# Two measuring scales, one Weyl product

Two symbols can have very different derivative scales. Their product under Weyl quantization couples these scales through the symplectic form. The useful small parameter measures that coupling; it need not control either scale separately. We first identify the precise compatibility conditions, then derive the quantization formula and apply the quadratic multiplier theorem of AN03-U002. This order keeps the geometric hypotheses visible when the analytic estimate is used.

The local symbol spaces and bounded compactly supported approximation are those of AN03-U001. We use all the conclusions of AN03-U002, with its Fourier convention \(D=-i\partial\). The additional prerequisites are finite-dimensional linear algebra; completeness, compact smooth approximation and integration in \(L^2\); and the tempered Schwartz kernel theorem: every continuous linear map \(\mathcal S(V)\to\mathcal S'(V)\) has a unique kernel in \(\mathcal S'(V\times V)\). We prove the needed quadratic minimization, core property and unitary-group formulas below. The kernel theorem is used only for the converse classification of all Schwartz-to-distribution operators; the displayed operators and kernel identities are constructed directly. External prerequisite import closure is a separate course obligation.

Throughout, \(V\) is a real vector space of dimension \(n\geq1\), \(W=V\oplus V^*\), and
\[
\sigma((x,\xi),(y,\eta))=\langle\xi,y\rangle-\langle x,\eta\rangle.
\tag{W1}
\]
Measures on \(V\) and \(V^*\) are dual for Fourier inversion with coefficient \((2\pi)^{-n}\). Multiplying one measure by a positive constant divides the other by that constant; the phase-space measure and all quantizations below are unchanged.

## AN03-WP-001 — Quadratic forms and the cross parameter

For a positive-definite quadratic form \(q\) on \(W\), define
\[
q^\sigma(T)=\sup_{S\ne0}\frac{|\sigma(T,S)|^2}{q(S)}.
\tag{W2}
\]
This is positive definite. Ordinary quadratic duality, composed with the invertible linear map induced by \(\sigma\), gives
\[
(q^\sigma)^\sigma=q,\qquad (c q)^\sigma=c^{-1}q^\sigma,
\qquad q\leq C r\ \Longleftrightarrow\ r^\sigma\leq Cq^\sigma.
\tag{W3}
\]
For example, write \(q(T)=T^tQT\) in coordinates and let \(J\) represent \(\sigma\). Then \(q^\sigma(T)=T^tJQ^{-1}J^tT\); substitution proves the first identity, and inversion of positive matrices proves the order reversal. Alternatively, all three follow by mapping the unit ellipsoid to its polar ellipsoid and using that taking the polar twice returns the original ellipsoid.

We will need an exact formula for the dual of a sum. If \(F_1,F_2\) are positive quadratic forms on a finite-dimensional space and primes denote ordinary dual forms, then
\[
(F_1+F_2)'(\zeta)
=\min_{\zeta_1+\zeta_2=\zeta}
\big(F_1'(\zeta_1)+F_2'(\zeta_2)\big).
\tag{W4}
\]
To prove it, represent the forms by positive matrices \(A_1,A_2\). Put \(v=(A_1+A_2)^{-1}\zeta\) and \(\zeta_j=A_jv\). Every other decomposition is \((\zeta_1+e,\zeta_2-e)\). Expanding its objective gives the value \(v^t(A_1+A_2)v\), plus \(e^t(A_1^{-1}+A_2^{-1})e\); the mixed terms cancel. The latter quantity is nonnegative and vanishes only at \(e=0\). This proves both the minimum and its value. In particular, if \(g=(g_1+g_2)/2\),
\[
g^\sigma(T)=2\min_{T_1+T_2=T}
\big(g_1^\sigma(T_1)+g_2^\sigma(T_2)\big).
\tag{W5}
\]
The factor two comes from the factor one half in the mean; it is not optional.

At one point of phase space, set
\[
h_j^2=\sup_{T\ne0}\frac{g_j(T)}{g_j^\sigma(T)},\qquad
H^2=\sup_{T\ne0}\frac{g_1(T)}{g_2^\sigma(T)}
=\sup_{T\ne0}\frac{g_2(T)}{g_1^\sigma(T)}.
\tag{W6}
\]
The two expressions for \(H\) are equal by (W3): each is the least constant in one of the equivalent inequalities \(g_1\leq H^2g_2^\sigma\), \(g_2\leq H^2g_1^\sigma\). These are also equivalent to
\[
|\sigma(T,S)|^2\leq H^2g_1^\sigma(T)g_2^\sigma(S).
\tag{W7}
\]
Indeed, divide by \(g_2^\sigma(S)\), take the supremum over \(S\), and use (W3).

For the mean metric, write \(h_g^2=\sup g/g^\sigma\). Then
\[
\max(h_1^2,h_2^2,H^2)\leq4h_g^2
\leq h_1^2+h_2^2+2H^2.
\tag{W8}
\]
For the first inequality, \(g_j\leq2g\) and \(g^\sigma\leq2g_k^\sigma\) show that \(g_j(T)/g_k^\sigma(T)\leq4g(T)/g^\sigma(T)\), for both choices of \(j,k\). For the second, \(2g=g_j+g_k\leq(h_j^2+H^2)g_j^\sigma\). Dualizing gives \(2g_j\leq(h_j^2+H^2)g^\sigma\). Add the two inequalities. No uncertainty inequality has been assumed in this argument.

## AN03-WP-002 — A transport test for compatible metrics

Let \(g_1,g_2\) be slowly varying metrics on \(W\), in the sense of AN03-U001. Write \(q_j(X)=g_{j,X}^\sigma\). Each metric is assumed symplectically temperate, meaning that for fixed constants \(C,N\), uniformly in \(X,Y,T\),
\[
q_j(X)(T)\leq Cq_j(Y)(T)
\big(1+q_j(Y)(X-Y)\big)^N.
\tag{W9}
\]
By (W3), this is equivalent to the primal inequality
\(g_{j,Y}\leq Cg_{j,X}(1+q_j(Y)(X-Y))^N\).
Constants can be enlarged so that a common \(C\geq1\), \(N\geq0\) works for both metrics. The bases of the quadratic distances matter.

The cross tests are
\[
\begin{split}
q_1(X)(T)&\leq Cq_1(Y)(T)
\big(1+q_2(X)(Y-X)\big)^N,\\
q_2(X)(T)&\leq Cq_2(Y)(T)
\big(1+q_1(X)(Y-X)\big)^N.
\end{split}
\tag{W10}
\]
We call the pair compatible when these hold. They require more than temperateness of each metric separately.

**Transport lemma.** Under (W9)–(W10), there are common constants \(C,L\) such that, for all \(j,k\in\{1,2\}\),
\[
q_j(X)\leq Cq_j(Y)\big(1+q_k(Y)(X-Y)\big)^L,
\quad
g_{j,Y}\leq Cg_{j,X}\big(1+q_k(Y)(X-Y)\big)^L.
\tag{W11}
\]
Moreover, with \(g=(g_1+g_2)/2\), there are \(C',L'\) such that
\[
g_{j,Y}\leq C'g_{j,X}
\big(1+g_Y^\sigma(X-Y)\big)^{L'},\qquad j=1,2.
\tag{W12}
\]
Consequently \(g\) is symplectically temperate.

**Proof.** For \(j=k\), the first estimate in (W11) is (W9). For \(j\ne k\), (W9) gives
\[
1+q_k(X)(X-Y)\leq C_0\big(1+q_k(Y)(X-Y)\big)^{N+1}.
\]
Insert this into (W10). Dualizing gives the second estimate in (W11).

Fix \(X,Y\), choose any intermediate point \(Z\), and put
\[
R=1+q_1(Y)(X-Z)+q_2(Y)(Z-Y).
\]
Use (W11) first on the segment from \(Z\) to \(Y\), with \(k=2\). It gives
\[
g_{j,Y}\leq Cg_{j,Z}R^L,
\qquad q_1(Z)(X-Z)\leq Cq_1(Y)(X-Z)R^L\leq CR^{L+1}.
\]
Use (W11) again on the segment from \(X\) to \(Z\), now with \(k=1\). Thus
\[
g_{j,Z}\leq Cg_{j,X}(1+q_1(Z)(X-Z))^L
\leq C_1g_{j,X}R^{L(L+1)}.
\]
Multiplication gives an exponent \(L^2+2L\). Minimize \(R\) using (W5); its minimum is \(1+g_Y^\sigma(X-Y)/2\). This proves (W12). Adding the two primal inequalities proves temperateness of the mean. It is also slowly varying: if \(g_X(Y-X)\) is sufficiently small, then \(g_{j,X}(Y-X)\leq2g_X(Y-X)\) is small for both metrics, and their slow-variation comparisons can be added. ∎

A positive weight \(m\) is symplectically temperate for \(g\) if it is \(g\)-continuous locally and
\[
m(Y)\leq Cm(X)\big(1+g_Y^\sigma(X-Y)\big)^N.
\tag{W13}
\]
Its reciprocal has the same property, with possibly larger exponent. In fact, apply (W13) with \(X,Y\) exchanged and use (W9) to replace the resulting distance based at \(X\) by a power of the distance based at \(Y\). Local comparability also reverses. Products and positive real powers are handled in the same way.

Suppose \(m_j\) is initially temperate for its own \(g_j\). The additional weight tests are
\[
m_1(Y)\leq Cm_1(X)(1+q_2(X)(X-Y))^N,
\quad
m_2(Y)\leq Cm_2(X)(1+q_1(X)(X-Y))^N.
\tag{W14}
\]
For a compatible pair, these are equivalent to both weights being temperate for the mean \(g\). Here is the full reduction. Own temperateness and (W14), with the distance-base conversion used above, give
\(m_j(Y)\leq Cm_j(X)(1+q_k(Y)(X-Y))^L\) for either \(k\). Repeat the two-segment proof of (W12), replacing its primal metric factor by \(m_j\); the estimate for \(q_1(Z)\) is unchanged. Minimization proves (W13) for the mean. Local \(g\)-continuity follows from \(g_j\)-continuity and \(g_j\leq2g\).

Conversely, \(g_Y^\sigma\leq2q_k(Y)\), followed by the distance-base conversion for \(g_k\), turns mean temperateness into (W14). If only mean temperateness and \(g_j\)-continuity were given at the outset, the same inequality with \(k=j\) also proves own temperateness. This explains the exact weight hypotheses used later.

## AN03-WP-003 — Why the tests are exactly the diagonal conditions

On \(W\times W\), use the product metric and weight
\[
G_{(Y,Z)}(T,S)=g_{1,Y}(T)+g_{2,Z}(S),\qquad
M(Y,Z)=m_1(Y)m_2(Z).
\tag{W15}
\]
The product metric is slowly varying and the product weight is locally \(G\)-continuous, directly from the separate local comparisons.

Use dual coordinates \((p,q;r,s)\) and the auxiliary quadratic phase
\(A_0(p,q;r,s)=2(q\cdot r-p\cdot s)\).
Its symmetric map in the convention of AN03-GAU-001 sends
\((p,q;r,s)\) to \((-s,r;q,-p)\). Substitution in that unit's definition of the phase-dual form gives
\[
G_{(Y,Z)}^{A_0}(T,S)=q_2(Z)(T)+q_1(Y)(S).
\tag{W16}
\]
Consequently \(G_{(X,X)}\leq H(X)^2G_{(X,X)}^{A_0}\), and \(H(X)^2\) is the least possible constant. Notice the interchange of the two metrics in (W16).

**Diagonal criterion.** The product metric is uniformly \(A_0\)-temperate at the points \((X,X)\) if and only if (W10) holds. If the individual weights are temperate for their own metrics, the product weight is uniformly \(A_0,G\)-temperate there if and only if (W14) holds.

**Proof of necessity.** In dual form, metric temperateness on the diagonal says
\[
q_1(X)(T)+q_2(X)(S)
\leq C\big(q_1(Y)(T)+q_2(Z)(S)\big)\mathcal D^L,
\quad
\mathcal D=1+q_2(Z)(X-Y)+q_1(Y)(X-Z).
\tag{W17}
\]
Set \(Z=X,S=0\) to obtain the first cross test. Set \(Y=X,T=0\) for the second. The analogous product-weight inequality is
\(m_1(Y)m_2(Z)\leq Cm_1(X)m_2(X)\mathcal D^L\).
The same substitutions and cancellation of a positive factor give (W14).

**Proof of sufficiency.** We supply the distance comparison that is needed for this implication. For any single temperate metric \(g\), write \(q(U)=g_U^\sigma\), and let
\[
\mathcal E=1+q(Y)(X-Z)+q(Z)(X-Y),\qquad P=Y+Z-X.
\]
Because \(P-Z=Y-X\) and \(P-Y=Z-X\), temperateness gives
\[
q(P)(X-Y)\leq C\mathcal E^{L+1},\qquad
q(P)(X-Z)\leq C\mathcal E^{L+1}.
\]
For example, the first is the comparison of \(q(P)\) with \(q(Z)\), whose controlling distance \(P-Z\) is exactly \(-(X-Y)\). Compare \(q(Y)\) with \(q(P)\); the controlling distance \(Y-P=X-Z\) is bounded by the second estimate. Do the analogous comparison for \(q(Z)\). We obtain
\[
q(Y)(X-Y)+q(Z)(X-Z)\leq C'\mathcal E^{(L+1)^2}.
\tag{W18}
\]

Now take the mean metric from AN03-WP-002. Its dual is at most twice each \(q_j\), so \(\mathcal E\leq2\mathcal D\). Apply (W12) to \(g_{1,Y}/g_{1,X}\) and \(g_{2,Z}/g_{2,X}\), and then (W18). This yields
\[
g_{1,Y}(T)+g_{2,Z}(S)
\leq C''\big(g_{1,X}(T)+g_{2,X}(S)\big)\mathcal D^{L''}.
\]
It is precisely the primal phase-temperateness condition from AN03-GAU-005. Dualization gives (W17). The weights are temperate for the mean by the preceding weight equivalence, so applying (W13) and (W18) also proves the product-weight condition. ∎

Taking \(g_1=g_2=g\) makes the cross tests consequences of ordinary symplectic temperateness and its distance-base conversion. Thus one temperate metric and any two temperate weights always satisfy the diagonal conditions, with parameter \(H=h_g\). This includes metrics and weights with no initial ordinary continuity.

## AN03-WP-004 — Kernels fix the normalization

For \(a\in\mathcal S(W)\) and \(\tau\in\mathbb R\), define
\[
\operatorname{Op}_\tau(a)u(x)
=(2\pi)^{-n}\iint e^{i\langle x-y,\xi\rangle}
a((1-\tau)x+\tau y,\xi)u(y)\,dy\,d\xi.
\tag{W19}
\]
The choices \(\tau=0,1,1/2\) are left, right and Weyl quantization. We write \(a^w=\operatorname{Op}_{1/2}(a)\).

These definitions extend to every tempered distribution \(a\). For example, put
\[
K_a(z+t/2,z-t/2)
=(2\pi)^{-n}\int e^{i\langle t,\xi\rangle}a(z,\xi)\,d\xi,
\qquad
a(z,\xi)=\int e^{-i\langle t,\xi\rangle}
K_a(z+t/2,z-t/2)\,dt.
\tag{W20}
\]
Both equalities are identities of tempered distributions: partial Fourier transformation and the invertible linear coordinate map \((z,t)\mapsto(z+t/2,z-t/2)\) are continuous isomorphisms of Schwartz space and its dual. The absolute determinant of this coordinate map is one.

For \(u,v\in\mathcal S(V)\), define \(\langle a^wu,v\rangle\) by applying \(K_a\) to \(u(y)\overline{v(x)}\). This is a continuous map \(\mathcal S\to\mathcal S'\). Indeed, each Schwartz seminorm of the test tensor is bounded by a product of finitely many seminorms of \(u,v\); a tempered distribution is bounded by a finite sum of such seminorms. The same construction works for every \(\tau\). It also gives the usual weak integral interpretation of (W19). The kernel theorem contract supplies the converse: all continuous maps \(\mathcal S\to\mathcal S'\) are obtained uniquely in this way.

For a finite polynomial in frequency, \(a(x,\xi)=\sum_\alpha a_\alpha(x)\xi^\alpha\), left and right quantization put the coefficients on the corresponding sides:
\[
\operatorname{Op}_0(a)u=\sum_\alpha a_\alpha D^\alpha u,
\qquad
\operatorname{Op}_1(a)u=\sum_\alpha D^\alpha(a_\alpha u).
\]
This remains valid for coefficients \(a_\alpha\in\mathcal S'(V)\) and \(u\in\mathcal S(V)\). Indeed, the inverse Fourier transform of \(\xi^\alpha\) is \(D_t^\alpha\delta(t)\). In the left kernel the coefficient is evaluated at \(x\), so pairing this delta derivative against the input differentiates \(u\); in the right kernel it is evaluated at \(y\), so the output derivative acts on the entire product \(a_\alpha u\). These kernels are defined by tensor products followed by invertible linear coordinate changes, and multiplication of a tempered coefficient by the smooth Schwartz input is well defined. No product of two arbitrary distributions is used.

Taking the complex conjugate and exchanging \(x,y\) in (W20) proves
\[
(a^w)^*=(\overline a)^w.
\tag{W21}
\]
Here the adjoint is the distributional sesquilinear adjoint on Schwartz functions. A real symbol therefore gives a symmetric Schwartz-domain operator whenever it has values in \(L^2\). Equation (W21) alone makes no assertion of selfadjoint closure for a general real symbol.

## AN03-WP-005 — Linear observables and their unitary groups

For smooth symbols define
\[
\{a,b\}=\partial_\xi a\cdot\partial_xb-
\partial_xa\cdot\partial_\xi b.
\tag{W22}
\]
When one factor is affine, this is also defined for distributional other factors. Directly from (W20), multiplication of the output by \(x_j\) and differentiation by \(D_{x_j}\) give
\[
x_j a^w=(x_ja+\tfrac i2\partial_{\xi_j}a)^w,
\qquad
D_{x_j}a^w=(\xi_ja+\tfrac1{2i}\partial_{x_j}a)^w.
\]
For the first formula, write the output coordinate as \(z_j+t_j/2\), and integrate the \(t_j\) factor by parts in \(\xi_j\). For the second, \(\partial_{x_j}=\frac12\partial_{z_j}+\partial_{t_j}\) on the kernel. These operations on distributions are legitimate by duality. Linearity gives, for every affine \(L\),
\[
L^w a^w=\big(La+\{L,a\}/(2i)\big)^w.
\tag{W23}
\]

**Proposition.** For real affine \(L(x,\xi)=b\cdot x+c\cdot\xi+d\), the operator \(L^w=b\cdot x+c\cdot D+d\), initially on \(\mathcal S(V)\), is essentially selfadjoint. Its unitary group is
\[
U_tu(x)=e^{itd+it b\cdot x+it^2 b\cdot c/2}u(x+tc),
\qquad U_t=(e^{itL})^w.
\tag{W24}
\]

**Proof.** We first obtain the \(L^2\) Fourier transform from the Schwartz Plancherel prerequisite. The normalized transform \(\mathcal F_0=(2\pi)^{-n/2}\mathcal F\) is an isometry on \(\mathcal S\). If \(u_j\to u\) in \(L^2\), with \(u_j\in\mathcal S\), then \(\mathcal F_0u_j\) is Cauchy in \(L^2\); completeness defines its limit, independently of the approximating sequence. Density therefore extends \(\mathcal F_0\) to an isometry on all of \(L^2\). The normalized inverse extends by the same argument. Their compositions are the identity on the dense Schwartz subspace, hence on \(L^2\) by continuity. Thus the extension is unitary, using precisely Schwartz Plancherel and the declared density/completeness prerequisites.

Multiplication by a real linear function \(\ell\) on \(L^2\), with domain \(\{u:\ell u\in L^2\}\), is selfadjoint. If a vector lies in the adjoint domain, testing against compactly supported smooth functions identifies its adjoint value as \(\ell u\), so it belongs to the displayed domain; the reverse inclusion follows by integration. Compact cutoff followed by smooth convolution approximates every domain vector in the graph norm: on a fixed compact set \(\ell\) is bounded, and the error in commuting convolution past \(\ell\) is bounded by the mollifier radius times \(|\nabla\ell|\|u\|_2\). Thus \(C_c^\infty\), and hence \(\mathcal S\), is a core. The unitary Fourier transform gives the same conclusion for \(c\cdot D\), because its Fourier transform is multiplication by \(c\cdot\xi\).

If \(c=0\), this already proves the assertion. If \(c\ne0\), choose orthonormal coordinates with \(c=(\gamma,0,\ldots,0)\), \(\gamma>0\), and put
\[
\phi(x)=\gamma^{-1}
\left(b_1x_1^2/2+x_1\sum_{j>1}b_jx_j\right).
\]
Multiplication by \(e^{i\phi}\) is unitary and preserves Schwartz space, as does its inverse. Since \(c\cdot\nabla\phi=b\cdot x\),
\[
e^{-i\phi}(c\cdot D)e^{i\phi}=c\cdot D+b\cdot x.
\]
Unitary conjugation preserves selfadjointness and the core property. Addition of the real constant \(d\) completes the closure argument.

The formula for \(U_t\) in (W24) is unitary, preserves \(\mathcal S\), and obeys \(U_tU_s=U_{t+s}\). Its derivative on \(\mathcal S\) is \(iL^wU_t\). It is also the exponential of the closure just constructed: conjugate the translation group for \(c\cdot D\) by \(e^{i\phi}\); the identity
\(\phi(x+tc)-\phi(x)=t b\cdot x+t^2b\cdot c/2\)
gives exactly (W24). The pure multiplication case is immediate. Strong continuity follows from continuity of translations in \(L^2\), first for compactly supported smooth functions and then by density and the unitary norm bound. Finally, substituting \(e^{itL}\) into the distributional kernel formula sets \(y=x+tc\) and gives the same phase. ∎

For later use write \(P=(p,q)\in W^*\) and let \(E_P(x,\xi)=e^{i(p\cdot x+q\cdot\xi)}\). The preceding calculation gives
\[
E_P^wu(x)=e^{ip\cdot(x+q/2)}u(x+q),\quad
E_P^wE_Q^w=e^{i\sigma(P,Q)/2}E_{P+Q}^w,
\quad \sigma(P,Q)=q\cdot r-p\cdot s
\tag{W25}
\]
for \(Q=(r,s)\). The last scalar factor follows by subtracting the phase of \(E_{P+Q}^w\) from the phase of the composite. Fourier inversion expresses a Schwartz symbol as a superposition of the \(E_P\)'s with integrable coefficient \((2\pi)^{-2n}\widehat a(P)\). Thus (W24), together with continuity in the symbol distribution, characterizes Weyl quantization. This claim uses the Fourier representation of arbitrary Schwartz symbols and its dual extension; it does not require pointwise Fourier integrability for every tempered distribution.

## AN03-WP-006 — The exact product before extension

Let
\[
\mathcal A(P,Q)=\sigma(P,Q)/2,
\qquad
C_j(a,b)(X)=\frac1{j!}
\left[(i\mathcal A(D_X,D_Y))^j(a(X)b(Y))\right]_{Y=X}.
\tag{W26}
\]
For \(a,b\in\mathcal S(W)\), (W25) and Fourier inversion give
\[
a\#b=\left[e^{i\mathcal A(D_X,D_Y)}(a\otimes b)\right]_{Y=X},
\qquad (a\#b)^w=a^wb^w.
\tag{W27}
\]
All integrals of Fourier coefficients in this derivation are absolutely convergent; the plane-wave operators have \(L^2\) norm one. The resulting symbol is Schwartz, since the quadratic multiplier preserves \(\mathcal S(W\times W)\), and restriction to the diagonal preserves Schwartz space.

An equivalent formula, with no Fourier transforms left in it, is
\[
(a\#b)(X)=\pi^{-2n}\iint
a(X+S)b(X+T)e^{2i\sigma(T,S)}\,dS\,dT.
\tag{W28}
\]
For clarity about the constant and sign, the distributional Fourier transform of the kernel \(\pi^{-2n}e^{2i\sigma(T,S)}\), evaluated on the Fourier exponentials \(e^{iP\cdot S+iQ\cdot T}\), is \(e^{i\sigma(P,Q)/2}\). Indeed, integration in \(S\) imposes \(T=(q/2,-p/2)\). Its Jacobian is \(2^{-2n}\), so the coefficient is \(\pi^{-2n}(2\pi)^{2n}2^{-2n}=1\); the remaining phase is \((q\cdot r-p\cdot s)/2\). This calculation is an identity of tempered distributions, tested against Schwartz functions; it can equally be justified by Gaussian regularization and passage to that topology. Since the integrand in (W28) has integrable absolute value for Schwartz factors, the identity gives the ordinary integral as written.

The first two coefficients are
\[
C_0(a,b)=ab,\qquad C_1(a,b)=\{a,b\}/(2i).
\tag{W29}
\]
In fact, replacing each \(D\) by \(-i\partial\) in \(i\mathcal A(D_X,D_Y)\) gives exactly the second expression. The sign agrees with the affine identity (W23).

There are two related quadratic phases in this proof. The auxiliary phase of (W16) is \(A_0=2\sigma\); the actual multiplier in (W27) is \(\mathcal A=\sigma/2=A_0/4\). With the convention \(A(\Xi)=\langle B\Xi,\Xi\rangle\) fixed in AN03-U002,
\[
G^{\mathcal A}=16G^{A_0},\qquad
h_{G,\mathcal A}(X,X)=H(X)/4.
\tag{W30}
\]
Scaling a quadratic phase by \(t\ne0\) scales its symmetric map by \(t\), hence its phase-dual form by \(t^{-2}\). This proves (W30) directly. Keeping these phases distinct prevents a normalization error in applying the Gauss estimate.

## AN03-WP-007 — The full product with distinct metrics

**Theorem.** Let \(g_1,g_2\) be symplectically temperate, compatible metrics. Suppose \(H(X)\leq1\) everywhere, with \(H\) as in (W6). Set \(g=(g_1+g_2)/2\). Let \(m_j\) be positive, \(g_j\)-continuous weights, both symplectically temperate for \(g\). Then (W27) has a unique weakly continuous bilinear extension
\[
\#:\ S(m_1,g_1)\times S(m_2,g_2)\longrightarrow S(m_1m_2,g).
\tag{W31}
\]
For every integer \(N\geq0\),
\[
R_N(a,b)=a\#b-\sum_{j<N}C_j(a,b)
\quad\hbox{belongs to}\quad S(H^Nm_1m_2,g).
\tag{W32}
\]
These maps are continuous in the respective Fréchet symbol topologies and continuous on products of bounded source sets with their local smooth topologies. For every target derivative order \(k\), there is a finite \(J\) such that
\[
p_k(R_N(a,b);H^Nm_1m_2,g)
\leq C_{N,k}
p_{\leq J}(a;m_1,g_1)p_{\leq J}(b;m_2,g_2).
\tag{W33}
\]
Constants depend on the fixed dimension, orders and structural metric/weight constants. Individual inequalities \(g_j\leq g_j^\sigma\) and the mean inequality \(g\leq g^\sigma\) are not hypotheses.

**Proof.** Use \(G,M\) from (W15). The weight equivalence in AN03-WP-002 implies the own and cross weight conditions needed by AN03-WP-003. That criterion gives uniform temperateness along the diagonal for the auxiliary phase. By (W30), the same inequalities hold for the actual phase: its dual distances are larger, so the right sides of the required upper bounds only increase. Also \(G\leq G^{\mathcal A}\) on the diagonal because \(H/4\leq1\).

The tensor map \((a,b)\mapsto a\otimes b\) is a continuous bilinear map into \(S(M,G)\). To check this without coordinate assumptions, split every direction in \(W\times W\) into its two components. If its \(G\)-length is at most one, each component has length at most one for its corresponding metric. The multilinear product rule has at most \(2^l\) terms at derivative order \(l\), so
\[
p_l(a\otimes b;M,G)
\leq2^l p_{\leq l}(a;m_1,g_1)p_{\leq l}(b;m_2,g_2).
\tag{W34}
\]
The tensor map also preserves boundedness and local smooth convergence.

Apply AN03-GAU-007 and AN03-GAU-008 on the linear diagonal subspace. The metric induced there by \(G\) is \(g_1+g_2=2g\), which defines the same symbol space as \(g\), with the factor \(2^{k/2}\) in order \(k\). The restricted weight is \(m_1m_2\), and the remainder factor in (W30) is \((H/4)^N\). The harmless fixed factor \(4^{-N}\) is absorbed in the seminorm bound. Equations (W31)–(W33) follow from the Gauss theorem and (W34).

The weight \(H\) is positive and locally \(g\)-continuous. Positivity follows from nondegeneracy in (W6); local comparison follows by comparing both metrics at nearby points and dualizing. It is also temperate for the mean: (W12) gives \(g_{j,Y}\leq Cg_{j,X}R^L\), where \(R=1+g_Y^\sigma(X-Y)\); dualizing the estimate for \(j=2\) gives \(q_2(Y)\geq C^{-1}q_2(X)R^{-L}\). Hence \(H(Y)\leq CH(X)R^L\). Thus the target weights in (W32) satisfy the asserted local and global conditions.

Finally, AN03-MET-007 gives compactly supported approximants to both input symbols that remain bounded and converge locally smoothly. For those approximants the definition agrees with (W27). Weak continuity passes to their limit and gives uniqueness. This is a bounded-set argument, rather than an incorrect claim of compact-support density in the full Fréchet topology. ∎

**Polynomial termination.** If either factor in the theorem is a polynomial of total degree less than \(N\), then \(R_N(a,b)=0\). Here is a justification that also covers a noncompact second factor. For compactly supported smooth \(u\) on the product space, Taylor's integral formula gives, for \(N\geq1\),
\[
T_{\mathcal A}u-\sum_{j<N}(i\mathcal A(D))^ju/j!
=\frac{i^N}{(N-1)!}\int_0^1(1-t)^{N-1}
T_{t\mathcal A}\big(\mathcal A(D)^Nu\big)\,dt.
\tag{W35}
\]
This identity extends to the bounded-symbol restriction on the diagonal. To see the needed continuity, expand \(\mathcal A(D)^N\) into its finitely many constant-coefficient directional derivatives. For a term \(\partial_{V_1}\cdots\partial_{V_{2N}}u\), use the weight
\(M_V(Z)=M(Z)\prod_\ell G_Z(V_\ell)^{1/2}\), as in AN03-GAU-007. It is locally continuous and phase-temperate along the diagonal by the product-metric comparisons already proved. The corresponding Gauss evaluation estimate is uniform for \(0< t\leq1\): its dual distance is \(t^{-2}G^{\mathcal A}\), so the same temperateness constants work and its uncertainty bound is no larger. At \(t=0\), use ordinary evaluation. These estimates give a common integrable bound in (W35). Bounded compactly supported approximation, weak continuity for each \(t\), and dominated convergence extend (W35) term by term. This argument needs only finitely many modified weights.

Every term of \(\mathcal A(D_X,D_Y)^N(a\otimes b)\) differentiates each factor exactly \(N\) times. Thus it is identically zero when either polynomial has degree less than \(N\). The right side of (W35) vanishes, proving termination. The assertion concerns a polynomial that belongs to the specified input symbol space; it does not assert membership for every polynomial and every weight.

**One metric and scalar parity.** Taking \(g_1=g_2=g\), \(H=h_g\), gives the full one-metric theorem under \(g\leq g^\sigma\). For scalar symbols, interchanging the two arguments changes \(\mathcal A\) to \(-\mathcal A\). Hence \(C_j(b,a)=(-1)^jC_j(a,b)\). Apply (W32) with \(N=3\) to the difference and with \(N=2\) to the sum. Using (W29) gives
\[
a\#b-b\#a-\{a,b\}/i\in S(h_g^3m_1m_2,g),
\qquad
a\#b+b\#a-2ab\in S(h_g^2m_1m_2,g).
\tag{W36}
\]
The even/odd cancellation uses commutativity of scalar multiplication. Applying it unchanged to matrix symbols would be incorrect.

Equations (W31)–(W36) construct a symbol product and its estimates. Equation \((a\#b)^w=a^wb^w\) has been proved here for Schwartz symbols. For arbitrary metric symbols the composition on the right needs a common operator domain and continuity theorem; that is the separate operator-calculus dependency AN03-E02. It is not inferred from two maps \(\mathcal S\to\mathcal S'\), whose composite need not be defined.

## AN03-WP-008 — Classical parameter ranges and change of quantization

Consider, for real \(\mu\),
\[
g_{(x,\xi)}(y,\eta)=\langle\xi\rangle^{2\delta}|y|^2+
\langle\xi\rangle^{-2\rho}|\eta|^2,
\qquad m(x,\xi)=\langle\xi\rangle^\mu,
\quad 0\leq\delta\leq\rho\leq1,\quad\delta<1.
\tag{W37}
\]
AN03-EX-001 proves slow variation and local weight continuity. Direct quadratic inversion gives
\[
g_{(x,\xi)}^\sigma(y,\eta)
=\langle\xi\rangle^{2\rho}|y|^2+
\langle\xi\rangle^{-2\delta}|\eta|^2,
\qquad h_g=\langle\xi\rangle^{\delta-\rho}\leq1.
\tag{W38}
\]

We verify the global condition, including the endpoint. Let \(r=\langle\eta\rangle\), \(s=\langle\xi\rangle\), \(d=|\eta-\xi|\). It suffices to bound
\[
(r/s)^{2\delta}+(s/r)^{2\rho}
\leq C(1+d^2r^{-2\delta})^L.
\tag{W39}
\]
If \(r/2\leq s\leq2r\), the left side is bounded. If \(r>2s\), the Lipschitz property of \(\langle\cdot\rangle\) gives \(d\geq r-s>r/2\), so the base on the right is at least a constant times \(r^{2(1-\delta)}\). Since \(s\geq1\), an exponent \(L\geq\delta/(1-\delta)\) controls the first ratio; the second is at most one. If \(s>2r\), then \(d>s/2\), and \(d^2r^{-2\delta}\geq(s/r)^2/4\) because \(r\geq1\) and \(\delta\leq1\). Now \(L\geq\rho\) works. This proves (W39) and metric temperateness. The same three cases control \((r/s)^\mu\): in the second case take \(2L(1-\delta)\geq\max(\mu,0)\), and in the third take \(2L\geq\max(-\mu,0)\). Thus every indicated real-order weight is temperate.

The class \(S(m,g)\) is exactly \(S^\mu_{\rho,\delta}\). The product theorem therefore has remainder order
\(\mu_1+\mu_2-N(\rho-\delta)\). When \(\delta=\rho<1\), it remains a continuous closure and finite-remainder theorem, but the order does not decrease with \(N\). This is not an asymptotic expansion in decreasing orders at that endpoint. At \((\rho,\delta)=(1,1)\), (W39) fails: take \(s=1\), \(r\to\infty\); its right-hand base stays bounded while the first ratio grows. The present argument therefore excludes type \((1,1)\).

For completeness the same metric calculation proves the full symbol change between the quantizations (W19). For fixed real \(\tau,s\), set
\[
b=\exp\big(i(\tau-s)\langle D_x,D_\xi\rangle\big)a.
\tag{W40}
\]
Then \(\operatorname{Op}_s(b)=\operatorname{Op}_\tau(a)\) as distributional kernels. One can verify the sign on a plane wave: in the kernel coordinates for \(s\), the old base point is \(z+(s-\tau)t\). Partial Fourier inversion sets \(t=-q\), giving the factor \(e^{i(\tau-s)p\cdot q}\), exactly the multiplier in (W40). Fourier transformation and invertible linear coordinate changes extend this identity to all tempered distributions.

For the auxiliary phase \(2\langle D_x,D_\xi\rangle\), the phase-dual metric is (W38). For the actual phase in (W40), its Gauss parameter is \(|\tau-s|h_g/2\). If this exceeds one, replace \(g\) by a fixed sufficiently small positive multiple; this leaves the symbol class unchanged, preserves the local and temperateness hypotheses, and makes that parameter at most one. AN03-GAU-008 now proves
\[
b-\sum_{|\alpha|<N}
\frac{(\tau-s)^{|\alpha|}}{\alpha!}
\partial_\xi^\alpha D_x^\alpha a
\ \in\ S^{\mu-N(\rho-\delta)}_{\rho,\delta}.
\tag{W41}
\]
The map and every remainder are continuous with finite source-seminorm control, and weakly continuous on bounded symbol sets. The zero phase is the identity and is handled directly. Constants may depend on \(\tau-s\); no uniformity over unbounded quantization parameters is claimed.

These classical symbols are tempered, since all coordinate derivatives have polynomial growth in \(\xi\), uniformly in \(x\). Bounded symbol approximants that converge locally smoothly converge also in \(\mathcal S'\): their common polynomial growth bound and the rapid decay of a Schwartz test function give dominated convergence. Thus the Gauss extension used in (W40) agrees with its distributional Fourier multiplier, since both are limits of the same compactly supported approximants, and the Gauss limit is locally smooth on the whole phase space. Their left quantizations map \(\mathcal S\) continuously into \(\mathcal S\): write the output as \((2\pi)^{-n}\int e^{ix\cdot\xi}a(x,\xi)\widehat u(\xi)\,d\xi\). Any output derivative introduces finitely many polynomial factors and symbol derivatives. To multiply by \(x^\alpha\), integrate by parts \(|\alpha|\) times in \(\xi\). Each term is bounded uniformly in \(x\) by finitely many Schwartz seminorms of \(u\), after choosing a sufficiently integrable negative power of \(\langle\xi\rangle\). This proves all output Schwartz seminorms. The other quantizations have the same mapping property by (W40).

In particular, the adjoint of a left quantization has left symbol
\(\exp(i\langle D_x,D_\xi\rangle)\overline a\), which is in the same class by (W40). Transposition of its continuous Schwartz action extends the original operator continuously to \(\mathcal S'\). This classical statement is proved using (W37); it does not close the separate arbitrary-metric operator dependency.

## AN03-WEX-001 — Scales that one-metric uncertainty would exclude

Take a standard symplectic Euclidean form \(Q\) with \(Q^\sigma=Q\), and choose constants \(L>1\), \(0<\varepsilon\leq1\). Let
\[
g_1=LQ,\qquad g_2=\varepsilon^2L^{-1}Q,\qquad m_1=m_2=1.
\tag{W42}
\]
All local and cross comparisons are constant. The duals are \(L^{-1}Q\) and \(L\varepsilon^{-2}Q\), so \(H=\varepsilon\). But \(h_1=L\), and
\(h_g=(L+\varepsilon^2/L)/2\), which exceeds one when \(L\) is large. Theorem (W31) still applies, with the \(\varepsilon^N\) remainder bound. Thus replacing its hypotheses by uncertainty of the mean would discard actual valid cases. Letting \(L\) grow also separates \(H\) from \(h_g\) by an arbitrarily large ratio.

As a normalization check in one dimension, (W23) gives
\(x\#\xi=x\xi+i/2\) and \(\xi\#x=x\xi-i/2\). Their difference is \(i\), agreeing with \([x,D]=i\). Since either factor has degree one, every remainder with \(N\geq2\) vanishes exactly.

## AN03-WPS-001 — Problems with solutions

**Problem 1.** On \(\mathbb R^2\), take \(g_1=9Q\), \(g_2=Q/36\). Compute \(H,h_1,h_2,h_g\). Which theorem applies to their product?

**Solution.** Their duals are \(Q/9\) and \(36Q\). Thus \(H^2=9/36=1/4\), \(H=1/2\), while \(h_1=9\), \(h_2=1/36\). The mean is \((325/72)Q\), so \(h_g=325/72\). The distinct-metric theorem applies with remainder weight \(2^{-N}\); its one-metric specialization applied to the mean does not meet its uncertainty hypothesis. Each chosen symbol must, of course, have the derivative bounds in its own stated metric.

**Problem 2.** Prove the sharp scalar version of (W5) for \(g_1(t)=a t^2\), \(g_2(t)=b t^2\), \(a,b>0\), using ordinary duality in place of \(\sigma\).

**Solution.** Minimize \(2((t-z)^2/a+z^2/b)\). Differentiation gives \(z=bt/(a+b)\). Substitution gives \(2t^2/(a+b)\), which is the ordinary dual of \((a+b)t^2/2\). The missing factor two would give the wrong dual even when \(a=b=1\).

**Problem 3.** Let \(a(x,\xi)=x^2\), \(b(x,\xi)=\xi^2\) in one dimension, with polynomial weights that contain them. Compute the exact Weyl product and its operator.

**Solution.** The zeroth term is \(x^2\xi^2\); the first is \(\{x^2,\xi^2\}/(2i)=2ix\xi\). The only second-order term differentiates twice in \(x\) in the first factor and twice in \(\xi\) in the second; its coefficient is \((i/2)^2/2!=-1/8\), so its value is \(-1/2\). All later terms vanish. Therefore \(x^2\#\xi^2=x^2\xi^2+2ix\xi-1/2\). Its Weyl operator equals \(x^2D^2\): apply (W23) twice to multiplication by \(x\), starting with \(\xi^2\). This polynomial calculation defines the composite on \(\mathcal S\) directly, so it does not require the unresolved general operator theorem.

**Problem 4.** For \(\rho=\delta=2/3\), what does an arbitrary number of terms in (W41) guarantee? Why can it not prove that its remainder is smoothing?

**Solution.** Every finite remainder belongs to the original order \(S^\mu_{2/3,2/3}\), with the indicated finite-seminorm continuity. The loss \(N(\rho-\delta)\) is zero for every \(N\). There is therefore no assertion of membership in lower orders, much less in their intersection. The theorem remains useful for stable quantization changes; the estimate alone supplies no smoothing conclusion.

**Problem 5.** Why is (W14) not replaceable by temperateness of \(m_j\) for \(g_j\) alone?

**Solution.** Necessity follows by setting the second input point equal to the diagonal point in the product-weight inequality: its remaining distance is measured by \(q_2(X)\), and its remaining ratio is \(m_1(Y)/m_1(X)\). This gives the first cross-weight test; exchanging the two roles gives the second. To show that own temperateness does not imply these tests, we give a compatible pair and a weight for which one test fails.

In one dimension put \(s=\langle\xi\rangle\) and
\[
g_{1,(x,\xi)}=s\,dx^2+s^{-1}d\xi^2,\qquad
g_2=dx^2+d\xi^2,\qquad
m_1(x,\xi)=\sqrt{1+s x^2},\qquad m_2=1.
\]
Here \(g_1^\sigma=g_1\); it is the temperate classical metric with \(\rho=\delta=1/2\). The second metric is constant. The first cross metric test follows from
\(\max(\langle\xi\rangle/\langle\eta\rangle,\langle\eta\rangle/\langle\xi\rangle)\leq C(1+|\xi-\eta|^2)\); the second is immediate because \(q_2\) is constant. Thus the metrics are compatible.

For \(X=(x,\xi)\), \(Y=(y,\eta)\), write \(r=\langle\eta\rangle\) and
\[
R=1+r(y-x)^2+r^{-1}|\eta-\xi|^2=1+g_{1,Y}^\sigma(X-Y).
\]
The same three cases used in (W39), now with \(\rho=\delta=1/2\), give \(r/s\leq C(1+|\eta-\xi|^2/r)\). The triangle inequality in \(\mathbb R^2\) therefore yields
\[
\frac{m_1(Y)}{m_1(X)}
\leq\max(1,\sqrt{r/s})+\sqrt r\,|y-x|
\leq C'\sqrt R.
\]
This proves own temperateness. For local continuity, a small \(g_{1,X}\)-displacement has a bounded \(g_{1,Y}\)-displacement by slow variation; the displayed bound and its reversed version then bound both weight ratios. Hence \(m_1\) is also \(g_1\)-continuous.

But for \(X=(0,\xi)\) and \(Y=(1,\xi)\), the weight ratio is \(\sqrt{1+\langle\xi\rangle}\), whereas \(1+q_2(X)(Y-X)=2\). The ratio is unbounded, so no fixed constants in (W14) can work. This example concerns the weight-compatibility criterion, which does not assume \(H\leq1\); that separate hypothesis is imposed only when the product estimate is applied.

## References and claim boundary

The antecedents are Lars Hörmander's [The Weyl calculus of pseudo-differential operators](https://onlinelibrary.wiley.com/doi/10.1002/cpa.3160320304), *Communications on Pure and Applied Mathematics* 32 (1979), 359–443, and the later treatment in Volume III of *The Analysis of Linear Partial Differential Operators*. Nicolas Lerner's author-hosted [chapter on metrics in phase space](https://webusers.imj-prg.fr/~nicolas.lerner/ch2booklerner.pdf), Theorems 2.3.7–2.3.8, and [Weyl–Hörmander lecture notes](https://webusers.imj-prg.fr/~nicolas.lerner/PSEUDO.2005.pdf), Theorems 3.2.4 and 4.1.1, provide further comparison references. The checked distinct-metric book theorem additionally imposes uncertainty on each input metric; example (W42) shows why that narrower statement cannot replace (W31). Its Fourier normalization also differs from the one fixed here. No reference prose, figures or proof text are imported.

The present unit gives full symbol-product statements at its declared prerequisites, including distinct metrics, scalar parity and endpoint behavior. Independent proof and correspondence review, external prerequisite closure, and rendered-artifact review remain required. General metric operator composition, continuity, positivity and the later elliptic and boundary results are separate course obligations.