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\begin{document}

\section{Localizing symbols with moving
metrics}\label{localizing-symbols-with-moving-metrics}

A symbol can be measured by a different ellipsoid at every point. Those
ellipsoids may be anisotropic, their coefficients need not vary
smoothly, and the weight may be discontinuous. We construct a locally
finite cover, a partition of unity, and smooth representatives of the
metric and weight. The resulting localization and reconstruction
estimates make no uncertainty, symplectic, or temperateness assumption;
later operator theorems require their own additional hypotheses.

We use finite-dimensional differential calculus, compactness, Lebesgue
volume under linear changes of variables, and the maximality principle
for partially ordered sets. The supporting arguments are collected in
\href{metric-foundation-bridges.pdf}{Metric and topological foundations}.
The covering and partition proofs appear here. A freely accessible
treatment of the wider calculus is {[}Lerner{]}.

\subsection{1. Data and symbol
seminorms}\label{data-and-symbol-seminorms}

Let \(E\) be a real vector space of finite dimension \(d\geq1\). At
every \(X\in E\), let \(g_X\) be a positive-definite quadratic form on
\(E\). Write

\[
B_X(t)=\{Y:g_X(Y-X)<t^2\}.
\]

We assume that there are \(r_*>0\) and \(C_g\geq1\) such that

\[
g_X(Y-X)\leq r_*^2
\quad\Longrightarrow\quad
C_g^{-1}g_X(T)\leq g_Y(T)\leq C_g g_X(T)
\quad(T\in E).                                      \tag{1}
\]

This is slow variation. A positive function \(m:E\to(0,\infty)\) is an
admissible weight for this \emph{local} discussion if, after possibly
decreasing \(r_*\),

\[
g_X(Y-X)\leq r_*^2
\quad\Longrightarrow\quad
C_m^{-1}m(X)\leq m(Y)\leq C_m m(X)                 \tag{2}
\]

for some \(C_m\geq1\). In this lesson, ``locally admissible'' means
exactly (2). We impose neither global temperateness nor an uncertainty
inequality. Neither continuity of \(X\mapsto g_X\) nor continuity of
\(m\) is required by (1)--(2).

For a smooth complex-valued function \(a\), define

\[
|a|_{k,g}(X)=
\sup_{g_X(T_j)\leq1}
\left|D^k a(X)[T_1,\ldots,T_k]\right|,
\qquad
p_k(a;m,g)=\sup_X\frac{|a|_{k,g}(X)}{m(X)}.       \tag{3}
\]

For \(k=0\), the numerator is \(|a(X)|\). The symbol space \(S(m,g)\)
consists of the smooth functions for which every \(p_k\) is finite. We
also write \(p_{\leq k}=\max_{j\leq k}p_j\). These norms of multilinear
derivatives avoid making the definition depend on a preferred basis.
Replacing them with the maximum of derivatives in a \(g_X\)-orthonormal
basis gives equivalent seminorms, with constants depending only on
\(d,k\).

For clarity about conventions, the usual one-sided slow-variation
assumption is sufficient. Suppose \(g_X(Y-X)\leq c\) implies
\(g_Y\leq Cg_X\). If \(g_X(Y-X)\leq c/C\), then \(g_Y(X-Y)\leq c\);
applying the same assumption with \(X,Y\) exchanged gives
\(g_X\leq Cg_Y\). Thus (1) results by shrinking the radius. We do not
add a stronger regularity assumption by using its symmetric form.

\subsection{2. A locally finite cover of moving
ellipsoids}\label{a-locally-finite-cover-of-moving-ellipsoids}

\textbf{Theorem 2.1 (Ellipsoid cover).} Choose any

\[
0<r<\frac{r_*}{4(1+C_g)},\qquad R_1=(1+C_g)r,
\qquad R_2=2R_1.
\]

There is an at most countable family of centers \(X_\nu\) such that:

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  the balls \(B_{X_\nu}(r)\) are pairwise disjoint;
\item
  the balls \(B_{X_\nu}(R_1)\) cover \(E\);
\item
  the larger balls \(B_{X_\nu}(R_2)\) form a locally finite family;
\item
  no point belongs to more than
\end{enumerate}

\[
N=\left\lceil C_g^d\left(1+R_2/r\right)^d\right\rceil               \tag{4}
\]

of those larger balls.

The particular numerical radii are only one convenient choice. All
constants are independent of the centers and of the eccentricities of
the ellipsoids.

\textbf{Proof.} Order the collections of pairwise disjoint balls
\(B_X(r)\) by inclusion. A chain has its union as an upper bound, since
any two balls in the union lie together in one member of the chain. The
maximality principle gives a maximal collection. Every ball in it is a
nonempty Euclidean open set. Choose a rational-coordinate point in each
ball, in any fixed linear coordinates; disjointness makes these chosen
points distinct. The collection is at most countable.

Maximality means that \(B_X(r)\) meets a selected ball \(B_{X_\nu}(r)\)
for every \(X\). Let \(Y\) belong to the intersection. Applying (1) with
centers \(X\) and \(X_\nu\) at \(Y\) gives

\[
g_{X_\nu}\leq C_g g_Y\leq C_g^2g_X.
\]

The triangle inequality in the norm \(g_{X_\nu}^{1/2}\) therefore gives

\[
g_{X_\nu}(X-X_\nu)^{1/2}
\leq g_{X_\nu}(X-Y)^{1/2}+g_{X_\nu}(Y-X_\nu)^{1/2}
<(C_g+1)r=R_1.
\]

This proves the covering assertion.

Fix \(X\), and consider centers with \(X\in B_{X_\nu}(R_2)\). Since
\(R_2<r_*\), (1) yields

\[
C_g^{-1}g_X\leq g_{X_\nu}\leq C_g g_X.
\]

Consequently the disjoint smaller balls \(B_{X_\nu}(r)\) all lie in the
\(g_X\)-ball centered at \(X\) of radius \(\sqrt{C_g}(R_2+r)\). Each has
volume at least \(C_g^{-d/2}r^d v_X\), where \(v_X\) is the volume of
the unit ball of \(g_X\). The containing ball has volume
\(C_g^{d/2}(R_2+r)^d v_X\). Dividing these two quantities bounds every
finite subcollection by the expression in (4), and hence bounds the
entire collection. This proves the multiplicity assertion without any
global comparison to the Euclidean metric.

For local finiteness we need more than a pointwise multiplicity
estimate. Choose \(0<\delta<r_*\), and consider all larger balls meeting
\(B_X(\delta)\). At an intersection point \(Y\), (1) first compares
\(g_Y\) with \(g_X\), then with \(g_{X_\nu}\). Thus

\[
C_g^{-2}g_X\leq g_{X_\nu}\leq C_g^2g_X.
\]

The centers lie within \(g_X\)-distance \(\delta+C_gR_2\) of \(X\).
Their disjoint smaller balls lie in the fixed ball of radius
\(\delta+C_g(R_2+r)\), and each has volume at least \(C_g^{-d}r^dv_X\).
Volume comparison bounds the number meeting this neighborhood by a
finite number. The family is therefore locally finite. \(\square\)

\textbf{The same centers at every permissible radius.} For every
\(0<R\leq r_*\), the family \(\{B_{X_\nu}(R)\}\) is locally finite and
has multiplicity at most \(\lceil C_g^d(1+R/r)^d\rceil\); it covers
whenever \(R\geq R_1\). These assertions hold for the same centers, not
a new selection for each \(R\). In the last two paragraphs of the proof
replace \(R_2\) by \(R\); the only radius condition used in the metric
comparison is \(R\leq r_*\). The covering assertion follows by inclusion
from the cover at \(R_1\).

In particular, given \(0<\varepsilon<1\), choose
\(r=r_*\sqrt\varepsilon/[8(1+C_g)]\). The same centers cover at every
radius satisfying \(\varepsilon r_*^2<R^2\), and their multiplicity for
all \(R^2<r_*^2\) is bounded by the one constant

\[
N_\varepsilon=\left\lceil C_g^d
\left(1+8(1+C_g)/\sqrt\varepsilon\right)^d\right\rceil.
\]

Indeed, \(R_1=r_*\sqrt\varepsilon/8\), which is smaller than every
radius in that covering range; the multiplicity estimate follows by
using \(R\leq r_*\).

\subsection{3. A partition of unity with uniform
derivatives}\label{a-partition-of-unity-with-uniform-derivatives}

\textbf{Theorem 3.1 (Uniform partition).} With the centers above, there
are smooth nonnegative functions \(\phi_\nu\), compactly supported in
\(B_{X_\nu}(R_2)\), such that

\[
\sum_\nu\phi_\nu=1,
\qquad
\sup_{X,\nu}|\phi_\nu|_{k,g}(X)\leq A_k
\quad(k\geq0).                                                   \tag{5}
\]

The constants depend only on \(d,k,r,C_g\) and on one fixed scalar
cutoff. The same assertion holds with \(g_X\) in each derivative norm
replaced by \(g_{X_\nu}\).

\textbf{Proof.} Choose a smooth function \(\chi:[0,\infty)\to[0,1]\)
equal to one on \([0,1]\), with support in \([0,4)\), and put

\[
\theta_\nu(X)=\chi\left(g_{X_\nu}(X-X_\nu)/R_1^2\right),
\quad H(X)=\sum_\nu\theta_\nu(X),
\quad\phi_\nu=\theta_\nu/H.                                     \tag{6}
\]

Local finiteness just proved makes all sums smooth. The cover gives
\(H\geq1\); multiplicity gives \(H\leq N\). A linear isometry from
\((E,g_{X_\nu})\) to Euclidean space turns \(\theta_\nu\) into the same
fixed radial cutoff at radius \(R_1\). Its derivatives of every order
have a bound independent of \(\nu\). On its support, (1) compares
\(g_X\) and \(g_{X_\nu}\), so these derivative bounds also hold in
\(g_X\), with an extra factor at most \(C_g^{k/2}\). Off the support,
all derivatives vanish.

At a given point only at most \(N\) terms can contribute derivatives to
\(H\). Therefore \(|H|_{k,g}\leq ND_k\) for constants \(D_k\). To
control the reciprocal, differentiate \(H\cdot H^{-1}=1\). In arbitrary
directions \(T_1,\ldots,T_k\), the term with all derivatives on
\(H^{-1}\) equals minus the sum of the other product-rule terms, divided
by \(H\). Since \(H\geq1\), induction on \(k\) gives a uniform bound for
\(|H^{-1}|_{k,g}\). No derivative of the metric occurs in this
calculation: the norm is evaluated at the fixed point after
differentiating scalar functions. Apply the product rule to
\(\theta_\nu H^{-1}\) to get (5). The same pointwise comparison of
metrics gives the assertion in the frozen metric. \(\square\)

\textbf{A partition uniform in the outer radius.} In the preceding
\(\varepsilon\)-parameter version, assume \(0<\varepsilon<1/2\). A
single partition can be chosen with compact support in \(B_{X_\nu}(R)\)
for \emph{every} \(R\) with \(2\varepsilon r_*^2<R^2<r_*^2\), and
derivative bounds depending on \(\varepsilon,r_*,C_g,d,k\) and the fixed
cutoff, not on \(R\). To prove this, take \(\chi=1\) on \([0,1]\) with
support in \([0,2)\), and use
\(\theta_\nu=\chi(g_{X_\nu}(X-X_\nu)/(\varepsilon r_*^2))\). The cover
at \(R_1=r_*\sqrt\varepsilon/8\) makes the sum at least one. All
supports lie within radius \(r_*\sqrt{2\varepsilon}<r_*\), so metric
comparison and the uniform bound \(N_\varepsilon\) apply. The derivative
and reciprocal argument above is unchanged. Since those compact supports
lie inside every ball in the stated range, the assertion follows. For
\(\varepsilon\geq1/2\) the range of such outer radii is empty.

\subsection{4. Localizing and reconstructing
symbols}\label{localizing-and-reconstructing-symbols}

\textbf{Theorem 4.1 (Analysis and reconstruction).} If \(a\in S(m,g)\),
the pieces \(a_\nu=\phi_\nu a\) are supported in \(B_{X_\nu}(R_2)\), and
for every \(k\)

\[
\sup_\nu\sup_X\frac{|a_\nu|_{k,g_{X_\nu}}(X)}{m(X_\nu)}
\leq C_k p_{\leq k}(a;m,g).                                     \tag{7}
\]

Conversely, suppose smooth functions \(b_\nu\) are supported in these
balls and satisfy

\[
\sup_\nu\sup_X |b_\nu|_{k,g_{X_\nu}}(X)/m(X_\nu)\leq L_k
\quad\text{for every }k.                                       \tag{8}
\]

Then \(b=\sum_\nu b_\nu\) is a smooth symbol and

\[
p_k(b;m,g)\leq NC_m C_g^{k/2}L_k.                             \tag{9}
\]

\textbf{Proof.} On the support of a piece, (1)--(2) compare the metric
and weight at \(X\) with their values at the center. Apply the product
rule and (5) to obtain (7). For reconstruction, local finiteness makes
the sum smooth. At every point at most \(N\) derivative tensors occur. A
direction of \(g_X\)-length at most one has \(g_{X_\nu}\)-length at most
\(\sqrt{C_g}\). Estimate each tensor by (8), multiply by the
weight-comparison constant \(C_m\), and add the at most \(N\)
contributions. This gives exactly (9). \(\square\)

This result is a quantitative interface: a later operator estimate can
be proved on compactly supported symbols measured in one fixed
ellipsoid, and then combined using a separately justified summability
estimate. Local bounded overlap alone does not justify summing the
\emph{images} of pieces under a nonlocal operator.

The reconstruction assertion also holds for the same centers and any
support radius \(0<R<r_*\), with the multiplicity constant at that
radius replacing \(N\). The all-radius part of Theorem 2.1 supplies
local finiteness and multiplicity, and (1)--(2) still compare center
data with point data. Thus the proof of (9) applies verbatim at the new
radius. In the \(\varepsilon\)-parameter version, \(N_\varepsilon\)
gives a uniform bound over all such radii.

\subsection{5. Smooth metrics and weights with the same symbol
space}\label{smooth-metrics-and-weights-with-the-same-symbol-space}

\textbf{Theorem 5.1 (Smooth representatives).} There is a positive
smooth function \(M\) such that

\[
C_m^{-1}m\leq M\leq C_m m,
\qquad M\in S(m,g).                                           \tag{10}
\]

There is also a smooth field of positive-definite forms \(G_X\) such
that

\[
C_g^{-1}g_X\leq G_X\leq C_g g_X,                              \tag{11}
\]

and, for every \(k\),

\[
\left|D_X^kG_X(V,W)[T_1,\ldots,T_k]\right|
\leq B_k\sqrt{g_X(V)g_X(W)}\prod_{j=1}^k\sqrt{g_X(T_j)}.       \tag{12}
\]

Here the same notation denotes a quadratic form and its symmetric
bilinear form. The symbol spaces \(S(M,G)\) and \(S(m,g)\) agree, with
equivalent seminorms.

\textbf{Proof.} Define

\[
M(X)=\sum_\nu\phi_\nu(X)m(X_\nu),\qquad
G_X=\sum_\nu\phi_\nu(X)g_{X_\nu}.                            \tag{13}
\]

Both sums are locally finite. On the support of \(\phi_\nu\), (1)--(2)
apply between \(X_\nu\) and \(X\). Since the coefficients are
nonnegative and sum to one, weighted averaging immediately proves (10)
without its derivative assertion, and proves (11). In particular the
average form is positive definite.

Differentiating (13) differentiates only \(\phi_\nu\), since the center
values are constants. At a given point there are at most \(N\) nonzero
terms, and (5) bounds the differentiated coefficients. It follows that
\(|M|_{k,g}\leq NA_kC_m m\). For the metric, the Cauchy--Schwarz
inequality for the positive form \(g_{X_\nu}\), followed by (1), gives

\[
|g_{X_\nu}(V,W)|
\leq\sqrt{g_{X_\nu}(V)g_{X_\nu}(W)}
\leq C_g\sqrt{g_X(V)g_X(W)}.
\]

Together with (5) this proves (12) with \(B_k=NA_kC_g\). Finally
(10)--(11) compare each defining seminorm in either direction. Thus
equality and equivalence of the symbol spaces follow directly from (3),
without differentiating the weight or the metric. \(\square\)

The new metric is itself slowly varying. Indeed, if
\(G_X(Y-X)\leq r_*^2/C_g\), then \(g_X(Y-X)\leq r_*^2\); inserting (11)
on both sides of (1) compares \(G_Y\) and \(G_X\) with constant
\(C_g^3\). The new weight is locally comparable on these balls by (2)
and (10). Thus regularization keeps us within the same class of local
data.

\textbf{Corollary 5.2 (Recognition of weights).} For fixed \(g\) and two
locally admissible weights \(m_1,m_2\), the inclusion
\(S(m_1,g)\subset S(m_2,g)\) as sets holds if and only if \(m_1/m_2\) is
bounded. The reverse implication follows from (3). For the forward
implication, construct \(M_1\) by (10). It lies in \(S(m_1,g)\), hence
in \(S(m_2,g)\); its zeroth seminorm gives \(M_1\leq Cm_2\), and (10)
then gives the claimed ratio bound. In particular, equality of these
spaces is equivalent to comparability of the weights in both directions.

\subsection{6. Completeness, products and
reciprocals}\label{completeness-products-and-reciprocals}

The pointwise differential inequalities in this section are local
statements: for fixed \(k\), they apply to complex-valued \(C^k\)
functions on a neighborhood of the point, measured in any fixed
positive-definite form \(Q\). In particular the product rule gives \[
|ab|_{k,Q}(X)\leq\sum_{j=0}^k\binom{k}{j}|a|_{j,Q}(X)|b|_{k-j,Q}(X).
\] No global symbol membership is needed for this inequality or for the
pointwise reciprocal bound (18). Global smoothness and symbol seminorms
are imposed only for the Fréchet-space and symbol-membership
conclusions.

\textbf{Proposition 6.1.} The seminorms (3) make \(S(m,g)\) a Fréchet
space. For locally admissible weights \(m_1,m_2\), multiplication is
continuous and satisfies

\[
p_k(ab;m_1m_2,g)
\leq\sum_{j=0}^k\binom{k}{j}
p_j(a;m_1,g)p_{k-j}(b;m_2,g).                                \tag{14}
\]

If \(a\in S(m,g)\) and \(|a(X)|\geq c\,m(X)\) for some \(c>0\), then
\(a^{-1}\in S(m^{-1},g)\). Its seminorms are bounded recursively by

\[
b_0=c^{-1},\qquad
b_k=c^{-1}\sum_{j=1}^k\binom{k}{j}p_j(a;m,g)b_{k-j},
\qquad p_k(a^{-1};m^{-1},g)\leq b_k.                         \tag{15}
\]

\textbf{Proof.} At any fixed point \(X\), (1)--(2) give a neighborhood
on which the metric is comparable to \(g_X\) and the weight is bounded
above and bounded away from zero. A finite subcover gives such bounds on
a compact set. In fixed coordinates this means that convergence in all
the seminorms (3) implies convergence of every derivative uniformly on
compact sets.

If \((a_n)\) is Cauchy in every seminorm, its coordinate derivatives of
every order have uniform limits on compact sets. These limits are the
derivatives of a single smooth function: on a coordinate box apply the
fundamental theorem of calculus to \(a_n\) along each coordinate
segment, pass to the uniform limit, and repeat for each already obtained
derivative. At a fixed \(X\), pass to the limit in

\[
|a_n-a_\ell|_{k,g}(X)\leq\varepsilon m(X)
\]

for sufficiently large \(n,\ell\). The result is the same bound for
\(a_n-a\), uniformly in \(X\); hence convergence holds in every \(p_k\),
and \(a\in S(m,g)\). The topology is Hausdorff because \(p_0\) separates
points. A countable family of seminorms is metrized, for example, by
\(\sum_{k\geq0}2^{-k-1}\min(1,p_k(a-b))\). This establishes the Fréchet
assertion.

For multiplication, apply the multilinear product rule. There are
\(\binom{k}{j}\) ways to place \(j\) of the \(k\) derivatives on the
first factor, giving (14). The product weight satisfies (2), with the
product of the two comparison constants and the smaller common radius.

The reciprocal is smooth because \(a\) has no zero. Differentiate
\(aa^{-1}=1\). Move the term with no derivative on \(a\) to the other
side, divide by \(a\), and use \(|a|^{-1}\leq c^{-1}m^{-1}\). Induction
gives (15): each product involving \(j\) derivatives of \(a\) has weight
\(m\), while the derivative of its reciprocal has weight \(m^{-1}\); the
weights cancel before the final division by \(a\). \(\square\)

The same argument works for matrices of any fixed finite size, using an
operator norm and the assumption \(\|a(X)^{-1}\|\leq c^{-1}m(X)^{-1}\).
The identity is differentiated with the order of matrix factors
preserved. This is a symbol estimate; it is not the construction of an
inverse to a quantized operator.

There is also a useful scalar pointwise bound that does not involve a
global weight. Fix \(k\geq1\), a positive-definite form \(Q\), and a
\(C^k\) function \(a\) near a point \(X\) with \(a(X)=1\). Then

\[
|a^{-1}|_{k,Q}(X)
\leq C_k\left(\sum_{j=1}^k |a|_{j,Q}(X)^{1/j}\right)^k.       \tag{18}
\]

To verify it, differentiate the scalar composite \(t\mapsto t^{-1}\)
with \(t=a\). Its \(k\)-th multilinear derivative is the sum over set
partitions \(\pi\) of \(\{1,\ldots,k\}\), with term

\[
(-1)^{|\pi|}|\pi|!\,a(X)^{-|\pi|-1}
\prod_{B\in\pi}D^{|B|}a(X)[T_j:j\in B].
\]

This formula follows by induction: differentiating the scalar power adds
a singleton block, while differentiating a derivative factor appends the
new index to its block; these exhaust the set partitions with their
indicated coefficients. Put \(H=\sum_{j=1}^k |a|_{j,Q}(X)^{1/j}\). Each
factor of order \(j\) is at most \(H^j\) on unit directions. Each
partition term therefore has size at most \(|\pi|!H^k\); summing proves
(18) with \(C_k=\sum_\pi|\pi|!\). If \(H=0\), every positive-order
derivative factor vanishes and the same conclusion holds. For any
nonzero \(a(X)\), the estimate keeps the relative derivative factors of
\(a/a(X)\) together with \(|a(X)|^{-1}\). The partition calculation
below shows every inverse power explicitly.

\subsubsection{The reciprocal bound from every original partition
term}\label{the-reciprocal-bound-from-every-original-partition-term}

For the scalar pointwise estimate, keep the original function \(a\),
point \(X\), positive-definite form \(Q\) and all inverse powers.
Suppose \(a\) is \(C^k\) near \(X\), \(k\geq1\), and \(a(X)\ne0\);
continuity gives a neighborhood on which its reciprocal is defined. The
full partition formula above applies there. Define the actual quantities

\[
q_j=|a|_{j,Q}(X),\qquad
H_a=\sum_{j=1}^k
\left(\frac{q_j}{|a(X)|}\right)^{1/j},\qquad
C_k=\sum_{\pi\in\mathcal P_k}|\pi|!,
\tag{19}
\]

where \(\mathcal P_k\) is the finite set of partitions of
\(\{1,\ldots,k\}\). All denominators use the original nonzero value; no
new function replaces \(a\). For \(Q\)-unit directions, the exact term
of partition \(\pi\) has absolute value at most

\[
|\pi|!\,|a(X)|^{-1-|\pi|}
\prod_{B\in\pi}q_{|B|}
=|\pi|!\,|a(X)|^{-1}
\prod_{B\in\pi}\frac{q_{|B|}}{|a(X)|}
\leq |\pi|!\,|a(X)|^{-1}H_a^k.
\tag{20}
\]

The equality displays one original inverse factor for each block and the
remaining factor \(|a(X)|^{-1}\). The last inequality follows because
each summand in \(H_a\) is nonnegative, so \(q_j/|a(X)|\leq H_a^j\), and
the block sizes sum to \(k\). Sum the finitely many exact partition
terms and take the supremum over all original unit directions. This
proves

\[
|a^{-1}|_{k,Q}(X)
\leq C_k|a(X)|^{-1}
\left(\sum_{j=1}^k
\left(\frac{|a|_{j,Q}(X)}{|a(X)|}\right)^{1/j}\right)^k.
\tag{21}
\]

If \(H_a=0\), every \(q_j\) is zero, every positive-order partition term
vanishes, and both sides of (21) are zero. The order-zero formula
remains exactly \(|a^{-1}|_{0,Q}(X)=|a(X)|^{-1}\). With \(a(X)=1\), (21)
gives the earlier (18) with the same \(C_k\); the earlier formula
remains its stated special case. The induction for the full partition
identity is unchanged: a derivative of the actual power \(a^{-1-|\pi|}\)
creates a singleton block with coefficient \(-1-|\pi|\), while a
derivative of an existing factor appends the new labeled direction to
that block. These two cases produce every partition once and retain its
sign \((-1)^{|\pi|}\), factorial \(|\pi|!\) and actual inverse power.
This proves the general estimate directly from the original function and
each full derivative term.

\subsection{7. Approximation on compact
sets}\label{approximation-on-compact-sets}

\textbf{Proposition 7.1.} Every \(a\in S(m,g)\) is the limit in
\(C^\infty_{\mathrm{loc}}\) of a sequence \(a_N\in C_c^\infty(E)\)
bounded in every seminorm of \(S(m,g)\).

\textbf{Proof.} First \(C_c^\infty(E)\subset S(m,g)\), by the
compact-set comparison in Proposition 6.1. Enumerate the partition and
set \(a_N=\sum_{\nu\leq N}\phi_\nu a\). Each sum has compact support.
The proof of (7)--(9) gives bounds independent of \(N\). On any compact
set only finitely many partition supports meet it; after all these
indices have been included, \(a_N=a\) on a neighborhood of that compact
set. This proves the asserted convergence. \(\square\)

This is deliberately a bounded approximation statement. It does not
assert convergence in the symbol seminorms. For the constant Euclidean
metric and \(m=1\), the symbol \(a=1\) has \(p_0(1-b;1,g)\geq1\) for
every compactly supported \(b\). Thus \(C_c^\infty\) need not be dense
in the Fréchet topology. Later extension theorems must specify which
topology their continuity uses.

\subsection{8. An anisotropic model}\label{an-anisotropic-model}

On \(E=\mathbb R_x^n\times\mathbb R_\xi^n\), take arbitrary real
\(\delta,q\), take \(\rho\leq1\), and set

\[
\langle\xi\rangle=(1+|\xi|^2)^{1/2},\qquad
g_{(x,\xi)}(y,\eta)
=\langle\xi\rangle^{2\delta}|y|^2
+\langle\xi\rangle^{-2\rho}|\eta|^2,
\qquad m(x,\xi)=\langle\xi\rangle^q.                         \tag{16}
\]

If \(g_{(x,\xi)}(y,\eta)\leq r_*^2\), then
\(|\eta|\leq r_*\langle\xi\rangle^\rho\leq r_*\langle\xi\rangle\). Since
the function \(\xi\mapsto\langle\xi\rangle\) is one-Lipschitz,

\[
(1-r_*)\langle\xi\rangle\leq\langle\xi+\eta\rangle
\leq(1+r_*)\langle\xi\rangle.
\]

Choose \(r_*<1/2\). Raising this ratio to each of the fixed real powers
in (16) proves both (1) and (2). From (3), or by testing coordinate
directions and expanding arbitrary directions, membership in \(S(m,g)\)
is equivalent to

\[
|\partial_x^\beta\partial_\xi^\alpha a(x,\xi)|
\leq C_{\alpha\beta}
\langle\xi\rangle^{q+\delta|\beta|-\rho|\alpha|}.             \tag{17}
\]

The present localization theorem therefore includes all these parameter
choices, not just \((\rho,\delta)=(1,0)\). It does not claim that all of
them support the same composition or continuity theorem. In particular,
those conclusions cannot be inferred merely from the proof of slow
variation.

\subsection{9. Exercises and solutions}\label{exercises-and-solutions}

\textbf{Problem 1: an unsmoothed weight.} For the Euclidean metric on
\(\mathbb R^d\), let \(m(X)=2+\mathbf1_{\{X_1\geq0\}}\). Verify the
hypotheses and identify \(S(m,g)\). Explain what the regularization
theorem supplies.

\textbf{Solution.} The metric comparison constant is one at every
radius. The weight lies between two and three, so any pair of points
satisfies (2) with \(C_m=3/2\). Formula (3) gives
\(\frac13\sup|D^ka|\leq p_k(a;m,g)\leq\frac12\sup|D^ka|\). Thus the
space consists exactly of smooth functions with all derivatives bounded,
with its usual topology. Theorem 5.1 constructs a positive smooth
representative comparable to the discontinuous weight. It does not claim
to make the original weight continuous or to approximate its jump
uniformly by continuous functions.

\textbf{Problem 2: the missing local-finiteness argument.} The intervals
\((2^{-j},3\cdot2^{-j-1})\), \(j\geq1\), have multiplicity one. Are they
locally finite in \(\mathbb R\)? Which part of Theorem 2.1 prevents the
analogous failure?

\textbf{Solution.} They are pairwise disjoint, but every neighborhood of
zero meets infinitely many. They are therefore not locally finite. In
Theorem 2.1, slow variation compares \emph{every} center whose large
ball meets a fixed small neighborhood with the metric at that
neighborhood's center. It gives a common positive lower bound for the
volumes of the disjoint small balls, inside one bounded ellipsoid. That
neighborhood-volume argument, rather than the pointwise bound alone,
proves local finiteness.

\textbf{Problem 3: a matrix symbol with a nonconstant scale.} For a
weight \(m\), let \(M\) be its smooth representative and let
\(B\in S(1,g;\mathbb C^{r\times r})\) satisfy \(\sup_X\|B(X)\|\leq1/2\).
Show that \(A=M(I+B)\) belongs to \(S(m,g)\) and that
\(A^{-1}\in S(m^{-1},g)\).

\textbf{Solution.} Product estimate (14) gives the symbol bound for
\(A\). For each \(X\), the geometric series \(\sum_{j\geq0}(-B(X))^j\)
converges in operator norm, multiplies \(I+B(X)\) to the identity, and
has norm at most two. Hence \(\|A(X)^{-1}\|\leq2/M(X)\leq2C_m/m(X)\).
The matrix form of (15) now bounds every derivative of the inverse.
Differentiating the infinite geometric series is unnecessary and would
require a separate convergence argument.

\textbf{Problem 4: a distinction in approximation.} Give an
approximating sequence for the constant symbol one which is bounded in
\(S(1,g)\), converges locally smoothly, and fails to converge in
\(p_0\), for Euclidean \(g\).

\textbf{Solution.} Choose \(\psi\in C_c^\infty\) equal to one near the
origin and set \(a_N(X)=\psi(X/N)\). Derivatives of order \(k\) have
supremum at most \(N^{-k}\sup|D^k\psi|\), so the sequence is bounded in
every seminorm. On each compact set it is eventually identically one.
But each \(a_N\) vanishes outside a compact set, so
\(p_0(a_N-1;1,g)\geq1\). This explicitly separates the two topologies
used in Proposition 7.1.

\subsection{References}\label{references}

\begin{itemize}
\tightlist
\item
  {[}Lerner{]} Nicolas Lerner, \emph{Introduction to the Weyl--Hörmander
  Calculus of Pseudodifferential Operators}, lecture notes.
\end{itemize}

\end{document}