}.
 \tag{EC3}
\]

Here \(\operatorname{Tr}(I)=2\), \(\operatorname{Tr}(C_F)=5\), and
\(C_F^2\) has diagonal \(-6,19\), whose sum is13. The two simple real
blocks are \(B=I_2\). Direct multiplication gives

\[
 S^T B S=H_{F,1},\qquad
 S^{-T}H_{F,1}S^{-1}=I_2,\qquad
 S^TH_{F,1}S=\begin{pmatrix}25&68\\68&185\end{pmatrix}.
 \tag{EC4}
\]

The last equality records the original erroneous congruence's actual
value, rather than silently interpreting \(S\) as its inverse.

Congruence preserves positive and negative counts: the spectral proof
identifies the positive count with the largest dimension of a subspace
on which the original quadratic form is positive definite. A larger
subspace would meet the nonpositive eigenspaces nontrivially by the
dimension formula. An invertible map preserves that maximum; apply the
same argument to the negative form. Hence \[
\operatorname{sig}H_{F,h}
=\sum_{\ell=1}^{r}\operatorname{sign}h(r_\ell).
\tag{HP9}
\] Each distinct real root contributes once; all original
multiplicities, complex blocks and null directions remain in the
calculation.

The signature can be computed using only rational coefficient operations
and sign tests. Permute any nonzero diagonal entry \(a\) first. For
\(H=\left(\begin{smallmatrix}a&v^T\\v&W\end{smallmatrix}\right)\),
multiplication gives \[
\begin{pmatrix}1&0\\-v/a&I\end{pmatrix}
H\begin{pmatrix}1&-v^T/a\\0&I\end{pmatrix}
=\begin{pmatrix}a&0\\0&W-vv^T/a\end{pmatrix}.
\tag{HP10}
\] Its signature is
\(\operatorname{sign}a+\operatorname{sig}(W-vv^T/a)\). If all diagonal
entries vanish but an off-diagonal \(b\ne0\) exists, permute that pair
first, with \[
A=\begin{pmatrix}0&b\\b&0\end{pmatrix},
\qquad A^{-1}=\begin{pmatrix}0&1/b\\1/b&0\end{pmatrix}.
\tag{HP11}
\] For
\(H=\left(\begin{smallmatrix}A&V\\V^T&W\end{smallmatrix}\right)\), the
invertible block change
\(\left(\begin{smallmatrix}I&-A^{-1}V\\0&I\end{smallmatrix}\right)\)
gives \(\operatorname{diag}(A,W-V^TA^{-1}V)\). The block \(A\) has
eigenvalues \(b,-b\), so signature zero. An all-zero matrix has
signature zero. Each step decreases dimension by one or two and covers
empty matrices. Every branch tests a rational coefficient \(p/q\),
\(q\ne0\): equality means \(p=0\), and positivity means \(pq>0\), since
\((p/q)q^2=pq\) with its positive factor \(q^2\). Thus each possible
signature has a finite Boolean description by polynomial signs in the
original coefficients.

For a finite family of real polynomials \(q_1,\ldots,q_k\), define \[
T_e=\operatorname{sig}H_{F,\prod_{j=1}^{k}q_j^{e_j}},
\quad e\in\{0,1,2\}^k,\qquad
N_\sigma=\#\{r:F(r)=0,\ (\operatorname{sign}q_j(r))_j=\sigma\},
\quad \sigma\in\{-1,0,1\}^k.
\tag{HP12}
\] Roots in the count are distinct, and \(0^0=1\). HP9 gives \[
T_e=\sum_{\sigma\in\{-1,0,1\}^k}
\left(\prod_{j=1}^{k}\sigma_j^{e_j}\right)N_\sigma.
\tag{HP13}
\] In the order \(e=0,1,2\), \(\sigma=-1,0,1\), the one-coordinate
matrix is \[
V=\begin{pmatrix}1&1&1\\-1&0&1\\1&0&1\end{pmatrix},
\quad W=V^{-1}=
\begin{pmatrix}0&-1/2&1/2\\1&0&-1\\0&1/2&1/2\end{pmatrix},
\qquad \det V=2.
\tag{HP14}
\] Direct multiplication proves both inverse identities. Successive
multiplication in each coordinate yields \[
N_\sigma=\sum_{e\in\{0,1,2\}^k}
\left(\prod_{j=1}^{k}W_{\sigma_j,e_j}\right)T_e.
\tag{HP15}
\] Each of the \(3^k\) signatures lies in the finite integer interval
\([-d,d]\); enumerating its possibilities proves that \(N_\sigma>0\) is
a finite Boolean polynomial condition on all coefficients. For \(k=0\),
the empty product is one and its single query counts the distinct real
roots. A nonzero constant \(F\) has no roots and all counts zero.

\subsubsection{10.2. Eliminating real polynomial quantifiers with all
degree
branches}\label{eliminating-real-polynomial-quantifiers-with-all-degree-branches}

A semialgebraic set is one described by a finite Boolean combination of
real polynomial equalities and inequalities. Take such a formula
\(\Phi(x,t)\) expressed by the signs of its full original family
\(f_1(x,t),\ldots,f_k(x,t)\). Partition parameter space into finite
degree branches: an identically zero input has all its
\(t\)-coefficients zero; degree \(j\) has all higher coefficients zero
and its coefficient of \(t^j\) nonzero. These are polynomial conditions
in the retained coordinates \(x\). Each zero input still contributes its
zero sign to \(\Phi\). On a branch form \[
F_x(t)=\prod_{\{j:f_j(x,\cdot)\ne0\}}f_j(x,t).
\tag{HP16}
\] Nonzero constant factors are included and an empty product is one.
All original inputs and signs remain in \(\Phi\); this auxiliary product
only locates their roots.

At roots of \(F_x\) and its nonzero derivative \(F_x'\), HP12--HP15 test
every possible original input sign pattern. A zero or constant
derivative has no root candidates. The multiplication matrices have
their actual branch degrees and leading coefficients. Their rational
entries and finite signature branches give semialgebraic conditions for
every tested pattern.

Between consecutive distinct real roots of \(F_x\), each nonzero \(f_j\)
has constant nonzero sign by continuity and the intermediate value
theorem. Rolle's theorem supplies a root of \(F_x'\) strictly inside
every such bounded interval, so that pattern is tested. The unbounded
intervals are tested by the leading coefficients: {\fontsize{7.5}{9}\selectfont
\[
\operatorname{sign}f_j(x,t)=\operatorname{sign}a_j(x)
\quad(t\hbox{ sufficiently large positive}),\qquad
\operatorname{sign}f_j(x,t)=(-1)^{d_j}\operatorname{sign}a_j(x)
\quad(t\hbox{ sufficiently large negative}),
\tag{HP17}
\]
} where \(d_j\) is the actual degree and \(a_j(x)\ne0\) its leading
coefficient. Divide the entire finite polynomial by its leading power
and choose the threshold so the sum of the absolute values of all lower
terms is smaller than \(|a_j(x)|\). All finitely many thresholds can be
met together, and zero inputs still have sign zero. With no real roots,
these tail patterns also test the whole line; with a constant product,
they are the only candidates.

Consequently \(\exists t\,\Phi(x,t)\) is equivalent, on every degree
branch, to the finite disjunction of its acceptable patterns at roots of
\(F_x,F_x'\) and its two tails. This proves the projection theorem by
finite coefficient conditions, with every degree drop and repeated root
covered. Starting from an innermost quantified variable and repeating
eliminates any finite number of quantifiers. Universal quantifiers are
eliminated through \(\forall t\,\Phi=\neg\exists t\,\neg\Phi\). Every
resulting formula is still a finite Boolean polynomial formula.

\subsubsection{10.3. The actual power bound for an unbounded
semialgebraic
function}\label{the-actual-power-bound-for-an-unbounded-semialgebraic-function}

Suppose \(g:(R_0,\infty)\to(0,\infty)\) has semialgebraic graph and
\(g(r)\to\infty\). Choose a Boolean polynomial formula for its graph,
retaining the fixed truth values of identically zero inputs. Let
\(Q_1,\ldots,Q_N\) be its nonzero polynomial inputs, including nonzero
constants. At each \((r,g(r))\), some \(Q_j\) vanishes: otherwise all
signs would stay fixed in a small open rectangle, forcing the whole
rectangle into the graph, whereas a graph has only one ordinate for a
fixed abscissa. Thus \[
Q(r,g(r))=0,\qquad Q(r,s)=\prod_{j=1}^{N}Q_j(r,s)\ne0.
\tag{HP18}
\] The nonzero product follows by successively multiplying leading
coefficients in the polynomial ring in each variable. All factors are
retained; this product is not substituted for the original graph
formula.

Write its complete coefficient expansion \[
Q(r,s)=\sum_{i=0}^{I}r^ia_i(s),\qquad
a_I(s)=\sum_{\ell=0}^{J}b_{I\ell}s^\ell,\quad b_{IJ}\ne0.
\tag{HP19}
\] Choose \(S\geq1\) with
\(\sum_{\ell<J}|b_{I\ell}|S^{\ell-J}\leq |b_{IJ}|/2\); then
\(|a_I(s)|\geq |b_{IJ}|s^J/2\) for \(s\geq S\). For nonzero \(a_i\),
\(i<I\), let \(J_i=\deg a_i\) and \(B_i\) be the absolute sum of all its
coefficients. A zero \(a_i\) contributes \(B_i=0\). Put \[
L=\max\big(\{0\}\cup\{J_i-J:i<I,\ a_i\ne0\}\big),
\quad \epsilon=\frac1{2(L+1)},\quad B=\sum_{i<I}B_i.
\tag{HP20}
\] For \(r\geq1\), \(S\leq s\leq r^\epsilon\), every lower contribution
is bounded by \[
\left|\sum_{i<I}r^ia_i(s)\right|
\leq B r^{I-1}s^{J+L}
\leq B r^{I-1+\epsilon L}s^J,\qquad
|r^Ia_I(s)|\geq \frac{|b_{IJ}|}{2}r^Is^J.
\tag{HP21}
\] For \(I\geq1\), choose \(R\geq1\) so that
\((2B/|b_{IJ}|)r^{-1+\epsilon L}<1\) for \(r>R\). Since
\(\epsilon L<1/2\), such an \(R\) exists. The leading term then strictly
exceeds the whole lower sum, forbidding \(Q(r,s)=0\) in that strip. For
\(I=0\), the lower sum is empty and the same nonvanishing is immediate.
Since \(g(r)\geq S\) eventually, HP18 proves \[
g(r)>r^\epsilon\quad\hbox{for all sufficiently large }r.
\tag{HP22}
\] The \(I=0\) case would contradict HP18 and \(g(r)\to\infty\), so it
cannot occur for this graph. The algebraic growth assertion has now been
proved.

\subsubsection{10.4. Local regularity forces the original complex zero
set
away}\label{local-regularity-forces-the-original-complex-zero-set-away}

Let \[
P(\zeta)=\sum_{|\alpha|\leq m}p_\alpha\zeta^\alpha,\quad
p_\alpha\in\mathbb C,\quad P\ne0,\qquad
P(D)=\sum_{|\alpha|\leq m}p_\alpha(-i)^{|\alpha|}\partial^\alpha
\tag{HP23}
\] be the original constant-coefficient operator. Suppose it is
hypoelliptic: on every open \(U\subset\mathbb R^n\), for every
distribution \(u\), smoothness of \(P(D)u\) implies smoothness of \(u\).
For a nonzero constant polynomial all nonzero-order derivatives vanish
and the reciprocal is constant. The zero-dimensional coordinate space
also has only constant polynomials. Hence assume \(n\geq1\) and actual
degree \(m\geq1\).

The original zero set \(Z=\{\zeta\in\mathbb C^n:P(\zeta)=0\}\) is closed
and nonempty. Its highest-degree homogeneous part is nonzero at some
real \(v\): a polynomial cannot vanish on all of \(\mathbb R^n\), by
induction in the number of variables, regarding it as a polynomial in
the last variable and using that a nonzero one-variable polynomial has
at most its degree many roots. The original \(P(tv)\) therefore has
degree \(m\) and a complex root by the proved root theorem. Fix one
original zero \(\zeta_0\).

On the original unit ball \(U=\{x:|x|<1\}\) with Lebesgue measure, put
\[
\mathcal E=\{u\in L^2(U):P(D)u=0\hbox{ as a distribution on }U\}.
\tag{HP24}
\] This is a closed Hilbert subspace. If \(u_k\to u\) in \(L^2(U)\),
Cauchy--Schwarz proves convergence against each test function and every
derivative appearing in HP23, so all the distributional equations pass
to the limit. Hypoellipticity makes every member smooth. The inclusion
\(\mathcal E\to C^\infty(U)\) has closed graph: simultaneous convergence
in \(L^2(U)\) and in all compact derivative seminorms has the same
distributional limit. Hence the two functions coincide almost
everywhere, with the smooth representative unique. The complete Fréchet
structure of \(C^\infty(U)\) is proved in the foundations. Their
closed-graph theorem supplies one constant \(A>0\) such that \[
\max_{1\leq j\leq n}|\partial_j u(0)|
\leq A\|u\|_{L^2(U)}\qquad(u\in\mathcal E).
\tag{HP25}
\] Indeed that maximum is one continuous target seminorm; finitely many
input seminorms of the Hilbert domain are all bounded by a constant
times its one original norm.

For \(\zeta=a+ib\in Z\), the actual
\(u_\zeta(x)=e^{i\langle x,\zeta\rangle}\) lies in \(\mathcal E\), since
\(D^\alpha u_\zeta=\zeta^\alpha u_\zeta\) and HP23 gives
\(P(D)u_\zeta=P(\zeta)u_\zeta=0\). Its derivative at zero is
\(i\zeta_j\), and \[
\|u_\zeta\|_{L^2(U)}^2
=\int_U e^{-2\langle x,b\rangle}\,dx
\leq |U|e^{2|b|},
\qquad
|\zeta|\leq\sqrt n\,A|U|^{1/2}e^{|b|}.
\tag{HP26}
\] The original sign, volume factor and every coordinate of \(\zeta\)
have been kept. Zeros with bounded imaginary part consequently have
bounded full norm.

For real \(\xi\), retain its exact distance \[
\delta(\xi)=\inf_{\zeta\in Z}|\xi-\zeta|
=\inf_{\zeta=a+ib\in Z}
\left(\sum_{j=1}^{n}(\xi_j-a_j)^2+\sum_{j=1}^{n}b_j^2\right)^{1/2}.
\tag{HP27}
\] The infimum is attained: using the competitor \(\zeta_0\), a
minimizing sequence can be restricted to a fixed closed bounded ball,
and \(Z\) is closed. Moreover \(\delta(\xi)\to\infty\) as
\(|\xi|\to\infty\). Otherwise some \(\xi_k\) tending to infinity and
\(\zeta_k\in Z\) would have \(|\xi_k-\zeta_k|\leq M\) for a fixed \(M\).
Then \(|\operatorname{Im}\zeta_k|\leq M\) but
\(|\zeta_k|\geq|\xi_k|-M\to\infty\), contradicting HP26. This is a
uniform statement outside large real balls.

Now define the actual minimum \[
g(r)=\min_{\substack{|\xi|=r,\ \xi\in\mathbb R^n\\
                    \zeta=a+ib\in Z}}
\left(\sum_{j=1}^{n}(\xi_j-a_j)^2+\sum_{j=1}^{n}b_j^2\right),
\qquad r>0.
\tag{HP28}
\] The sphere is nonempty and compact. The competitor \(\zeta_0\) bounds
this cost by \((r+|\zeta_0|)^2\). Restricting to costs no larger than
that also bounds \(|\zeta|\leq r+(r+|\zeta_0|)\), so a closed bounded
feasible set attains the minimum. HP27 implies \(g(r)\to\infty\).

The graph of \(g\) is semialgebraic by the projection theorem just
proved. Its exact formula says there exist \(\xi,a,b\) such that \[
\sum_j\xi_j^2=r^2,\quad
\operatorname{Re}P(a+ib)=0,\quad
\operatorname{Im}P(a+ib)=0,\quad
s=\sum_j(\xi_j-a_j)^2+\sum_jb_j^2,
\tag{HP29}
\] and for all \(\xi',a',b'\) satisfying the same sphere and both zero
equations, the full corresponding squared cost is at least \(s\). The
real and imaginary equations are real polynomials obtained by expanding
all of HP23, with every coefficient retained. Quantifier elimination
applies to this precise formula. Since \(g(r)>0\) eventually, HP22 gives
\(\epsilon>0,R\geq1\) such that \[
\delta(\xi)^2\geq g(|\xi|)>|\xi|^\epsilon\qquad(|\xi|>R).
\tag{HP30}
\] In particular \(P(\xi)\ne0\) there; the separation is at least
\(|\xi|^{\epsilon/2}\).

\subsubsection{10.5. Every derivative of the original
polynomial}\label{every-derivative-of-the-original-polynomial}

Fix such a real \(\xi\), and write \(\delta=\delta(\xi)>0\). For a
complex unit vector \(\eta\), retain the whole restriction
\(P(\xi+t\eta)\). If its actual degree \(d_\eta\geq1\), its original
factorization is \[
P(\xi+t\eta)=a_\eta\prod_{\ell=1}^{d_\eta}(t-t_\ell),
\quad
P(\xi)=a_\eta\prod_{\ell=1}^{d_\eta}(-t_\ell),
\quad
|t_\ell|\geq\delta,\quad d_\eta\leq m.
\tag{HP31}
\] Indeed \(\xi+t_\ell\eta\in Z\) and \(|t_\ell\eta|=|t_\ell|\). Here
\(a_\eta\) is the actual leading coefficient of the restriction.
Comparing both full products, with that coefficient present, yields \[
|P(\xi+t\eta)|
\leq |P(\xi)|\prod_{\ell=1}^{d_\eta}
\left(1+\frac{|t|}{|t_\ell|}\right)
\leq |P(\xi)|(1+|t|/\delta)^m.
\tag{HP32}
\] If the restriction is constant, it equals \(P(\xi)\) and the same
bound holds with its empty product. For \(w\ne0\), take
\(t=|w|,\eta=w/|w|\); the value \(w=0\) is immediate. On the full
polydisc \(|w_j|\leq\delta/(2\sqrt n)\), this gives \(|w|\leq\delta/2\)
and \(|P(\xi+w)|\leq(3/2)^m|P(\xi)|\).

Apply the positively oriented Cauchy coefficient formula successively on
the \(n\) original coordinate circles of radius
\(r_\xi=\delta/(2\sqrt n)\). The polynomial is entire and continuous on
their compact product, so finite-circle Fubini justifies the iterated
integration. The full formula is \[
\partial^\alpha P(\xi)=
\frac{\alpha!}{(2\pi i)^n}
\int_{|w_1|=r_\xi}\cdots\int_{|w_n|=r_\xi}
\frac{P(\xi+w)}{\prod_{j=1}^{n}w_j^{\alpha_j+1}}\,
dw_n\cdots dw_1,
\qquad
\left|\frac{\partial^\alpha P(\xi)}{P(\xi)}\right|
\leq\alpha!(3/2)^m(2\sqrt n)^{|\alpha|}\delta^{-|\alpha|}.
\tag{HP33}
\] The \(n\) circle lengths \((2\pi r_\xi)^n\) cancel the displayed
\((2\pi)^n r_\xi^n\) denominator exactly. Set \(c=\epsilon/2>0\), and
choose the finite common constant \[
C=\max\left(\{1\}\cup
\{\alpha!(3/2)^m(2\sqrt n)^{|\alpha|}:1\leq|\alpha|\leq m\}\right).
\tag{HP34}
\] HP30 and HP33 prove \[
\left|\frac{\partial^\alpha P(\xi)}{P(\xi)}\right|
\leq C|\xi|^{-c|\alpha|}
\qquad(\alpha\ne0,\ |\xi|>R).
\tag{HP35}
\] Derivatives with \(|\alpha|>m\) are exactly zero and are included.
This is the quantitative hypoelliptic polynomial implication for the
original operator and its original local regularity definition.

\subsubsection{10.6. The reciprocal, its full ordered derivatives and
its
cutoffs}\label{the-reciprocal-its-full-ordered-derivatives-and-its-cutoffs}

Choose \(p_{\alpha_*}\ne0\) with \(|\alpha_*|=m\). Then
\(\partial^{\alpha_*}P=\alpha_*!p_{\alpha_*}\), so \[
|P(\xi)|^{-1}\leq D_0|\xi|^{-cm},\qquad
D_0=C/|\alpha_*!p_{\alpha_*}|\quad(|\xi|>R).
\tag{HP36}
\] The selected original coefficient supplies only this bound; every
coefficient remains in \(P\).

For every multiindex \(\alpha\), the full derivative identity is \[
\partial^\alpha(P^{-1})=
\sum_{k=0}^{|\alpha|}(-1)^kP^{-k-1}
\!\!\sum_{\substack{\alpha_1+\cdots+\alpha_k=\alpha\\|\alpha_j|\geq1}}
\frac{\alpha!}{\alpha_1!\cdots\alpha_k!}
\prod_{j=1}^{k}\partial^{\alpha_j}P.
\tag{HP37}
\] For \(k=0\), the inner sum has one empty tuple if \(\alpha=0\), with
coefficient and product one, and is zero otherwise. To prove the
identity at a nonzero point \(P(\xi)\), expand the entire finite Taylor
polynomial of \(P(\xi+w)-P(\xi)\). Modulo monomials of degree greater
than \(|\alpha|\), multiply \[
\sum_{k=0}^{|\alpha|}(-1)^kP(\xi)^{-k-1}
\big(P(\xi+w)-P(\xi)\big)^k
\tag{HP38}
\] by the full \(P(\xi+w)\). The product is
\(1+(-1)^{|\alpha|}P(\xi)^{-|\alpha|-1}
(P(\xi+w)-P(\xi))^{|\alpha|+1}\); its last term has no coefficient of
degree at most \(|\alpha|\). The coefficients in HP38 are therefore the
reciprocal Taylor coefficients, by recursive comparison in the identity
\(P\cdot P^{-1}=1\). The coefficient of \(w^\alpha\), multiplied by
\(\alpha!\), gives exactly HP37, including every ordered tuple and sign.

Apply HP35 to each derivative factor, and HP36 to the remaining
\(P^{-1}\). Then \[
|\partial^\alpha(P^{-1})(\xi)|
\leq D_\alpha|\xi|^{-cm-c|\alpha|},\qquad
D_\alpha=D_0\sum_{k=0}^{|\alpha|}C^k
\!\!\sum_{\substack{\alpha_1+\cdots+\alpha_k=\alpha\\|\alpha_j|\geq1}}
\frac{\alpha!}{\alpha_1!\cdots\alpha_k!}.
\tag{HP39}
\] The empty tuple convention remains and zero higher derivatives remain
zero in the exact identity. Put \(\rho=\min(c,1)>0\). For
\(|\xi|\geq1\), the full comparison
\(\langle\xi\rangle/|\xi|\leq\sqrt2\) gives \[
|\partial^\alpha(P^{-1})(\xi)|
\leq 2^{\rho(m+|\alpha|)/2}D_\alpha
\langle\xi\rangle^{-\rho m-\rho|\alpha|}
\qquad(|\xi|>R).
\tag{HP40}
\] Only the estimate uses \(\rho\); the original polynomial and its
\(c\)-estimates have not been replaced.

Choose \(R_2>R_1>R\) and an actual smooth cutoff \(\theta\) zero for
\(|\xi|\leq R_1\) and one for \(|\xi|\geq R_2\). Define \(a=\theta/P\)
on the nonzero domain outside the inner ball, and define it as zero on a
neighborhood of that ball. The definitions agree smoothly because the
cutoff vanishes there. Every derivative is \[
\partial^\alpha a=
\sum_{\beta\leq\alpha}\binom{\alpha}{\beta}
(\partial^\beta\theta)\,\partial^{\alpha-\beta}(P^{-1}).
\tag{HP41}
\] The term \(\beta=0\) has HP40's frequency bound. Every positive-order
cutoff derivative is supported in the compact nonzero annulus, where
every reciprocal derivative and weight is continuous and bounded. On the
inner zero region all derivatives are zero. Thus
\(a\in S^{-\rho m}_{\rho,0}\), with every cutoff contribution retained.
It is independent of \(x\), so its positive-order \(x\)-derivatives
vanish exactly. A nonzero constant \(P\) has its constant reciprocal in
\(S^0_{\rho,0}\) for any allowed \(\rho\). No reciprocal is asserted
across real zeros of a nonconstant \(P\).

The finite sign-count inversion is the classical Tarski-query mechanism.
Cohen and Mahboubi, \emph{Formal proofs in real algebraic geometry: from
ordered fields to quantifier elimination},
\href{https://arxiv.org/abs/1201.3731v2}{arXiv:1201.3731v2}, original
part5.tex, lines 226--328, gives the query/count relation in its own row
order. Original part6.tex, lines 1--158 and 379--459, describes its
formal quantifier-elimination setting. The present proof supplies the
full trace-form calculation, projection argument and analytic receiving
estimates; no omitted formal proof is imported.

Their row vectors are \(n=(N_+,N_-,N_0)\), \(t=(T_1,T_2,T_0)\), with
\(t=nS\) and \[
S=\begin{pmatrix}1&1&1\\-1&1&1\\0&0&1\end{pmatrix},
\quad R=\begin{pmatrix}0&1&0\\0&0&1\\1&0&0\end{pmatrix},
\quad L=\begin{pmatrix}0&0&1\\1&0&0\\0&1&0\end{pmatrix}.
\tag{HP42}
\] Our column vectors are \(N=Rn^T\), \(T=Lt^T\), and direct
multiplication gives \(V=LS^TR^T\), with \(\det S=\det V=2\). This
retains the source's factor two and exact ordering maps. Their article
supplies historical context for the query mechanism; the complete trace,
projection, growth and operator argument has been proved above.

Here is the resulting adapter to the present calculus. If \(P\) has
degree \(d>0\), replace \(c\) by \(\rho=\min(c,1)>0\). A derivative of
total degree \(d\) is a nonzero constant for some multiindex, so (E48)
yields \(|P(\xi)|^{-1}\leq C|\xi|^{-d\rho}\). Repeated differentiation
of \(P^{-1}P=1\) expresses each derivative of \(P^{-1}\) as a finite sum
of \[
P^{-1}\prod_{\nu=1}^k
\frac{\partial^{\alpha_\nu}P}{P},
\qquad \alpha_\nu\ne0,\qquad
\alpha_1+\cdots+\alpha_k=\alpha,
\tag{E49}
\] with numerical coefficients. Induction follows by differentiating one
factor at a time, using the quotient rule; total derivative degree
increases by one in each resulting term. Equation (E48) bounds (E49) by
\(C_\alpha\langle\xi\rangle^{-d\rho-\rho|\alpha|}\). A smooth cutoff
\(\theta\) that is zero on a sufficiently large ball and one outside a
larger ball therefore gives
\(\theta(\xi)/P(\xi)\in S^{-d\rho}_{\rho,0}\), independent of \(x\).
Cutoff derivatives occur only in a compact region where \(P\ne0\) and do
not change the conclusion. A nonzero constant polynomial has its
constant reciprocal in \(S^0_{\rho,0}\) for any allowed \(\rho\). No
reciprocal is asserted across real zeros of a nonconstant polynomial.
The quantitative hypoelliptic implication and the full reciprocal
adapter are now proved.

\textbf{Editorial completion: the exact converse for the original
polynomial.} Retain the nonzero polynomial \(P\), all its complex
coefficients, \(D=-i\partial\), and the original complex zero set
\(Z=\{\zeta\in\mathbb C^n:P(\zeta)=0\}\). For \(n\ge1\) and
\(m=\deg P\ge1\), the following three statements are equivalent: \[
\begin{array}{ll}
\text{(i)}&P(D)u\in C^\infty(U)\Longrightarrow u\in C^\infty(U)
       \text{ for every open }U\text{ and }u\in\mathcal D'(U);\\
\text{(ii)}&\delta(\xi)=\inf_{\zeta\in Z}|\xi-\zeta|\longrightarrow\infty
       \text{ as real }|\xi|\longrightarrow\infty;\\
\text{(iii)}&\text{There are }R,C,c>0\text{ such that, for }|\xi|>R,\ P(\xi)\ne0\text{ and}\\
&|\partial^\alpha P(\xi)|\le C|P(\xi)|\,|\xi|^{-c|\alpha|}
       \quad\text{for every nonzero multiindex }\alpha.
\end{array}
\tag{HC1}
\] Derivatives above the full original degree are zero, and remain part
of (iii). Statement (i) implies (ii) by HP23--HP27: the complete
closed-graph estimate on the original unit-ball solution space bounds
every zero's real part by its imaginary part through the actual
exponential \(e^{i\langle x,\zeta\rangle}\) and the original ball
volume. The attained distance argument there gives (ii). No reciprocal
has been evaluated at a real zero.

The proof HP28--HP35 needs only (ii) after that point. Specifically,
retain its exact minimum of the squared distance on the sphere of real
radius \(r\), all real and imaginary coordinates of \(\zeta\), and its
full quantifier formula. Statement (ii) makes that minimum tend to
infinity. The proved projection and graph-growth argument HP16--HP22
supplies a positive power lower bound. The full restriction
\(P(\xi+t\eta)\), its leading coefficient and every root give
HP31--HP32, and its original polydisc estimate gives HP33--HP35. Thus
(ii) implies (iii) with the original constants and derivative factors,
without requiring (i) as an extra hypothesis.

In this polynomial application, the circle step can also be checked
without importing a general analytic theorem. Retain the finite
expansion
\(P(\xi+w)=\sum_{|\beta|\le m}\partial^\beta P(\xi)w^\beta/\beta!\). On
each positively oriented circle \(w_j=re^{it_j}\), its original
derivative is \(dw_j=ire^{it_j}dt_j\); direct integration gives
\(\int_{|w_j|=r}w_j^{\beta_j-\alpha_j-1}dw_j=2\pi i\) if
\(\beta_j=\alpha_j\), and zero otherwise, including negative nonzero
exponent differences. Integrating the finite expansion in the original
coordinate order therefore proves \[
\partial^\alpha P(\xi)=\frac{\alpha!}{(2\pi i)^n}
 \int_{|w_1|=r}\cdots\int_{|w_n|=r}
 \frac{P(\xi+w)}{w_1^{\alpha_1+1}\cdots w_n^{\alpha_n+1}}
 \,dw_n\cdots dw_1.
\tag{HC2}
\] For \(|\alpha|>m\), no matching monomial occurs and both sides are
zero. At the original radius \(r=\delta(\xi)/(2\sqrt n)\), HP32 bounds
the numerator by \((3/2)^m|P(\xi)|\). The full circle lengths
\((2\pi r)^n\), denominator \(r^{|\alpha|+n}\), and prefactor
\(\alpha!/(2\pi)^n\) give exactly
\(\alpha!(3/2)^m(2\sqrt n)^{|\alpha|}|P(\xi)|\delta(\xi)^{-|\alpha|}\).
There are finitely many nonzero original derivatives through degree
\(m\), so one maximum of these explicit constants gives the common \(C\)
in (iii). The earlier general circle proof remains a separate valid
analytic result.

We prove (iii) implies (i), including the actual local distribution map.
HP36--HP41 is the full reciprocal calculation for precisely (iii). A
nonzero derivative of degree \(m\) supplies HP36's lower growth bound
for \(|P|\); HP37 retains every ordered reciprocal tuple, sign and
factorial. With \(\rho=\min(c,1)\), it proves that the actual cutoff
reciprocal of HP41 belongs to \(S^{-\rho m}_{\rho,0}\). Denote that same
symbol by \(q(\xi)=\theta(\xi)/P(\xi)\) on its original nonzero domain
and by zero on the inner neighborhood, and put \(r(\xi)=1-\theta(\xi)\).
The actual cutoff has \(R_2>R_1>R\), is zero for \(|\xi|\le R_1\), and
is one for \(|\xi|\ge R_2\). Consequently \(r\in C_c^\infty\), and on
the whole frequency space, including the original real zeros, \[
Pq=qP=\theta,\qquad
P(D)q(D)=q(D)P(D)=\theta(D),\qquad
I=q(D)P(D)+r(D).
\tag{HC3}
\] These are actual Fourier multiplier identities on \(\mathcal S\) and
\(\mathcal S'\), using the unchanged inverse factor \((2\pi)^{-n}\).
Polynomial multiplication, the smooth multiplier with polynomially
bounded derivatives, and their transpose actions are all defined on
these domains. Equality of their full multiplier products proves both
orders in (HC3).

Let \(K_q=\mathcal F^{-1}q\) and \(K_r=\mathcal F^{-1}r\), as tempered
distributions with that factor. The first kernel is smooth off zero.
Here is the complete estimate. For an arbitrary multiindex \(\beta\),
choose an integer \(M\ge0\) with \(2M\rho>n-\rho m+|\beta|\). The
distributional Fourier differentiation identities give, for \(z\ne0\),
\[
\partial_z^\beta K_q(z)=(2\pi)^{-n}|z|^{-2M}
 \int e^{i\langle z,\xi\rangle}(-\Delta_\xi)^M[(i\xi)^\beta q(\xi)]\,d\xi.
\tag{HC4}
\] Indeed multiplication by \(|z|^{2M}\) is inverse Fourier
transformation of \((-\Delta_\xi)^M\), and the amplitude on the right is
integrable by the following full expansion: \[
\begin{aligned}
(-\Delta_\xi)^M[(i\xi)^\beta q]
={}&(-1)^Mi^{|\beta|}
 \sum_{|\gamma|=M}\frac{M!}{\gamma!}
 \sum_{\substack{\nu\le2\gamma\\\nu\le\beta}}
 \binom{2\gamma}{\nu}\frac{\beta!}{(\beta-\nu)!}
 \xi^{\beta-\nu}\partial^{2\gamma-\nu}q.
\end{aligned}
\tag{HC5}
\] Every displayed term has large-frequency bound with power
\(|\beta|-\rho m-2M\rho-(1-\rho)|\nu|\le|\beta|-\rho m-2M\rho<-n\). The
bounded frequency region is smooth and finite. The full original
coefficients, all zeros of higher polynomial derivatives, and all
compact cutoff terms have therefore been retained. This proves absolute
integrability and continuity of the right side of (HC4) on the punctured
space. The distributional identity identifies it with the corresponding
distributional derivative there, so it does not assume vanishing of an
unproved cutoff boundary term.

For clarity, continuous distributional derivatives of every order imply
classical smoothness here. On any smaller ball whose closure avoids
zero, convolve locally with a compact mollifier of integral one. Each
continuous derivative converges uniformly on yet smaller compact sets to
its displayed derivative. For a coordinate segment in such a set, the
fundamental theorem of calculus for the convolutions, followed by
uniform convergence of the function and its first derivative, gives the
same theorem for the limiting function. It is therefore differentiable
with that derivative. Apply the argument to every derivative in turn.
This proves the asserted smoothness with all derivatives in (HC4). The
kernel \(K_r\) is smooth everywhere by the absolutely convergent
integrals of \((i\xi)^\beta r(\xi)\) for every \(\beta\), retaining
\((2\pi)^{-n}\).

We also need the precise receiving map for a compact distribution,
rather than a kernel picture. If \(g\) has compact support disjoint from
a relatively compact open neighborhood \(W\), choose
\(\eta\in C_c^\infty\) equal to one near \(\operatorname{supp}g\), with
support disjoint from \(\overline W\). On \(W\) the actual multiplier
\(q(D)g\) is the function \[
(q(D)g)(x)=\langle g(y),\eta(y)K_q(x-y)\rangle.
\tag{HC6}
\] The pairing is complex-linear, with no conjugation. A compact
distribution has a finite derivative-order bound on the fixed compact
support of \(\eta\). All derivatives of \(K_q\) are continuous on the
resulting compact set of differences away from zero. Difference
quotients and their Taylor integral remainders converge uniformly
through that finite order. Applying the bound for \(g\) proves arbitrary
differentiability in \(x\), with derivative obtained by replacing
\(K_q\) by \(\partial_x^\beta K_q\) in (HC6). To identify the function
with the multiplier, pair it with a test function supported in \(W\).
Moving its compact integral through the continuous distribution pairing
gives the distribution convolution \(K_q*g\); Fourier transformation of
convolution with a compact distribution is multiplication of its Fourier
transform by \(q\), with the original inverse factor. This identity
follows first for compact smooth approximants of \(g\) by the full
Fourier integral, and then by their distribution limit; multiplication
by \(q\) is continuous on \(\mathcal S'\). Thus (HC6) is the required
restriction of the actual multiplier. The same argument for \(K_r\)
proves that \(r(D)g\) is smooth everywhere for every compact
distribution \(g\), without a separation condition.

Finally let \(u\in\mathcal D'(U)\), with \(f=P(D)u\in C^\infty(U)\). Fix
\(x_0\in U\); choose \(\chi\in C_c^\infty(U)\) equal to one on an open
neighborhood \(V\) of \(x_0\), and extend \(v=\chi u\) by zero to its
actual compact tempered distribution. Writing the unchanged original
polynomial as \(P(\xi)=\sum_{|\alpha|\le m}p_\alpha\xi^\alpha\), the
full Leibniz formula is \[
\begin{aligned}
P(D)v&=\chi f+g,\\
g&=\sum_{|\alpha|\le m}p_\alpha
 \sum_{0<\beta\le\alpha}\binom\alpha\beta
 (D^\beta\chi)D^{\alpha-\beta}u.
\end{aligned}
\tag{HC7}
\] Each term is formed on \(U\), has compact support inside \(U\), and
then extends by zero. The full factors
\(D^\beta\chi=(-i)^{|\beta|}\partial^\beta\chi\) and
\(D^{\alpha-\beta}=(-i)^{|\alpha-\beta|}\partial^{\alpha-\beta}\)
remain. Every term of \(g\) is supported outside \(V\). Choose a smaller
neighborhood \(W\) of \(x_0\) with closure in \(V\). Applying (HC3) to
the actual \(v\) gives \[
v=q(D)(\chi f)+q(D)g+r(D)v.
\tag{HC8}
\] The first term is Schwartz, because \(\chi f\) is compactly supported
smooth and the polynomial derivative bounds of \(q\) preserve every
Schwartz seminorm under multiplication in Fourier space. The second is
smooth on \(W\) by (HC6), and the third is smooth everywhere by its
proved compact-distribution map. Therefore \(u=v\) is smooth near
\(x_0\). Since \(U,u,x_0\) were arbitrary, (i) follows. This proves all
directions of (HC1), without assuming the local regularity conclusion or
suppressing the actual commutator contributions.

A nonzero constant polynomial is separately hypoelliptic: \(P(D)u=p_0u\)
and its actual inverse is multiplication by \(p_0^{-1}\). Its zero set
is empty, so (HC1)'s nonconstant distance statement is not used for it.
In dimension zero the space has one point and the same constant argument
applies. The zero polynomial is excluded throughout. These are editorial
consequences of the original full polynomial calculation, with no
novelty claim; its original forward statement and historical attribution
remain identifiable.

\subsection{11. Worked examples}\label{worked-examples}

For a bounded smooth matrix function \(B(x)\) with bounded derivatives
and a constant matrix \(M\), take \(a(\xi)=M\xi_j\), \(b(x,\xi)=B(x)\).
Formula (E21) terminates because \(a\) is frequency-linear: \[
a\circ b=M\xi_jB+M D_jB,
\qquad b\circ a=BM\xi_j.
\tag{E46}
\] One can also prove termination directly from
\(MD_j(Bu)=MB D_ju+M(D_jB)u\); kernel injectivity identifies the
symbols. The leading commutator contains \((MB-BM)\xi_j\), which has
order one. Scalar Poisson-bracket cancellation cannot remove it.

For a scalar example,
\(a(x,\xi)=2+\tfrac14\sin x_1+\langle\xi\rangle^2\) is uniformly
elliptic of order two. The initial inverse is its ordinary reciprocal,
of order minus two, but it is usually not the inverse symbol for
composition. The first correction term in \(a\circ a^{-1}-1\) is
\(\sum_j(\partial_{\xi_j}a)D_j(a^{-1})\), which the general calculus
places in order at most minus one. In this particular example its base
derivatives give the sharper order minus three. The theorem corrects the
error recursively without imposing any periodic-boundary or compact-base
condition.

The loss of gain at equality is visible in one dimension. Fix \(0<r<1\),
let \(R_j=2^{Lj}\) for a fixed large integer \(L\), and put
\(\lambda_j=R_j^r\). Start the sequence at an index large enough that
intervals of radius \(3\lambda_j\) about \(R_j\) are disjoint. Choose
\(\varphi,\omega\in C_c^\infty(\mathbb R)\), supported in \((-2,2)\) and
\((-1/10,1/10)\), respectively, with \(\omega(0)=1\) and
\(\varphi(1)\ne\varphi(0)\). Define \[
a(\xi)=\sum_j\varphi((\xi-R_j)/\lambda_j),
\qquad
b(x,\xi)=\sum_j e^{i\lambda_j x}
\omega((\xi-R_j)/\lambda_j).
\tag{E47}
\] Disjoint supports and \(\langle\xi\rangle\asymp R_j\) on each term
prove \(a,b\in S^0_{r,r}\): a base derivative costs \(\lambda_j\), and a
frequency derivative costs \(\lambda_j^{-1}\). On the support of the
\(j\)-th term of \(b\), the exact frequency-shift calculation after
(E22) gives
\((a\circ b)(x,\xi)=e^{i\lambda_jx}a(\xi+\lambda_j)\omega((\xi-R_j)/\lambda_j)\).
At \(\xi=R_j\), the difference from \(ab\) has absolute value
\(|\varphi(1)-\varphi(0)|>0\). It therefore belongs to no strictly
negative symbol order. Nevertheless (E28), (E37) and (E40) apply to both
operators. This example separates boundedness from decreasing-order
asymptotics.

\subsection{12. Exercises and solutions}\label{exercises-and-solutions}

\textbf{Problem 1.} Let \(m_{2j}=-j\), \(m_{2j+1}=3-j\), and choose
arbitrary \(a_j\in S^{m_j}_{\rho,\delta}\). What order is guaranteed for
the remainder after the first four terms? Does summing the even terms
first preserve the same meaning?

\textbf{Solution.} The remaining orders start with \(m_4=-2\),
\(m_5=1\), \(m_6=-3\), \(m_7=0\); their maximum is one. Thus (E6) gives
remainder order one after four terms. A genuine permutation of the
natural numbers preserves the expansion modulo smoothing. The
instruction ``all even terms first, then all odd terms'' is not a
permutation indexed by the natural numbers, since no odd term occurs at
a finite position. One may separately sum both subsequences and add the
resulting two symbols; each tail still has orders tending to minus
infinity, and (E6) verifies that the resulting sum has the original
expansion. This is an additional two-sum construction, not a nonexistent
enumeration.

\textbf{Problem 2.} In one dimension compute the exact left symbol of
\(D^2 M_{e^{3ix}}\), where \(M_f\) denotes multiplication by \(f\).
Check the coefficient of the first derivative term.

\textbf{Solution.} Since \(D(e^{3ix}u)=e^{3ix}(D+3)u\), the symbol is
\(e^{3ix}(\xi+3)^2=e^{3ix}(\xi^2+6\xi+9)\). In (E21), the first
derivative term is \((2\xi)D_xe^{3ix}=6\xi e^{3ix}\); the second is
\((2)D_x^2e^{3ix}/2!=9e^{3ix}\). Higher frequency derivatives of
\(\xi^2\) vanish. The sign is positive in the frequency shift, which
agrees with (E13) and (E22).

\textbf{Problem 3.} Show explicitly why the proof of (E28) gives no
uniform constant as \(\delta\uparrow1\), even when \(\delta\leq\rho\).

\textbf{Solution.} With fixed Taylor order \(K\), the error norms in
(E32) sum to at most a constant times \(\sum_{j\geq1}2^{-jK(1-\delta)}
=2^{-K(1-\delta)}/(1-2^{-K(1-\delta)})\). This tends to infinity. The
finite-band part remains bounded, but the estimate for the errors no
longer sums at \(\delta=1\). This is a precise failure of this proof's
uniform bound; it is not a proof that every particular type-\((1,1)\)
operator is unbounded.

\textbf{Problem 4.} Construct a distribution in \(B^0_{2,\infty}\) that
does not belong to \(H^0\), and explain why Schwartz density must not be
used in the \(B^0_{2,\infty}\) proof.

\textbf{Solution.} For every \(j\geq1\), choose a smooth Fourier
function \(f_j\) supported strictly inside \(A_j\), normalized so that
the inverse Fourier transform has \(L^2\) norm one. The distribution
with Fourier transform \(\sum_jf_j\) is tempered: on each annulus
Cauchy--Schwarz bounds its pairing with a Schwartz function by that
function's annular \(L^2\) norm, and those norms form a summable rapidly
decreasing sequence. Every dyadic \(L^2\) norm is one, so its
\(B^0_{2,\infty}\) norm is one, while its squared \(H^0\) norm is
\(\sum_j1=\infty\). The tails of its finite annular sums all have Besov
norm one. For any Schwartz function the annular norms tend to zero, so
its distance from this distribution in that norm is at least one. The
proof after (E39) instead uses convergence in a strictly weaker Sobolev
space and estimates each fixed output band; it remains valid at the
sequence endpoint.

\textbf{Problem 5.} Suppose \(a\in S^m_{\rho,\delta}\), \(\delta<\rho\),
is square-matrix elliptic, \(b\in S^{-m}_{\rho,\delta}\), and
\(a\circ b-I\) is smoothing. A proposed proof of \(b\circ a-I\)
interchanges \(a\) and \(b\) in every differential coefficient. Identify
the problem and give a valid replacement.

\textbf{Solution.} Matrix factors in (E21) do not commute, and the
derivatives fall on different factors in the two orders. Even their
leading products need not agree. Use (E43) to obtain a uniform inverse
bound for the pointwise matrix \(a\), construct a two-sided parametrix
\(c\) by (E44)--(E45), and compute
\(b-c=c\circ(a\circ b-I)-(c\circ a-I)\circ b\). Both terms are smoothing
because it is a two-sided ideal. Hence
\(b\circ a-I=(b-c)\circ a+(c\circ a-I)\) is smoothing. Associativity and
the ideal property replace commutativity.

\textbf{Problem 6.} Let \(a(x,\xi)=f(x)g(\xi)\), where
\(f\in C^{n+1}(\mathbb R^n)\) and all its derivatives through order
\(n+1\) are integrable, while \(g\) is only bounded and measurable.
Prove \(L^2\) boundedness and state whether (E21) follows.

\textbf{Solution.} Condition (E34) holds with
\(M=\|g\|_\infty\sum_{|\alpha|\leq n+1}\|D^\alpha f\|_1\), so the
rough-frequency theorem gives the asserted bound. It does not imply a
differentiated composition formula: \(g\) need not have a first
frequency derivative, whereas (E21) is a theorem about smooth symbols
with all the specified derivative bounds. The simpler factorization is
multiplication by \(f\) after the bounded Fourier multiplier \(g(D)\);
the hypotheses also imply \(f\) is bounded, as follows from the
integrable Fourier bound (E35).

\subsection{Further questions}\label{further-questions}

The next structural questions have distinct hypotheses: coordinate
changes introduce a lower bound on \(\delta\) relative to \(1-\rho\);
positivity estimates depend on more than boundedness; and boundary
kernels require additional structure near the normal-frequency
directions. The endpoint example (E47) is a useful test for any proposed
extension that claims a decreasing symbolic error at \(\delta=\rho\).

The classical summation results cited below do not themselves supply the
entire \(\delta>\rho\) range or the Besov sequence endpoints proved
here.

\subsection{References}\label{references}

\begin{itemize}
\tightlist
\item
  Nicolas Lerner,
  \href{https://webusers.imj-prg.fr/~nicolas.lerner/ch1booklerner.pdf}{first
  chapter}, Lemma 1.1.24 and Remark 1.1.28;
  \href{https://webusers.imj-prg.fr/~nicolas.lerner/ch2booklerner.pdf}{metric
  chapter}, Lemma 2.2.18 and Theorems 2.3.7 and 2.5.1. These are the
  author-hosted September 2009 chapter files used for the earlier source
  comparison.
\item
  Richard Melrose,
  \href{https://math.mit.edu/~rbm/18.157-F05.pdf}{Pseudodifferential-operator
  lectures}, version 0.7E, 29 November 2006.
\end{itemize}

\end{document}