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\begin{document}

\section{Quadratic Fourier multipliers at a moving
scale}\label{quadratic-fourier-multipliers-at-a-moving-scale}

A quadratic Fourier multiplier spreads a localized function in
directions set by its quadratic phase. The local Taylor polynomial and
the tail require different estimates. This lesson proves both and
combines them for symbols measured by a slowly varying metric.

\href{metric-localization.pdf}{Localizing symbols with moving metrics}
supplies the symbol class, partitions and bounded compactly supported
approximation. \href{prerequisite-bridges.pdf}{Fourier transforms, finite
spectra and convex separation} supplies Fourier inversion, Plancherel,
the real spectral theorem and convex separation, with the same transform
convention. Basic references are {[}Lerner 2005{]}; all estimates used
below are proved in this lesson or those two prerequisite lessons.

\subsection{1. The phase and its dual
distance}\label{the-phase-and-its-dual-distance}

Let \(E=\mathbb R^d\), with dual pairing \(\langle X,\Xi\rangle\). We
use

\[
\widehat u(\Xi)=\int e^{-i\langle X,\Xi\rangle}u(X)\,dX,\qquad
u(X)=(2\pi)^{-d}\int e^{i\langle X,\Xi\rangle}\widehat u(\Xi)\,d\Xi,
\qquad D=-i\partial.
\]

Let \(A\) be any real quadratic form on \(E^*\). Its associated
symmetric linear map \(B:E^*\to E\) is specified by
\(A(\Xi)=\langle B\Xi,\Xi\rangle\); in particular \(A'(\Xi)=2B\Xi\).
Define

\[
T_Au=\mathcal F^{-1}(e^{iA}\widehat u).
\]

Multiplication by \(e^{iA}\) preserves Schwartz space: every derivative
of that function is a polynomial times it, so the product rule controls
all Schwartz seminorms. Thus \(T_A\) acts on Schwartz functions, and by
duality on tempered distributions. It commutes with constant-coefficient
derivatives.

For a positive-definite quadratic form \(Q\) on \(E\), set

\[
Q^A(Z)=\sup_{\Xi:\,Q(B\Xi)<1}|\langle Z,\Xi\rangle|^2.
                                                               \tag{G1}
\]

The value is \(+\infty\) unless \(Z\in\operatorname{ran}B\). Indeed, if
\(Z\) does not annihilate \(\ker B\), a multiple of a kernel vector
makes the numerator arbitrarily large with zero denominator. Conversely,
on the quotient \(E^*/\ker B\), the form \(Q(B\Xi)\) is positive
definite, so its dual is a finite positive-definite form on the
annihilator \((\ker B)^\perp=\operatorname{ran}B\). This also proves
that (G1), restricted to that range, is a quadratic form.

In coordinates in which \(Q\) is Euclidean and \(B\) is diagonal with
eigenvalues \(\beta_j\),

\[
Q^A(Z)=\sum_{\beta_j\ne0}\frac{Z_j^2}{\beta_j^2}
\quad\text{if }Z_j=0\text{ whenever }\beta_j=0.
                                                               \tag{G2}
\]

Off that subspace it is infinite. Consequently

\[
h_Q=\sup_{\Xi\ne0}\frac{|A(\Xi)|}{Q^{-1}(\Xi)}
=\max_j|\beta_j|,
\qquad
Q(Z)\leq h_Q^2Q^A(Z).                                         \tag{G3}
\]

Equivalently \(h_Q^2=\sup_{Z\ne0}Q(Z)/Q^A(Z)\), where the ratio is zero
off \(\operatorname{ran}B\). Formula (G2) proves this by maximizing on
the eigen-directions with \(\beta_j\ne0\); when \(B=0\), every ratio in
this convention is zero.

Here \(Q^{-1}\) is the ordinary dual quadratic form. If \(B=0\), take
\(h_Q=0\); the inequality means \(Q(0)=0\) on the finite domain of
\(Q^A\), with the off-domain case interpreted directly rather than as
the undefined product \(0\cdot\infty\).

If \(Q_1\leq M Q_2\), their phase-dual forms obey
\(Q_1^A\geq M^{-1}Q_2^A\), because the defining ellipsoid for the first
supremum contains a suitable dilation of the ellipsoid for the second.
Both are infinite off the same range, and the assertion on that range is
the usual reversal of positive quadratic forms.

\subsection{2. A Taylor estimate for the local
multiplier}\label{a-taylor-estimate-for-the-local-multiplier}

\textbf{Theorem 2.1 (uniform Taylor estimate).} Let \(K=\{X:Q(X)<1\}\),
\(u\in C_c^\infty(K)\), and let \(s\) be an integer with \(s>d/2\). For
every integer \(N\geq0\),

\[
\sup_X\left|T_Au(X)-\sum_{j<N}\frac{(iA(D))^ju(X)}{j!}\right|
\leq \frac{C_{d,s}}{N!}
\max_{0\leq j\leq s}\sup_{Y\in K}|A(D)^Nu|_{j,Q}(Y).          \tag{G4}
\]

The constant is independent of \(A\) and \(Q\); for \(N=0\) the sum is
empty.

\textbf{Proof.} Make a linear change of variables that sends \(Q\) to
the Euclidean metric. The Fourier Jacobian and inverse-transform
Jacobian cancel, so it suffices to prove the same estimate on the
Euclidean unit ball. For real \(t\), Taylor's integral remainder gives

\[
\left|e^{it}-\sum_{j<N}(it)^j/j!\right|\leq |t|^N/N!.
\]

For \(N=0\) this is simply \(|e^{it}|=1\). Fourier inversion therefore
bounds the left side of (G4) by
\((2\pi)^{-d}\|\widehat{A(D)^Nu}\|_{L^1}/N!\). For any smooth \(v\)
supported in the unit ball, Cauchy--Schwarz gives

\[
\|\widehat v\|_{L^1}
\leq \left(\int\langle\Xi\rangle^{-2s}\,d\Xi\right)^{1/2}
\|\langle\Xi\rangle^s\widehat v\|_{L^2}
\leq C_{d,s}\sum_{|\alpha|\leq s}\|\partial^\alpha v\|_{L^2}.
\]

The integral is finite exactly for \(s>d/2\). Plancherel proves the last
estimate, and the fixed volume of the support bounds the \(L^2\) norms
by derivative suprema. This proves (G4), including its uniformity. \(\square\)
\textbf{Editorial calculation in the original coordinates.} Here are all
the Jacobian, derivative and norm factors in that comparison. The given
form remains \(Q(X)=X^{\mathsf T}GX\), with the original positive matrix
\(G\), original measure \(dX\), phase \(A\) and function \(u\). Choose
an invertible comparison map \(L\) with \(L^{\mathsf T}GL=I_d\), by the
finite Gram construction in
\href{stable-prerequisite-bridges.pdf}{Polynomial and contour tools}. Put
\(J=|\det L|\), so \(J^2\det G=1\) and \(G^{-1}=LL^{\mathsf T}\). Define
the auxiliary pullbacks \(u_L(y)=u(Ly)\) and
\(A_L(\eta)=A(L^{-\mathsf T}\eta)\); these are comparison maps, and do
not replace the original working objects. The full affine change of
measure gives \[
 \widehat u(\Xi)=J\widehat{u_L}(L^{\mathsf T}\Xi),\quad
 d\Xi=J^{-1}d\eta\ (\eta=L^{\mathsf T}\Xi),\quad
 A_L(\eta)=\langle L^{-1}BL^{-\mathsf T}\eta,\eta\rangle.
 \tag{GJ1}
\] Consequently the complete comparison of the actual Fourier operators
is \[
 (T_Au)(Ly)=(2\pi)^{-d}\int e^{i\langle y,\eta\rangle}
 e^{iA(L^{-\mathsf T}\eta)}J\widehat{u_L}(\eta)J^{-1}\,d\eta
 =(T_{A_L}u_L)(y).
 \tag{GJ2}
\] Both Jacobians and the original inverse coefficient are displayed.
The chain rule gives \(D_y=L^{\mathsf T}D_X\) on pullbacks, hence
\(A_L(D_y)u_L=(A(D_X)u)_L\) and the same identity for every recursively
applied power. Write \(v=A(D_X)^Nu\), \(v_L=v\circ L\), and let \(T_j\)
be the actual columns of \(L\). They satisfy \(Q(T_j)=1\). Every mixed
derivative \(\partial_y^\alpha v_L\) is the pullback of the ordered
original directional derivative
\(\partial_{T_1}^{\alpha_1}\cdots\partial_{T_d}^{\alpha_d}v\), with all
its factors retained.

For the integer \(s>d/2\), put
\(I_{d,s}=\int_{\mathbb R^d}(1+|\eta|^2)^{-s}\,d\eta\), and
\(c_{s,\alpha}=s!/(\alpha!(s-|\alpha|)!)\) for \(|\alpha|\leq s\). The
complete multinomial identity is
\((1+|\eta|^2)^s=\sum_{|\alpha|\leq s}c_{s,\alpha}\eta^{2\alpha}\). Its
coefficient sum is \((1+d)^s\). The integral is finite: on
\(|\eta|\leq1\) the integrand is at most one; on
\(2^k<|\eta|\leq2^{k+1}\) it is at most \(2^{-2sk}\), and the volume is
at most \(\omega_d2^{d(k+1)}\). The resulting geometric series converges
because \(2s>d\). Plancherel with the original factor \((2\pi)^d\) and
both affine measures now gives \[
 \begin{aligned}
 \int(1+Q^{-1}(\Xi))^{-s}\,d\Xi&=J^{-1}I_{d,s},\\
 \int(1+Q^{-1}(\Xi))^s|\widehat v(\Xi)|^2\,d\Xi
 &=J\int(1+|\eta|^2)^s|\widehat{v_L}(\eta)|^2\,d\eta\\
 &=J(2\pi)^d\sum_{|\alpha|\leq s}c_{s,\alpha}
       \|\partial_y^\alpha v_L\|_{L^2(dy)}^2\\
 &=JJ^{-1}(2\pi)^d\sum_{|\alpha|\leq s}c_{s,\alpha}
       \|\partial_{T_1}^{\alpha_1}\cdots\partial_{T_d}^{\alpha_d}v\|_{L^2(dX)}^2.
 \end{aligned}
 \tag{GJ3}
\] Here \(Q^{-1}(\Xi)=|L^{\mathsf T}\Xi|^2\). The original support has
volume \(\lambda_E(K)=J\omega_d\), by the same affine law. If
\(P=\max_{0\leq j\leq s}\sup_K|v|_{j,Q}\), each directional norm squared
in (GJ3) is at most \(J\omega_dP^2\). Cauchy--Schwarz in the original
frequency variable and the original Taylor bound therefore give \[
 \begin{aligned}
 \sup_X\left|T_Au(X)-\sum_{j<N}(iA(D_X))^ju(X)/j!\right|
 &\leq\frac{(2\pi)^{-d}}{N!}\|\widehat v\|_{L^1(d\Xi)}\\
 &\leq\frac{(2\pi)^{-d}}{N!}
 (J^{-1}I_{d,s})^{1/2}
 \bigl(JJ^{-1}(2\pi)^dJ\omega_d(1+d)^sP^2\bigr)^{1/2}\\
 &=\frac{(2\pi)^{-d/2}}{N!}
 \bigl(I_{d,s}\omega_d(1+d)^s\bigr)^{1/2}P.
 \end{aligned}
 \tag{GJ4}
\] Thus (G4) holds for the original form and measure with this explicit
constant; no determinant or inverse-transform factor was omitted. The
two directions of the pullback comparison are the inverse maps \(X=Ly\)
and \(y=L^{-1}X\), and preserve precisely the full support and the
displayed derivative identities. In dimension zero take the sole-point
measure, determinant and \(\omega_0\) to be one. The phase and \(B\) are
zero, \(h_Q=0\) is assigned directly, \(T_A=I\), and the Taylor
remainder is zero for \(N\geq1\) and the identity for \(N=0\); no empty
spectral maximum or undefined zero-times-infinity product is used.

The same reasoning at a fixed support radius \(a>0\) gives a constant
depending also on \(a\). Combining (G3) with diagonalization yields the
useful consequence

\[
\|T_Au-\sum_{j<N}(iA(D))^ju/j!\|_\infty
\leq C_{d,s,N,a}\,h_Q^N
\max_{j\leq s+2N}\sup|u|_{j,Q},\qquad
\operatorname{supp}u\subset\{Q<a^2\}.                         \tag{G5}
\]

To see the coefficient bound explicitly, in orthonormal
eigen-coordinates \(A(D)=\sum_j\beta_jD_j^2\). Expanding its \(N\)-th
power has the sum of absolute coefficients bounded by
\((\sum_j|\beta_j|)^N\leq d^Nh_Q^N\). Applying at most \(s\) further
derivatives gives (G5).

\subsection{3. Repeated integration by parts away from
support}\label{repeated-integration-by-parts-away-from-support}

\textbf{Lemma 3.1 (affine reciprocal).} Let \(L\) be real affine and
nonzero on \(\{Q<R^2\}\), with \(R>1\). Then for \(Y\in K\) and every
integer \(k\geq0\),

\[
|L(0)/L|_{k,Q}(Y)\leq k!R(R-1)^{-k-1}.                      \tag{G6}
\]

\textbf{Proof.} The assumption implies \(L(0)\ne0\). In
\(Q\)-orthonormal coordinates, write \(L(Y)/L(0)=1-\ell\cdot Y\). Its
zero hyperplane misses the open ball of radius \(R\), so
\(|\ell|\leq R^{-1}\). Direct differentiation gives

\[
\partial_{T_1}\cdots\partial_{T_k}(1-\ell\cdot Y)^{-1}
=k!\frac{\prod_j(\ell\cdot T_j)}{(1-\ell\cdot Y)^{k+1}}.
\]

For unit directions and \(|Y|<1\), its absolute value is at most
\(k!R^{-k}(1-R^{-1})^{-k-1}\), which is (G6). \(\square\)

For affine \(L\), the Fourier multiplication/derivative identities give
the exact commutator

\[
[T_A,L]=T_A\langle A'(D),L'\rangle.                           \tag{G7}
\]

For example, Fourier transforming multiplication by \(X_j\) gives
\(i\partial_{\Xi_j}\); subtracting in the order \(T_AX_j-X_jT_A\) leaves
\(A'_{\Xi_j}e^{iA}\), confirming the sign in (G7). Constant terms
commute.

Fix an observation point \(X\), choose \(\eta\in E^*\), and put
\(L(Y)=\langle Y-X,\eta\rangle\). If it is nonzero on a neighborhood of
\(\operatorname{supp}u\), the function \(L^{-1}u\), extended by zero, is
smooth and compactly supported. Evaluating (G7) at \(X\), where
\(L(X)=0\), gives

\[
T_Au(X)=2T_A\big(\langle B\eta,D\rangle(L^{-1}u)\big)(X).
                                                               \tag{G8}
\]

In the following estimate, assume in addition that \(L\ne0\) throughout
\(\{Q<R^2\}\) for a fixed \(R>1\). Iteration preserves the support.
Using (G6), the product rule and the fact that a directional derivative
along \(B\eta\) costs \(Q(B\eta)^{1/2}\), induction proves

\[
|T_Au(X)|
\leq C_{k,R,d,s}
\left(\frac{Q(B\eta)^{1/2}}{|L(0)|}\right)^k
\max_{j\leq s+k}\sup_{Y\in K}|u|_{j,Q}(Y).                    \tag{G9}
\]

For the induction, define \(Vv=\langle B\eta,D\rangle(L^{-1}v)\).
Formula (G6) bounds derivatives of \(L^{-1}\) by a constant divided by
\(|L(0)|\); therefore
\(\max_{j\leq l}|Vv|_{j,Q}\leq C_{l,R}Q(B\eta)^{1/2}|L(0)|^{-1}\max_{j\leq l+1}|v|_{j,Q}\).
Apply this \(k\) times and then (G4) with \(N=0\) to \(V^ku\). This
proves (G9) with no assumption that \(A\) is invertible or definite.

\subsection{4. Decay at phase distance}\label{decay-at-phase-distance}

\textbf{Theorem 4.1 (off-support decay).} Under the hypotheses of
Theorem 2.1, for \(R>1\) and \(k\geq0\),

\[
|T_Au(X)|\leq C_{k,R,d,s}
\left(1+\inf_{Y\in RK}Q^A(X-Y)\right)^{-k/2}
\max_{j\leq s+k}\sup_K|u|_{j,Q}.                              \tag{G10}
\]

For \(k>0\), an infinite infimum makes the right side zero. For \(k=0\),
the factor is defined to be one.

\textbf{Proof.} Denote the square root of the infimum by \(d_A\). The
open unit ball of \(Q^A\) is a bounded convex ellipsoid inside
\(\operatorname{ran}B\), and its support function at \(\eta\) is
\(Q(B\eta)^{1/2}\), by duality on the quotient used in (G1). The support
function of \(RK\) is \(R Q^{-1}(\eta)^{1/2}\).

Choose \(0<a<d_A\). The point \(X\) is outside the open convex set
\(RK+\{Z:Q^A(Z)<a^2\}\). Separation supplies a nonzero \(\eta\), with
orientation chosen so that

\[
\langle X,\eta\rangle
\geq RQ^{-1}(\eta)^{1/2}+aQ(B\eta)^{1/2}.                    \tag{G11}
\]

This separation statement also applies when the second ellipsoid is
lower dimensional, since \(RK\) makes the sum open in \(E\). Formula
(G11) shows that \(L(Y)=\langle Y-X,\eta\rangle\) has no zero in \(RK\),
and \(Q(B\eta)^{1/2}/|L(0)|\leq a^{-1}\). If \(Q(B\eta)=0\), (G8)
already gives \(T_Au(X)=0\), so no division by it is needed. Otherwise
(G9) bounds the left side by \(C a^{-k}\) times the indicated derivative
norm.

Let \(a\) tend to \(d_A\). When \(d_A=\infty\), let \(a\) tend to
infinity, obtaining zero for \(k>0\). For \(d_A\leq1\), use the global
estimate (G4) with \(N=0\). Combining that bound with the one for
\(d_A>1\) changes the constant by at most \(2^{k/2}\) and proves (G10).
\(\square\)

Translations and fixed rescalings of \(K\) give the same estimate for a
function supported in \(B_Z(a)\), with its tail measured from a larger
ball \(B_Z(b)\), \(0<a<b\). Constants depend on the two radii and their
gap. Applying (G10) to a derivative gives the corresponding
differentiated estimate, since \(T_A\) commutes with that derivative.

\subsection{5. Weights measured from the observation
point}\label{weights-measured-from-the-observation-point}

Let \(g\) be slowly varying and let \(m\) be a positive \(g\)-continuous
weight in the sense of \href{metric-localization.pdf}{Localizing symbols
with moving metrics}. Fix an observation point \(X\). The required
additional assumptions are

\[
g_X\leq g_X^A,                                                \tag{G12}
\] \[
g_Y(T)\leq C\,g_X(T)(1+g_Y^A(X-Y))^{N_g},\qquad
m(Y)\leq C\,m(X)(1+g_Y^A(X-Y))^{N_m}.                         \tag{G13}
\]

These hold for every \(Y,T\), with fixed finite exponents
\(N_g,N_m\geq0\) and positive constants. This normalization loses no
generality: any valid negative exponent can be enlarged to zero because
the base \(1+g_Y^A(X-Y)\) is at least one. They are conditions at \(X\);
at this stage no such estimate is assumed at another observation point.
In particular the phase-dual form on the right is based at
\textbf{\(Y\)}. Infinite values impose no comparison across the
corresponding directions; the inequalities are required on their
finite-distance domain.

Reversal of quadratic forms, as in Section 1, gives

\[
g_X^A(T)\leq C\,g_Y^A(T)(1+g_Y^A(X-Y))^{N_g};
\]

applying this to \(T=X-Y\) yields

\[
1+g_X^A(X-Y)\leq C'(1+g_Y^A(X-Y))^{N_g+1}.                  \tag{G14}
\]

All useful estimates here are on the common finite domain
\(\operatorname{ran}B\); elsewhere the extended-valued inequality has
its evident meaning. If \(B\) is invertible, dualizing once more shows
the equivalence of the metric condition in (G13) and the displayed
phase-dual comparison. We make no such equivalence claim for a
degenerate \(B\), since a restriction to \(\operatorname{ran}B\) need
not determine a quadratic form on all of \(E\).

Write \(h(Y)=h_{g_Y}\), using (G3). When \(A\ne0\), \(h\) is positive.
It is \(g\)-continuous because comparable metrics give comparable
\(h\)'s. More precisely, if \(g_Y\leq M g_X\), then
\(g_Y^{-1}\geq M^{-1}g_X^{-1}\), and the first formula in (G3) gives
\(h(Y)\leq Mh(X)\). Thus (G13) implies

\[
h(Y)\leq C h(X)(1+g_Y^A(X-Y))^{N_g}.                         \tag{G15}
\]

This argument does not differentiate the possibly nonsmooth metric.
Assumption (G12) says \(h(X)\leq1\).

\subsection{6. How many localization balls are
near?}\label{how-many-localization-balls-are-near}

Choose a partition \((\phi_\nu)\) from
\href{metric-localization.pdf}{Localizing symbols with moving metrics},
with support balls \(B_{X_\nu}(a)\), and choose fixed larger radii
\(0<a<b<c<r_*\). Write \(U_\nu=B_{X_\nu}(b)\) and
\(U'_\nu=B_{X_\nu}(c)\). These three families have uniformly bounded
multiplicity and are locally finite. Set

\[
d_\nu(X)=\inf_{Y\in U_\nu}g_{X_\nu}^A(X-Y).                  \tag{G16}
\]

\textbf{Lemma 6.1 (counting balls).} Under (G12) and the metric part of
(G13), there are constants \(C,L\), depending only on dimension, the
three radii and metric structural constants, for which

\[
\#\{\nu:d_\nu(X)\leq t\}\leq Ct^L\quad(t\geq1).
                                                               \tag{G17}
\]

Consequently \(\sum_\nu(1+d_\nu(X))^{-M}\leq C_M\) for any \(M>L\),
uniformly when the structural constants are uniform. Terms with
\(d_\nu=\infty\) are zero.

\textbf{Proof.} Use linear coordinates in which \(g_X\) is Euclidean.
For each index being counted choose \(Y_\nu\in U_\nu\) with
\(g_{X_\nu}^A(X-Y_\nu)\leq2t\); using \(2t\) avoids any need for
attainment. Slow variation compares \(g_{Y_\nu}\) with \(g_{X_\nu}\),
hence also their phase-dual forms. Thus
\(g_{Y_\nu}^A(X-Y_\nu)\leq C_1t\). Equations (G13)--(G14) and (G12) now
imply

\[
g_{X_\nu}\leq C_2t^{N_g}g_X,\qquad
g_X(X-Y_\nu)\leq C_3t^{N_g+1}.                              \tag{G18}
\]

Every Euclidean ball centered at \(Y_\nu\) of radius
\(\kappa t^{-N_g/2}\), with a sufficiently small structural
\(\kappa>0\), lies in \(U'_\nu\): its \(g_{X_\nu}\)-radius is less than
the gap \(c-b\). These balls have the bounded multiplicity of the
\(U'_\nu\)'s. Their centers, by (G18), lie in a Euclidean ball of radius
\(C_4t^{(N_g+1)/2}\), and the small balls lie in a fixed constant
enlargement of it. Integrating their characteristic functions and
dividing by their common volume gives

\[
\#\{\nu:d_\nu(X)\leq t\}
\leq C_5t^{d(2N_g+1)/2}.
\]

This argument first applies to any finite subcollection and therefore
also bounds the entire set. It proves (G17) with \(L=d(2N_g+1)/2\).
Split indices into \(d_\nu\leq1\) and the dyadic bands
\(2^j<d_\nu\leq2^{j+1}\). The band contribution is at most
\(C2^{(j+1)L}2^{-jM}\); the resulting geometric series converges for
\(M>L\). \(\square\)

\subsubsection{Counting along the original phase
range}\label{counting-along-the-original-phase-range}

The ambient-dimensional count above remains valid. There is also a
stronger count which uses the rank of the original phase map and keeps
the original metric and volume. Put \(V=\operatorname{ran}B\) and
\(r=\dim V\). Retain the original \(E=\mathbb R^d\), coordinate measure
\(dX\), quadratic phase \(A(\Xi)=\langle B\Xi,\Xi\rangle\), localization
sets and metric \(g_X\). This argument does not replace any of them.
Write \(\mu\) for the overlap bound of the original larger sets
\(U'_\nu\). In Lemma 6.1 take \(h_*=1\); the following proof also allows
the original comparison \(g_X\leq h_*^2g_X^A\) on its finite domain for
any fixed \(0<h_*<\infty\), as in (G24).

Fix \(t\geq1\) and a finite collection of indices with
\(d_\nu(X)\leq t\). Select the same near-minimizers \(Y_\nu\in U_\nu\)
used in (G18). Their phase distance is finite. The exact finite domain
proved in (G1), for every positive-definite form, therefore gives

\[
Y_\nu-X\in V,\qquad
g_{X_\nu}\leq C_2t^{N_g}g_X,\qquad
g_X(Y_\nu-X)\leq C_3h_*^2t^{N_g+1}.
\tag{GR1}
\]

The first form estimate is the original first estimate in (G18). For the
last one, (G14) bounds the original \(g_X^A(X-Y_\nu)\) by a structural
multiple of \(t^{N_g+1}\); the actual factor \(h_*^2\) is then supplied
by the comparison of \(g_X\) with \(g_X^A\). Enlarge the same structural
\(C_4\) if necessary so that \(C_4^2\geq C_3\), and choose the original
structural \(\kappa>0\) with \(\sqrt{C_2}\kappa<c-b\). Put

\[
\rho=\kappa t^{-N_g/2},\qquad
R=C_4h_*t^{(N_g+1)/2},\qquad
K_\nu=\{Z:g_X(Z-Y_\nu)<\rho^2\}.
\tag{GR2}
\]

For \(Z\in K_\nu\), (GR1) gives
\(g_{X_\nu}(Z-Y_\nu)^{1/2}<\sqrt{C_2}\kappa\). The triangle inequality
for the original positive form and \(Y_\nu\in U_\nu\) give
\(g_{X_\nu}(Z-X_\nu)^{1/2}<b+\sqrt{C_2}\kappa<c\). Thus
\(K_\nu\subset U'_\nu\), and their indicator sum is at most \(\mu\).
Every center belongs to the original affine subspace \(X+V\) and has
\(g_X\)-distance at most \(R\) from \(X\).

Use the bilinear form obtained by polarizing the original \(g_X\), and
let \(W=V^{\perp_{g_X}}\). There is an exact direct sum \(E=V\oplus W\):
in a basis of \(V\), its positive Gram matrix determines the unique
projection of any vector onto \(V\), and subtracting that projection
gives its orthogonal component. The two projection maps satisfy

\[
Z-X=v+w,\quad v\in V,\ w\in W,
\qquad g_X(Z-X)=g_X(v)+g_X(w).
\tag{GR3}
\]

Both projection norms are therefore at most the original norm. Project
\(Z-Y_\nu\) for \(Z\in K_\nu\). The parallel component has norm less
than \(\rho\), while the perpendicular component has the same bound.
Adding the center only changes the parallel component. Consequently the
whole union of the original ellipsoids lies in

\[
\mathcal T_X=\{X+v+w:v\in V, w\in W,
g_X(v)<(R+\rho)^2, g_X(w)<\rho^2\}.
\tag{GR4}
\]

Here is its volume with every original matrix factor retained. Let
\(G_X\) be the positive matrix of \(g_X\) in the original coordinates.
Choose any basis matrices \(C\) of \(V\) and \(D\) of \(W\), and write
\(J=[C\ D]\), \(G_V=C^{\mathsf T}G_XC\), \(G_W=D^{\mathsf T}G_XD\).
Orthogonality gives the exact full matrix and determinant identities

\[
J^{\mathsf T}G_XJ=
\begin{pmatrix}G_V&0\\0&G_W\end{pmatrix},
\qquad
(\det J)^2\det G_X=\det G_V\det G_W.
\tag{GR5}
\]

The coordinate map \((u,z)\mapsto X+Cu+Dz\) has Jacobian \(|\det J|\).
The finite-coordinate ellipsoid volume formula follows with every
original factor from
\href{metric-foundation-bridges.pdf\#section*.7}{Metric
and topological foundations, Section 6} and
\href{stable-prerequisite-bridges.pdf}{Polynomial and contour tools,
Sections 10.4--10.5 and 10.10}. For any original positive
\(m\)-dimensional Gram matrix \(H\), those finite Gram residuals give an
invertible comparison map \(L_H\) satisfying
\(L_H^{\mathsf T}HL_H=I_m\). Taking its complete determinant and
applying the original affine-volume law gives

{\fontsize{8.5}{10}\selectfont
\[
(\det L_H)^2\det H=1,\qquad
\{u:u^{\mathsf T}Hu<s^2\}=L_H\{v:|v|<s\},\qquad
\lambda_m\{u:u^{\mathsf T}Hu<s^2\}
=|\det L_H|\omega_m s^m
=\frac{\omega_m s^m}{\sqrt{\det H}}.
\tag{GR6a}
\]
}

Here \(s>0\), and \(\omega_j\) is the volume of the Euclidean unit ball
in dimension \(j\), with \(\omega_0=1\). The map is used to prove the
comparison; every receiving ellipsoid still has its original Gram
matrix. A zero-dimensional determinant and its sole-point measure are
one. Apply (GR6a) to the three original matrices and the product-measure
theorem proved in \href{banach-foundation-bridges.pdf}{Banach estimates,
quotient spaces and compact parameter arguments}. Then

\[
\begin{aligned}
\lambda_E(K_\nu)&=\frac{\omega_d\rho^d}{\sqrt{\det G_X}},\\
\lambda_E(\mathcal T_X)
&=|\det J|\,
 \frac{\omega_r(R+\rho)^r}{\sqrt{\det G_V}}
 \frac{\omega_{d-r}\rho^{d-r}}{\sqrt{\det G_W}}\\
&=\frac{\omega_r\omega_{d-r}(R+\rho)^r\rho^{d-r}}
 {\sqrt{\det G_X}}.
\end{aligned}
\tag{GR6}
\]

The second equality uses (GR5); the Jacobian and both Gram determinants
have first been displayed. The original matrix \(G_X\) remains in both
ambient-volume formulas. For a finite index collection \(I\), integrate
its indicator sum, use the actual overlap bound and (GR4), and obtain

\[
|I|\frac{\omega_d\rho^d}{\sqrt{\det G_X}}
\leq\mu\lambda_E(\mathcal T_X)
=\mu\frac{\omega_r\omega_{d-r}(R+\rho)^r\rho^{d-r}}
 {\sqrt{\det G_X}}.
\tag{GR7}
\]

Dividing the two positive ambient volumes cancels exactly the common
original \(\sqrt{\det G_X}\) factor. It gives

\[
\begin{aligned}
|I|&\leq\mu\frac{\omega_r\omega_{d-r}}{\omega_d}
\left(1+\frac{C_4h_*}{\kappa}
t^{(2N_g+1)/2}\right)^r\\
&\leq K_r t^{L_r},\qquad
K_r=\mu\frac{\omega_r\omega_{d-r}}{\omega_d}
\left(1+\frac{C_4h_*}{\kappa}\right)^r,
\qquad L_r=\frac{r(2N_g+1)}2.
\end{aligned}
\tag{GR8}
\]

The last step uses \(t\geq1\) and the original \(N_g\geq0\). The bound
holds for every finite collection; if the full counted set had more
indices than its finite bound, a finite subset of that size would
contradict (GR8). Thus it bounds the full set without assuming that an
infimum is attained. When \(r=0\), every selected center equals \(X\),
so all selected original larger sets contain \(X\) and the count is
directly at most \(\mu\). Formula (GR8) agrees with this, using
\(\omega_0=1\) and \(L_0=0\). When \(r=d\), \(W=\{0\}\), and the product
measure has its zero-dimensional factor one; the bound recovers the
original ambient exponent. The zero phase still has the exact identity
action stated after (G22).

\begin{figure}
\centering
\pandocbounded{\includegraphics[keepaspectratio,alt={Five original metric ellipsoids inside their phase-range tube}]{../figures/original-phase-range-tube.png}}
\caption{Five original metric ellipsoids inside their phase-range tube}
\end{figure}

The displayed finite configuration has \(d=2\), \(A(\Xi)=\Xi_1^2\),
\(B=\operatorname{diag}(1,0)\), \(g_X(x,y)=4x^2+y^2\), \(X=(0,0)\),
\(\rho=1\), \(R=6\), and centers
\((-3,0),(-1.5,0),(0,0),(1.5,0),(3,0)\). The original metric balls have
coordinate semiaxes \(1/2\) and \(1\); they are disjoint, so their
actual overlap is one. Their tube is \([-3.5,3.5]\times[-1,1]\). Its
coordinate measure is \(14\), while each ball has measure \(\pi/2\). The
original coordinate basis gives \(|\det J|=1\), \(\det G_V=4\),
\(\det G_W=1\), \(\det G_X=4\). The unit disk lies in a square of area
four, so \(\pi\leq4\) and \(5\pi/2\leq10\leq14\), as required by (GR7).
This is an exact chosen configuration for (GR3)--(GR8), rather than a
claim to display every localization family. The full general proof is
above; the affine-volume and Gram sources are the precise internal
sections cited at (GR6a). Its editable figure source retains all
coordinates and parameters.

The summation and all its actual receivers can now use \(L_r\) instead
of \(L\). For \(M>L_r\), separate \(d_\nu\leq1\) from
\(2^j<d_\nu\leq2^{j+1}\), \(j\geq0\). Infinite-distance terms remain
exactly zero. The full geometric sum yields

\[
\sum_\nu(1+d_\nu(X))^{-M}
\leq K_r+K_r\sum_{j\geq0}2^{(j+1)L_r-jM}
=K_r\left(1+\frac{2^{L_r}}{1-2^{L_r-M}}\right).
\tag{GR9}
\]

In (G20), pointwise extension therefore permits any integer \(k\) with
\(k/2-N_m>L_r\). For \(l\) directional derivatives, the exact modified
weight already used in (G21) has exponent \(N_m+lN_g/2\); it suffices
that \(k>2N_m+lN_g+r(2N_g+1)\). The scalar \(N\)-th far remainder in
Section 8 retains its complete exponent \(-k/2+N_m+N(N_g+1/2)\).
Applying it to that same derivative weight gives the sufficient choice

\[
k>2N_m+lN_g+2N(N_g+1/2)+r(2N_g+1).
\tag{GR10}
\]

No estimate of the original near-support derivatives is removed. Its
ambient Sobolev integer still satisfies \(s>d/2\), and the near part
requires input derivatives through \(l+s+2N\); the far part requires
them through \(l+s+k\). Thus the original bound (G22), or (G26) for
finite \(h_*\), holds with a sufficient upper input order
\(l+s+\max(2N,k)\). Every original \(h(X)^N\), metric-direction factor,
localization constant and bounded-approximation argument remains.
Uniform structural constants on \(E_0\) make (GR8)--(GR9) uniform there,
so the existing derivative-limit and weak-continuity proofs apply to
these same absolutely summable tails. In Problem 3 the scalar choice
becomes \(k>5r+26\), while the near order is still \(s+8\). This is a
proved consequence of the original phase-range and localization
formulas, with no optimality or literature-novelty claim; it preserves
the earlier ambient-dimensional result.

\subsection{7. Pointwise extension and derivative
control}\label{pointwise-extension-and-derivative-control}

\textbf{Theorem 7.1 (pointwise extension).} Under (G12)--(G13), the map
\(u\mapsto T_Au(X)\) on \(C_c^\infty(E)\) has a unique continuous
extension to \(S(m,g)\) which is continuous on each bounded subset when
that subset is given the \(C^\infty_{\rm loc}\) topology. For some
finite \(J\),

\[
|T_Au(X)|\leq C m(X)p_{\leq J}(u;m,g).                       \tag{G19}
\]

The constants and \(J\) depend only on the metric/weight structural
constants and dimension.

\textbf{Proof.} Put \(u_\nu=\phi_\nu u\). The localization estimate of
\href{metric-localization.pdf}{Localizing symbols with moving metrics}
and (G10), with the gap between the support radius \(a\) and \(b\), give

\[
|T_Au_\nu(X)|
\leq C_k m(X_\nu)p_{\leq s+k}(u;m,g)
(1+d_\nu(X))^{-k/2}.                                       \tag{G20}
\]

For finite \(d_\nu\), choose a point nearly attaining (G16). Local
comparison and the weight part of (G13) give
\(m(X_\nu)\leq C m(X)(1+d_\nu(X))^{N_m}\), with a harmless change of
constant. For infinite \(d_\nu\), (G10) makes the left side of (G20)
zero. Choose \(k\) so that \(k/2-N_m>L\), where \(L\) comes from (G17).
The series \(\sum_\nu T_Au_\nu(X)\) converges absolutely and gives
(G19).

Its majorant is uniform for \(u\) in a bounded set of \(S(m,g)\). Each
finite partial sum is continuous in \(C^\infty_{\rm loc}\), since each
\(\phi_\nu u\) has fixed compact support and (G4) gives a finite
derivative bound for its image at \(X\). Uniform convergence of these
continuous functions on the bounded set proves the stated continuity. If
\(u\) has compact support, the partition sum is finite and agrees with
the original \(T_Au(X)\). Finally, the bounded compactly supported
approximation in
\href{metric-localization.pdf\#section*.28}{Approximation
on compact sets} shows uniqueness: any extension with this bounded-set
continuity must take the limit of the same compactly supported partial
sums. This also proves independence of the chosen partition. \(\square\)

This bounded-set continuity is called weak continuity in this course's
symbol estimates. It is stronger than merely specifying a continuous
functional on the Fréchet symbol space: compactly supported functions
need not be dense in that Fréchet topology.

For fixed directions \(T_1,\ldots,T_l\), let
\(m_T(Y)=m(Y)\prod_jg_Y(T_j)^{1/2}\). If a direction is zero the
derivative statement below is trivial; otherwise \(m_T\) is positive and
satisfies local continuity and the weight condition (G13), with exponent
\(N_m+lN_g/2\). The function \(v=\partial_{T_1}\cdots\partial_{T_l}u\)
obeys \(p_j(v;m_T,g)\leq p_{j+l}(u;m,g)\). Using (G19) for \(v\) and
commuting derivatives first gives the following estimate for compactly
supported \(u\):

\[
|\partial_{T_1}\cdots\partial_{T_l}T_Au(X)|
\leq C_lm(X)\prod_jg_X(T_j)^{1/2}
p_{l\leq j\leq J_l}(u;m,g).                                \tag{G21}
\]

The notation on the right means the maximum of exactly that finite range
of seminorms. No lower derivative seminorm is needed.

For a general symbol at a single distinguished observation point, the
left side of (G21) denotes the jet value defined by
\(T_A(\partial_{T_1}\cdots\partial_{T_l}u)(X)\). No derivative of a
function defined only at one point is asserted. Actual tangential
derivatives are identified under the uniform subspace hypotheses in the
next paragraph.

If the hypotheses hold uniformly for \(X\) in a linear subspace \(E_0\),
these formulas define a smooth restriction \(T_Au|_{E_0}\) in
\(S(m|_{E_0},g|_{TE_0})\). Here is a justification of smoothness, which
is not implicit in pointwise extension. Bounded compactly supported
approximants to \(u\) have, by (G21), locally uniformly bounded
derivatives of every order on \(E_0\), and each derivative has a
pointwise limit by the scalar extension argument. On a compact
coordinate box, one additional derivative gives equicontinuity; a finite
grid then upgrades pointwise convergence to uniform convergence. The
fundamental theorem of calculus identifies the limits as derivatives of
one smooth function, successively in every order. Thus the restriction
has (G21). The same bounded-derivative argument applied to a convergent
net or sequence on a bounded source set proves weak continuity of the
restriction. On these bounded sets the usual \(C^\infty_{\rm loc}\)
topology is metrizable, so sequential continuity suffices.

\subsection{8. The remainder with a small
parameter}\label{the-remainder-with-a-small-parameter}

\textbf{Theorem 8.1 (global Taylor remainder).} Assume (G12)--(G13)
uniformly on a linear subspace \(E_0\), and suppose \(A\ne0\). For every
integer \(N\geq0\), define

\[
R_Nu=T_Au-\sum_{j<N}(iA(D))^ju/j!.
\]

Then \(R_N:S(m,g)\to S(mh^N,g)|_{E_0}\) is weakly continuous. For every
\(l\), some finite \(J_{N,l}\) satisfies

\[
|\partial_{T_1}\cdots\partial_{T_l}R_Nu(X)|
\leq C_{N,l}m(X)h(X)^N
\prod_jg_X(T_j)^{1/2}p_{\leq J_{N,l}}(u;m,g),\quad X\in E_0.
                                                               \tag{G22}
\]

Only dimension, \(N,l\) and the structural metric/weight constants
determine the seminorm and estimate. If \(A=0\), then \(R_N=0\) for
\(N\geq1\) and \(R_0\) is the identity restriction; this case is stated
directly, without treating a zero function as a positive weight.

\textbf{Proof.} It suffices initially to consider compactly supported
\(u\) and \(l=0\). Use the partition from Section 6. At a fixed \(X\),
only a bounded number of indices have \(X\in U'_\nu\). For these near
indices, slow variation compares \(g_{X_\nu}\) with \(g_X\), the weights
with \(m(X)\), and \(h(X_\nu)\) with \(h(X)\). Applying (G5) to
\(u_\nu\) in the frozen metric yields

\[
|R_Nu_\nu(X)|\leq C_Nm(X)h(X)^Np_{\leq s+2N}(u;m,g).
\]

Their sum has the same bound with a structural multiplicity factor.

For a far index \(X\notin U'_\nu\), every \(Y\in U_\nu\) satisfies
\(g_{X_\nu}(X-Y)^{1/2}\geq c-b\). Slow variation between \(Y\) and
\(X_\nu\) implies \(g_Y(X-Y)\geq c_0>0\). If \(g_Y^A(X-Y)\) is finite,
(G3) and (G15) give

\[
c_0\leq h(Y)^2g_Y^A(X-Y)
\leq C h(X)^2(1+g_Y^A(X-Y))^{2N_g+1}.
\]

Choose \(Y\) nearly attaining (G16) and compare its metric with the
center metric. For finite \(d_\nu(X)\) this proves

\[
1\leq C h(X)(1+d_\nu(X))^{N_g+1/2}.                        \tag{G23}
\]

Terms with infinite \(d_\nu\) are zero in (G20). Since \(u_\nu\)
vanishes on a neighborhood of \(X\) for every far index, all
Taylor-polynomial terms vanish there, and \(R_Nu_\nu(X)=T_Au_\nu(X)\).
Multiply the bound (G20) by the \(N\)-th power of (G23), using the right
side to bound one from above. After weight comparison it becomes

\[
|R_Nu_\nu(X)|
\leq C m(X)h(X)^Np_{\leq s+k}(u;m,g)
(1+d_\nu(X))^{-k/2+N_m+N(N_g+1/2)}.
\]

Choose \(k\) so that the negative exponent exceeds the counting exponent
\(L\). Summation by Lemma 6.1 proves (G22) for \(l=0\). Notice that the
gain was extracted from the fixed separation of the far support, not
from an unproved asymptotic summation theorem.

For derivatives, apply this scalar estimate to
\(v=\partial_{T_1}\cdots\partial_{T_l}u\) and the modified weight
\(m_T\) used in (G21). The operator \(R_N\) commutes with those
derivatives. The modified weight has controlled structural constants, so
this gives (G22) for arbitrary \(l\), uniformly in the directions.

Finally, \(h\) is positive, locally \(g\)-continuous, and satisfies
(G15), so \(mh^N\) is an allowed target weight on \(E_0\). The
continuous differential part of \(R_N\) is already defined on symbols.
The weak extension of \(T_A\) from Theorem 7.1, bounded approximation,
and (G22) extend the identity and estimate to every \(u\in S(m,g)\). The
locally bounded derivative argument there proves smoothness and weak
continuity in the target symbol class. \(\square\)

No conclusion here requires \(h(X)\to0\). If \(h=1\) on a region, the
remainder belongs to the same weight class there and the theorem gives
no decreasing-order filtration on that region. This is a quantitative
Taylor theorem for one operator, not a construction that sums every
prescribed formal sequence of symbols.

\subsubsection{A finite bound suffices without changing the
metric}\label{a-finite-bound-suffices-without-changing-the-metric}

The bound one in (G12) is stronger than this proof needs. Retain the
same phase, metric, weight, observation subspace and phase-dual form.
Suppose \(A\ne0\), retain (G13) uniformly on \(E_0\), and replace (G12)
by

\[
 g_X(T)\leq h_*^2g_X^A(T),\qquad X\in E_0,\qquad
 0<h_*<\infty.
 \tag{G24}
\]

As before, this comparison is required on the finite domain of the
phase-dual form. Every assertion of Theorems 7.1 and 8.1 still holds,
with constants also depending on \(h_*\). No metric or symbol is
replaced by a rescaled one.

Here is the complete change in the proof. Equation (G3) identifies (G24)
with \(h(X)\leq h_*\). In the counting proof, the first estimate in
(G18) is unchanged. Its second estimate becomes
\(g_X(X-Y_\nu)\leq C_3h_*^2t^{N_g+1}\), by (G14) and (G24). The small
Euclidean balls still have radius \(\kappa t^{-N_g/2}\), with the same
structural \(\kappa\), and lie in the same larger localization balls.
Their union now lies in a Euclidean ball of radius
\((C_4h_*+\kappa)t^{(N_g+1)/2}\). Dividing its volume, multiplied by the
unchanged multiplicity, by the small-ball volume gives

\[
 \#\{\nu:d_\nu(X)\leq t\}
 \leq C_5(1+h_*)^d t^{d(2N_g+1)/2},\qquad t\geq1.
 \tag{G25}
\]

The exponent is the original exponent. Thus all weighted tail sums used
in (G19) and (G21) converge under the same strict exponent inequalities.
Only their constants change. The local Taylor estimate (G5) already
permits every finite value of the frozen phase parameter. The near-index
proof therefore retains its actual factor \(h(X)^N\). For the far
indices, (G15), the fixed support gap and (G3) still prove (G23) without
using a bound of one. The identical tail argument gives

\[
 |\partial_{T_1}\cdots\partial_{T_l}R_Nu(X)|
 \leq C_{N,l,h_*}m(X)h(X)^N
       \prod_j g_X(T_j)^{1/2}p_{\leq J_{N,l}}(u;m,g).
 \tag{G26}
\]

The modified derivative weights, bounded approximation, identification
of derivative limits and uniqueness proofs use these same estimates.
They consequently prove the full continuity and smoothness assertions,
including finite-seminorm control. The case \(A=0\) remains the exact
identity case already stated after (G22). A finite bound on \(h\) does
not imply that successive remainders have smaller weights.
\textbf{Editorial extension without a uniform phase bound.} For a
nonzero original phase, every individual \(h(X)\) is finite and
positive, although its values on \(E_0\) need not have a common bound.
The same original formulas prove the following larger target classes
without (G12) or (G24). Retain the original \(A,B,g,m,E_0\), and suppose
(G13) holds with uniform structural constants on \(E_0\). Let
\(r=\operatorname{rank}B\) and define the additional positive weight \[
 w(X)=(1+h(X))^r,\qquad
 C_r=\mu\frac{\omega_r\omega_{d-r}}{\omega_d}
       \max(1,C_4/\kappa)^r,\qquad
 L_r=\frac{r(2N_g+1)}2.
 \tag{GW1}
\] The constants \(\mu,C_4,\kappa\) are exactly the structural constants
of (GR1)--(GR8). For a fixed observation point, use its actual \(h(X)\),
rather than a uniform \(h_*\), in the unchanged inequality
\(g_X\leq h(X)^2g_X^A\) on the finite phase domain from (G3). In
(GR1)--(GR7) the original ellipsoids, both Gram determinants, Jacobian,
radius \(\rho\), and overlap bound remain; only the displayed parallel
radius is \(R=C_4h(X)t^{(N_g+1)/2}\). Dividing the very same two
positive ambient volumes gives \[
 \begin{aligned}
 \#\{\nu:d_\nu(X)\leq t\}
 &\leq\mu\frac{\omega_r\omega_{d-r}}{\omega_d}
       \left(1+\frac{C_4h(X)}{\kappa}t^{(2N_g+1)/2}\right)^r\\
 &\leq C_rw(X)t^{L_r}\quad(t\geq1),\\
 \sum_\nu(1+d_\nu(X))^{-M}
 &\leq C_rw(X)\left(1+\frac{2^{L_r}}{1-2^{L_r-M}}\right)
       \quad(M>L_r).
 \end{aligned}
 \tag{GW2}
\] The second inequality uses \(1+ch\leq\max(1,c)(1+h)\) and \(t\geq1\),
with both terms kept. The third follows by the full original dyadic
bands and their geometric sum. No term with infinite phase distance is
inserted; those terms are zero. The proof first bounds every finite
collection and therefore the whole count, exactly as in (GR8).

Apply this sum to (G20) after the original weight comparison. Any
integer \(k\) with \(k/2-N_m>L_r\) proves the unique pointwise extension
and \[
 |T_Au(X)|\leq C m(X)w(X)p_{\leq s+k}(u;m,g).
 \tag{GW3}
\] Its bounded-set local continuity and independence of the partition
follow from the same finite terms and uniform-on-source-bounded-set tail
bound at this fixed point. Thus even at a single observation point a
common phase bound is unnecessary; its actual finite \(h(X)\) supplies
the full constant through (GW2).

For the remainder of order \(N\), the near indices still satisfy the
original bound \(C_Nm(X)h(X)^Np_{\leq s+2N}\), since local comparison
and (G5) never use \(h(X)\leq1\). Their bounded multiplicity gives the
same near sum. For the far indices, the full separation inequality (G23)
likewise uses no upper bound on \(h(X)\). Consequently their unchanged
exponent remains \(-k/2+N_m+N(N_g+1/2)\), with the actual \(h(X)^N\)
retained. Using (GW2), and applying the same reasoning to
\(v=\partial_{T_1}\cdots\partial_{T_l}u\) with its original weight
\(m_T\), proves \[
 \begin{aligned}
 |\partial_{T_1}\cdots\partial_{T_l}T_Au(X)|
 &\leq C_lm(X)w(X)\prod_jg_X(T_j)^{1/2}
          p_{l\leq j\leq l+s+k}(u;m,g),\\
 |\partial_{T_1}\cdots\partial_{T_l}R_Nu(X)|
 &\leq C_{N,l}m(X)h(X)^Nw(X)\prod_jg_X(T_j)^{1/2}
          p_{\leq l+s+\max(2N,k)}(u;m,g),\\
 k&>2N_m+lN_g+2N(N_g+1/2)+r(2N_g+1),\qquad s>d/2.
 \end{aligned}
 \tag{GW4}
\] This single sufficient choice also implies the first bound's weaker
requirement \(k>2N_m+lN_g+r(2N_g+1)\). The zero-direction case is zero
directly; otherwise the modified weight stays positive and its original
exponent is \(N_m+lN_g/2\). All input orders, the ambient Sobolev
condition, the rank exponent, the original phase gain and every
metric-direction factor remain. Constants depend on the structural data,
\(N,l,d\), and no uniform upper phase bound.

It remains to prove that these are actual smooth receiving maps, with
legitimate target weights. Slow variation gives two-sided comparison of
\(h\) at nearby points, as proved for (G15), so it gives two-sided
comparison of \(1+h\), of \(w\), and of \(mh^Nw\). No derivative of
\(g\) or \(h\) is taken. At finite phase distance, (G15) also gives the
full comparison \[
 \begin{aligned}
 1+h(Y)&\leq\max(1,C)(1+h(X))
                (1+g_Y^A(X-Y))^{N_g},\\
 w(Y)&\leq\max(1,C)^r w(X)
                (1+g_Y^A(X-Y))^{rN_g}.
 \end{aligned}
 \tag{GW5}
\] Together with the original comparisons for \(m\) and \(h^N\), this
proves local continuity of both new positive target weights. In
particular they are bounded above and below on compact coordinate boxes,
by the finite-cover argument in the metric prerequisite. Apply (GW4) to
the original bounded compactly supported approximants. On each such box
in \(E_0\), the upper weight and metric bounds give locally uniform
derivative bounds of all orders. One extra derivative and a finite grid
upgrade the already proved pointwise limits to uniform limits. The full
coordinate fundamental-theorem identities then identify those limits as
successive derivatives, exactly as in the original Section 7 proof. The
displayed global weighted estimates pass to these limits. The
finite-term and summable-tail argument on bounded source sets, followed
by this local derivative identification, proves weak continuity. Thus
the exact maps are \(T_A:S(m,g)\to S(mw,g)|_{E_0}\) and
\(R_N:S(m,g)\to S(mh^Nw,g)|_{E_0}\), on the original spaces and
observation subspace. The enlarged codomains are explicit; no claim of
the old smaller target is made when \(h\) is unbounded. If a common
\(h_*\) is available, \(w\leq(1+h_*)^r\), and the full original (G19),
(G21), (G22) and (G26) follow with precisely their old targets. The zero
phase keeps its separately stated exact identity and zero remainders; a
zero target weight is never presumed positive.

\subsection{9. A phase with a kernel}\label{a-phase-with-a-kernel}

Take \(E=\mathbb R^2\), \(Q=|dX|^2\) and \(A(\Xi)=\tau\Xi_1^2\), with
\(\tau\ne0\). Then \(Q^A(Z)=Z_1^2/\tau^2\) when \(Z_2=0\), and it is
infinite otherwise. The Fourier multiplier acts in the first variable
only. If a function is supported in the unit disk, the result vanishes
for \(|X_2|\geq1\). The infinite-distance case of (G10) captures this
exact support restriction rather than merely a large finite decay
factor. Meanwhile \(h_Q=|\tau|\); (G5) gives a remainder of order
\(|\tau|^N\), uniformly as \(\tau\to0\).

As a different example, \(g_X=Q\) constant and \(m=1\) satisfy the
global hypotheses whenever \(h_Q\leq1\). If \(h_Q=1\), (G22) still holds
but its factor \(h_Q^N\) is one for every \(N\). No successive order
gain can be read into that endpoint.

\subsection{10. Exercises and solutions}\label{exercises-and-solutions}

\textbf{Problem 1.} Identify exactly where real-valuedness of \(A\) is
used.

\textbf{Solution.} It makes \(|e^{iA}|=1\), so Fourier multiplication
preserves the relevant \(L^2\) modulus and the real-variable Taylor
remainder is bounded by \(|A|^N/N!\). It also makes \(B\) a real
symmetric map, permitting real orthogonal diagonalization and real
affine separating directions. For a complex phase the multiplier may
grow exponentially; the proof does not carry over merely by replacing
real eigenvalues with complex ones.

\textbf{Problem 2.} Why does the proof of (G17) use small balls around
\(Y_\nu\), instead of estimating the volume of each whole \(U_\nu\)?

\textbf{Solution.} Temperateness based only at the observation point
gives the upper form bound in (G18), which ensures a small Euclidean
ball fits near \(Y_\nu\). It need not give the two-sided comparison that
would control the entire shape of \(U_\nu\). The enlargement gap puts
each small ball inside \(U'_\nu\), whose multiplicity is known, and that
is sufficient for the integrated counting estimate.

\textbf{Problem 3.} In dimension \(d\), suppose the metric exponent in
(G13) is \(N_g=2\), the weight exponent is \(N_m=3\), and a remainder of
order \(N=4\) is wanted. Give a sufficient tail-iteration requirement.

\textbf{Solution.} The proof gives \(L=5d/2\). For the far remainder sum
it requires \(k/2-3-4(2+1/2)>5d/2\), or \(k>5d+26\). Any integer
satisfying that strict inequality works, with input derivative order at
most \(s+k\) in this part and \(s+8\) in the near part. This is a
sufficient structural choice, not an assertion of optimal derivative
count.

\textbf{Problem 4.} Why does uniqueness in Theorem 7.1 fail if the
bounded-set continuity requirement is simply omitted from the argument?

\textbf{Solution.} The compactly supported approximants are bounded and
converge locally smoothly, but need not converge in the Fréchet symbol
topology. A merely Fréchet-continuous functional therefore cannot be
passed through this approximation without a further argument. The proof
of uniqueness explicitly uses continuity on the bounded approximating
set. This explains the logical need in that proof; it does not by itself
construct a second extension in every possible symbol space.

The off-support proof uses affine commutators and convex separation.

\subsection{References}\label{references}

\begin{itemize}
\tightlist
\item
  {[}Lerner 2005{]} Nicolas Lerner,
  \href{https://webusers.imj-prg.fr/~nicolas.lerner/PSEUDO.2005.pdf}{Introduction
  to the Weyl--Hörmander Calculus}, lecture notes, 2005.
\end{itemize}

\end{document}