Complex-phase Fourier integral operators

Positive imaginary parts damp oscillation. They also allow the real zeros of a canonical ideal to be singular. Composition and continuity therefore have to be formulated using the complex tangent plane of the ideal at each real zero. This lesson proves transverse composition and the exact geometric criterion under which every order-zero operator associated with such an ideal is locally bounded on $L^2$.

The symplectic convention is $\Omega=d\xi\wedge dx$, extended complex bilinearly; a positive plane satisfies $i\Omega(\overline V,V)\ge0$. We use $D=-i\partial$, Fourier transform $\widehat u(\eta)=\int e^{-iy\cdot\eta}u(y)\,dy$, and inverse factor $(2\pi)^{-n}$. All frequency cones exclude zero. Phase domains and closed conic amplitude supports are localized near a marked ray. Bounded frequencies and smooth kernel terms are kept as smoothing remainders.

The preceding positive-ideal lesson supplies smooth complex graph division, flat-residue necessity and sufficiency, positive critical values, partial Legendre reduction, complex nonstationary estimates and the two-way oscillatory representation theorem. The positive-plane lesson supplies its exact Hermitian nullspace theorem, and the composition lesson supplies Fourier localization and support conventions. A symbol residue retains the bounded normalized coefficient families constructed in the positive-ideal lesson; ordinary smooth ideal membership alone does not give uniform symbol damping estimates. The real phase-equivalence theorem is not being applied to complex phases.

Section 11 needs the endpoint symbol class $S^0_{1/2,1/2}$, which the ordinary calculus with $\delta<\rho$ does not cover. The complete endpoint proof connects this case directly to the AN-03 packet estimate already included in the programme. The metric calculations remain explicit; the broader AN-03 metric route is recorded as comparison.

1. Twisted ideals, bundles and adjoints

Let $\iota(x,\xi,y,\eta)=(x,\xi,y,-\eta)$. It is an involution and takes $\Omega_X+\Omega_Y$ to $\Omega_X-\Omega_Y$. If $J'$ is a positive conic Lagrangian ideal on $T^*(X\times Y)\setminus0$, its twist $J$ is a positive conic canonical ideal. Its tangent plane is Lagrangian and positive for $\Omega_X-\Omega_Y$. The prime denotes this twist, not a derivative. Its real zero set is denoted $J_{\mathbb R}$; its complex tangent at a real zero $p$ is $T_pJ$.

Suppose

For finite-dimensional smooth bundles $E\to X$, $F\to Y$, an FIO of order $m$ has kernel

It takes compactly supported smooth sections of $\Omega_Y^{1/2}\otimes F$ to distribution sections of $\Omega_X^{1/2}\otimes E$. In coordinates the kernel half-density is $|dx\,dy|^{1/2}$. Multiplying it by $u(y)|dy|^{1/2}$ produces $|dx|^{1/2}|dy|$, which can be integrated in $y$. This explains the density type without an arbitrary choice of volume form. Smooth positive Hermitian metrics give the local $L^2$ norms used below; changing such a metric on a compact patch gives equivalent norms.

Write $n=n_X+n_Y$. In local trivializations an oscillatory representative is

Here $\phi$ is homogeneous of degree one, $\operatorname{Im}\phi\ge0$, and its critical differentials are independent over $\mathbb C$ at the marked real critical point. The extended twisted ideal has generators

Its functions independent of $\theta$ form $J$.

The sesquilinear adjoint uses phase $-\overline\phi$, whose imaginary part is $\operatorname{Im}\phi$. Swapping the base factors, its generators are

They are minus the conjugates of (1.4), in swapped order. Therefore

Here the stars on bundles denote the conjugate duals for the sesquilinear pairing; Hermitian identifications express the same adjoint as an operator from $E$ to $F$. A bilinear transpose is a different operation. The order is unchanged because both the total base dimension and $N$ are unchanged. In a trivialization the amplitude is the coefficient adjoint $a^*$. In particular, replacing the phase by $-\phi$ would be incorrect when there is damping.

2. Transversality and simultaneous normal coordinates

Consider matching real points of $J_1$ and $J_2$, with nonzero covectors in $T^*X,T^*Y,T^*Z$. Put

and let $L_1\subset V_X\times V_Y$, $L_2\subset V_Y\times V_Z$ be their complex tangent planes. Composition is transverse when $L=L_1\times L_2$ is transverse to

The ambient symplectic form is $\Omega_X-\Omega_{Y_1}+\Omega_{Y_2}-\Omega_Z$. The subspace $D$ is coisotropic, with

Since $L^\perp=L$, transversality is equivalent to

These vectors are complex. Indeed $(L+D)^\perp=L\cap D^\perp$, so (2.2) is exactly $L+D$ equal to the ambient space. The expected intersection dimension is $n_X+n_Z$. Projection $L\cap D\to V_X\times V_Z$ has kernel $L\cap D^\perp=0$. Its image is consequently Lagrangian. Positivity descends because the two middle symplectic terms cancel on $D$.

We need real base coordinates in which both ideals have momentum graph descriptions. Here is the simultaneous-coordinate argument, including its complex-plane step. If a positive Lagrangian plane in ordinary cotangent coordinates is parametrized by columns $(Q,P)$, then $P+iQ$ is invertible. A vector in its kernel has $p=-iq$; hence

which contradicts positivity unless the vector is zero. The determinant of a momentum projection after a symmetric cotangent shear is therefore a nonzero polynomial in the shear entries: the complex shear just exhibited gives a nonzero value.

For the first twisted plane use shear variables $S_X,S_Y$; for the second use $S_Y,S_Z$. Twisting converts them to block shears with opposite signs on the input factor. Each determinant polynomial is nonzero, since its own two blocks can be chosen to realize the preceding $iQ$ test. Their product is a nonzero complex polynomial in the shared collection $(S_X,S_Y,S_Z)$. A nonzero complex polynomial cannot vanish at every real tuple: apply the one-variable polynomial identity successively to the real coordinates. Thus one real choice makes both determinants nonzero.

At a marked nonzero covector a real base-coordinate change with first derivative identity can realize any symmetric shear. If $\xi_k\ne0$, prescribe its second derivatives in that base component so that their contraction with $\xi$ is the desired symmetric matrix, and set the other second derivatives to zero. A sufficiently small neighborhood preserves a diffeomorphism. Do this in $X,Y,Z$; the common $Y$ choice is used by both ideals. The positive-ideal momentum-graph theorem now gives phases

where each $H_j$ has degree one. This does not presume a smooth real canonical relation.

3. The composed ideal in normal phases

The graph generators for the two ideals are

Before restricting to the middle diagonal, adjoin $y-y'$ and $\eta-\eta'$. Real division by these diagonal equations is exact. Restriction gives the ideal

Its functions independent of $y,\eta$ define $J_1\circ J_2$. Equivalently, lift a function of $(x,\xi,z,\zeta)$ to the middle diagonal and require it to belong to the restricted sum of the two ideals. This intrinsic restriction-and-elimination definition makes sense even when the real matching set is singular.

In these coordinates (2.2) says that a vector $v\in\mathbb C^{n_Y}$ satisfying

is zero. Thus the $n_Y$ complex differentials $d(H_{1\eta}+H_{2\eta})$, in all of $(\xi,\eta,\zeta)$, are independent. Invertibility of the middle Hessian is sufficient, but is not the definition; the outer blocks in (3.2) matter.

Set $\Phi=\phi_1+\phi_2$ and use mixed parameters $(y,\xi,\eta,\eta',\zeta)$. The critical functions are

To prove their differential independence, take a linear relation. The $dx,dz$ coefficients remove the second and fifth groups. The $dy$ coefficients make the coefficients of the third and fourth groups equal. The remaining relation has the form

Write $d\eta'=d\eta+d(\eta'-\eta)$. Independence from (3.2) forces $b=0$, after which $a=0$. Also $\operatorname{Im}\Phi\ge0$. Section 5 below homogenizes the mixed parameters, so this is a nondegenerate positive homogeneous phase.

Its extended canonical generators add $\Phi_x-\xi_{\rm out}$ and $\Phi_z+\zeta_{\rm out}$ to (3.3). These simply identify the normal parameters $\xi,\zeta$ with the endpoint cotangent coordinates. Restriction by $\eta'-\eta=0$ then leaves exactly (3.1). Thus the phase defines the composed ideal, and the positive-phase ideal theorem makes it positive and conic. This proves existence and coordinate independence of the local composed ideal, rather than only a claim about its real set.

4. Arbitrary phases: rank and exact elimination

Let $\phi(x,y,\theta)$, $\chi(y,z,\tau)$ be any positive nondegenerate phases for the two ideals at the matching point. The sum

has critical equations $\Phi_y=0,\phi_\theta=0,\chi_\tau=0$. We first prove their independence directly.

Suppose $a\cdot d\Phi_y+b\cdot d\phi_\theta+c\cdot d\chi_\tau=0$, with complex coefficients. By symmetry of the Hessians, its $\theta,x$ components say that $(\delta x,\delta y,\delta\theta)=(0,a,b)$ is a critical tangent for $\phi$ with zero output covector variation. Its $\tau,z$ components say that $(\delta y,\delta z,\delta\tau)=(a,0,c)$ is a critical tangent for $\chi$ with zero input covector variation. The $y$ component identifies their intermediate twisted covector variations: $-d\phi_y=d\chi_y$. Equation (2.2) makes this common intermediate tangent zero, including $a=0$. The critical-map differential of either nondegenerate phase is injective over $\mathbb C$: if base variations and all cotangent variations are zero, a parameter vector annihilates every column of $d\phi_\theta$, and their independence makes it zero. Consequently $b=c=0$. This proves the missing rank assertion without complex phase equivalence.

We also need equality of smooth ideals, not merely equality of tangent planes. If $f(x,\xi,z,\zeta)$ belongs to the intrinsic composed ideal, its lifted local representation uses

Divide each coefficient except the one multiplying $\phi_y+\eta$ by that graph ideal in $\eta$; its graph Jacobian is the identity. Absorb the quotients into the exceptional coefficient. All the other coefficients become independent of $\eta$. Their residual difference $R$ is therefore independent of $\eta$ and belongs to $(\eta+\phi_y)$.

The flat-residue theorem gives, on a smaller fixed patch, for every derivative and every $L$,

For a smooth nonnegative function $G=\operatorname{Im}\phi$, Taylor's theorem on a compact neighborhood gives $|dG|^2\le CG$. Euler's identity gives $G=\theta\cdot\operatorname{Im}\phi_\theta$. On a normalized frequency patch this implies $|\operatorname{Im}\phi_y|^2\le C|\phi_\theta|$. Taking arbitrary powers in (4.3) yields derivative flatness with respect to $\phi_\theta$.

For completeness, the division is explicit away from $\phi_\theta=0$: write

Every derivative of a coefficient has only a finite additional denominator power; (4.3) with arbitrarily large $L$ controls it and makes its derivatives tend to zero at the zero set. Extension by zero is smooth and flat. Thus $R\in(\phi_\theta)$, and the exceptional $\eta$-term can be removed. What remains in (4.2) is exactly the extended ideal of (4.1).

We have proved inclusion of the intrinsic composed ideal in the phase ideal. Both are canonical ideals with $n_X+n_Z$ independent generators. Write one generating system in the other: its differential matrix at the common real zero is invertible, so its coefficient matrix is invertible on a smaller patch. The inclusion is equality. In symbol applications graph division and (4.4) retain the bounded normalized coefficient families of the positive-ideal lesson; no uniform estimate is inferred from bare smooth ideal membership.

5. Operator composition, supports and orders

Assume the following global conditions on the cones in question:

1. Every real zero of $J_1,J_2$ has both endpoint covectors nonzero.
2. The complex tangent composition is transverse at every real matching point.
3. The real matching set

projects injectively and properly to $T^*(X\times Z)\setminus0$.

If $A_1$ and $A_2$ are properly supported FIOs of orders $m_1,m_2$, then

The middle bundle is $F$; contraction composes $\operatorname{Hom}(F,E)$ with $\operatorname{Hom}(G,F)$, in that order. The middle half-densities produce $|dy|$. No clean excess is included in this theorem.

Here are the support and convergence details. Localize both kernels in base coordinates and by homogeneous frequency partitions. Smooth factors give smooth compositions: for example a compactly supported smooth output test applied to one kernel has empty compatible wavefront by the nonzero endpoint condition, and the other smooth factor can then be paired with it. Proper support keeps the integrated base variables compact on compact endpoint sets. The same argument, with derivatives of endpoint tests, gives smoothness of these remainders.

In a nonsmooth pair of phase patches, a critical real matching point of the sum must satisfy

In particular the two intermediate covectors agree and are nonzero. On small patches, the intermediate covector size is comparable to $|\theta|$ in the first patch and to $|\tau|$ in the second; hence a matching contribution has these parameter sizes comparable. Away from a compact neighborhood of (5.2), either one phase has positive damping bounded below on the unit patch, or a normalized critical derivative is bounded below. Compactness supplies a uniform lower bound. The positive complex nonstationary theorem, applied to the sum and tested also in endpoint variables, gives arbitrary inverse frequency powers there, with all endpoint derivatives. Dyadic integration then converges in every smooth seminorm. This deals both with unmatched rays and with the noncomparable-frequency regions; low frequencies are smooth separately.

The noncomparable-frequency assertion can be made quantitative without a compactness claim at a zero parameter. Decompose into shells $|\theta|\asymp2^j$, $|\tau|\asymp2^k$. If $j\ge k+C_0$, near the first phase's real critical set its nonzero intermediate covector gives $|\phi_y|\ge c2^j$, whereas $|\chi_y|\le C2^k$; take $C_0$ large enough to obtain $|\Phi_y|\ge(c/2)2^j$. Away from that critical set the first phase supplies damping or a normalized critical derivative bounded below. The complex nonstationary estimate in the dominant phase variables, with $\tau/2^k$ as a bounded parameter, gives $2^{-Lj}$ for arbitrary $L$. Every fixed endpoint derivative and the two shell volumes cost at most $2^{C_1j}$ for a finite $C_1$, by the symbol estimates and $k\le j$. Summing the at most $O(1+j)$ smaller shells is absorbed by choosing $L$ larger. The case $k\ge j+C_0$ is symmetric. Comparable unmatched shells have the compact normalized lower bound already described. Thus all unmatched pieces have arbitrary smooth decay after summation, including arbitrarily unequal frequency scales.

Properness in condition 3 supplies a finite cover of the matching points over a compact output cone. Indeed its inverse image of a compact normalized output neighborhood is compact. Injectivity gives a single matching point for each projected real point, so the local composed ideal germs obtained in Sections 3–4 agree wherever their real domains overlap. Restriction away from real zeros gives the entire local smooth ideal, as usual. A finite partition and cutoff construction consequently glues the local ideals and their oscillatory representations. The real set of $M$ was never assumed smooth.

On a matching patch use (4.1). To turn its mixed parameters into ordinary homogeneous parameters, set

This is a smooth invertible map on the combined nonzero cone. Scaling $\omega$ scales $\theta,\tau$ and leaves $y$ fixed, so $\Phi$ is homogeneous of degree one. Its block triangular derivative has determinant $R^{n_Y}$. Therefore $dy\,d\theta\,d\tau=R^{-n_Y}d\omega$.

Before this Jacobian, the product amplitude has order

On comparable cones $R$ is comparable to both parameter sizes, and differentiation of $y=\omega_y/R$ or either normalized direction has the appropriate symbol loss. Multiplication by $R^{-n_Y}$ thus gives order

The constants also match exactly:

This is the normalization for (5.1).

To justify the kernel identity, first impose compact frequency cutoffs on both integrals. Fubini applies and gives their actual composition integral. On the matching part the transformed amplitudes stay in a bounded set of the symbol class (5.4) and converge locally with every derivative; the positive oscillatory construction is continuous for such cutoff limits after testing and integrating by parts sufficiently many times. The unmatched parts converge to the smooth remainders just estimated, uniformly in every prescribed finite number of derivatives. These two limits identify the distributional composition and its representation. This proves (5.1) with its exact order, supports and bundle contraction.

6. What the universal L² theorem says

For a positive conic canonical ideal satisfying (1.1), the following are equivalent:

1. Every scalar order-zero FIO associated with $J$ is continuous $L^2_{\mathrm{comp}}(Y;\Omega_Y^{1/2})\to L^2_{\mathrm{loc}}(X;\Omega_X^{1/2})$.
2. At each real zero, $T_pJ$ contains no nonzero real vector of either form

The first statement has a universal quantifier over amplitudes and operators. Extra decay can make an individual operator bounded even if (6.1) fails. The condition permits complex one-sided vectors that are not real. Finite-rank bundle versions follow by local trivialization and matrix entries; scalar necessity embeds in one nonzero input and output fiber direction. Proper support or compact localization gives the corresponding bounded $L^2$ maps between the chosen patches, not an automatic uniform bound on a noncompact manifold.

We prove necessity in Section 7 and sufficiency in Sections 8–11. The distinction between the real condition here and the complex condition (2.2) is central to the proof.

7. Necessity: a quadratic zoom and a real transport obstruction

Fix a real zero. Section 2, for a single ideal, gives a normal phase

Translate the bases so that the marked point has $x=y=0$, and put $p_0=(\xi_0,\eta_0)$. Then $dH(p_0)=0$, and Euler's identity also gives $H(p_0)=0$. Write $n=n_X+n_Y$ and let $Q(q)=\tfrac12 H''(p_0)[q,q]$. Since $-\operatorname{Im}H\ge0$ has a minimum at $p_0$, $\operatorname{Im}Q\le0$.

Choose a smooth amplitude in $S^{-n/4}$, supported in a small cone about $p_0$, equal there at high frequency to

Use base cutoffs equal to one near $(0,0)$. By the positive representation theorem this gives an order-zero FIO $A$, because the normal phase has $N=n$. The universal hypothesis makes this chosen operator locally $L^2$ bounded, with a bound $M$ on fixed compact base patches.

For compactly supported smooth $u,v$, form norm-preserving packets

Their supports shrink into the regions where the base cutoffs are one. Their Fourier transforms are $t^{-n_Y/2}\widehat u((\eta-t^2\eta_0)/t)$ and the analogous expression for $v$. Thus, with $c=c_{n,n}$,

Homogeneity gives

The factor $1/2$ is already in $Q$; there is no additional factor from the zoom.

Here is a uniform domination for this limit. When $|q|\le c_0t$, with $c_0<|p_0|/2$, the frequency $t^2p_0+tq$ has size comparable to $t^2$, and the scaled amplitude is bounded. Otherwise $t\le C|q|$; boundedness of $a$ gives the majorant $C(1+|q|)^{n/2}$. On the amplitude support $|e^{-iH}|\le1$. The Schwartz Fourier transforms in (7.4) absorb this polynomial majorant. Dominated convergence therefore gives

The quadratic model $A_0$, with kernel

is consequently a bounded map $L^2(Y)\to L^2(X)$. Equation (7.6), density of compact tests and the Riesz representation theorem give this extension. Its distribution kernel is nonzero: $e^{-iQ}$ is a nonzero bounded smooth tempered function, and Fourier inversion is injective. Tensor products of compact tests detect every nonzero distribution kernel. Hence some compact smooth $u$ has nonzero $L^2$ output $f=A_0u$.

The tangent canonical plane is

Suppose a nonzero real $(b,c;0,0)$ belongs to $L$. Symplectic orthogonality to every graph vector gives

Applying $b\cdot D_x-c\cdot x$ to (7.7), the resulting integrand factor is $b\cdot\xi-c\cdot x=-c\cdot(x-Q_\xi)$. This is a frequency derivative of the exponential. Integration by parts, first with cutoff and then in the tempered distribution limit, yields

If $b=0$, multiplication by $c\cdot x$ forces the $L^2$ function $f$ to vanish off a measure-zero hyperplane. Nonzero $f$ therefore requires $c=0$, contradicting the assumed vector.

If $b\ne0$, choose real linear coordinates with $b=\lambda e_1$, $\lambda\ne0$. A real quadratic solution of $b\cdot\nabla S=c\cdot x$ is

Equation (7.10) says $b\cdot D(e^{-iS}f)=0$. This gauge is unitary on $L^2$. The Fourier transform of the gauged function is supported on $b\cdot\xi=0$, also a measure-zero hyperplane; an $L^2$ function with that support is zero by Plancherel. This again contradicts nonzero $f$.

There is thus no real output-only vector. The bounded adjoint of $A_0$ has conjugate inverse canonical plane; a real input-only vector of $L$ becomes a real output-only vector there. The same argument excludes it. This proves both sides of (6.1) at the marked point, and hence necessity everywhere.

8. Sufficiency begins with positive-plane geometry

Assume (6.1). Localize both bases compactly and microlocalize $A$ near one real ray. Pseudodifferential partitions and the positive representation theorem supply these localizations; smooth remainders on compact patches are bounded by their square-integrable kernels. Only finitely many localized terms occur for a compact input/output patch. It is enough to bound each term.

Let $L=T_pJ$. The positive-plane theorem says that its Hermitian nullspace is

In particular a one-sided vector in $L\cap\overline L$ is zero under (6.1): its real and imaginary parts would be forbidden real one-sided vectors.

Failure of transversality for $\overline{J^{-1}}\circ J$ would give a nonzero common intermediate $X$-vector with zero $Y$ component in both $L$ and $\overline L$. Equation (8.1) excludes it. The local sum-phase composition proof therefore applies to $B=A^*A$.

We need a particular graph projection of its composed plane. Write its vectors as $(t_Y',t_Y'')$, represented by

Suppose the left base variation of $t_Y'$ and the right covector variation of $t_Y''$ are zero. Each $Y$ self symplectic product is then zero. Positivity in $L$ and the reversed inequality in $\overline L$ force $i\Omega_X(\overline{t_X},t_X)=0$. Both pairs in (8.2) are therefore in the Hermitian nullspace and belong to $L\cap\overline L$. Their difference $(0,t_Y'-t_Y'')$ belongs to that space and is zero by (8.1) and (6.1). The equal $Y$-vectors now have both base and covector variations zero. Finally $(t_X,0)\in L\cap\overline L$ is also zero. We have proved injectivity of

where $z$ is the left $Y$ base and $\eta$ the right $Y$ covector. The dimensions agree, so this is an isomorphism.

The partial Legendre theorem for positive ideals now gives a defining phase

There is a local unique graph for the intermediate critical variables in these external coordinates. This also gives local injectivity of the real matching projection: a real matching point has its intermediate variables equal to this graph's values. With smaller closed conic supports, the intermediate variables and normalized directions stay in compact subsets of their patches; the localized matching projection is proper. Thus the local use of composition is consistent with the injectivity/properness hypotheses of Section 5. It makes no global assertion about other matching rays.

The order of $B$ is zero. The phase (8.4) has base dimension $2n_Y$ and $n_Y$ parameters, so its amplitude has ordinary order zero. Modulo the already bounded smooth terms, write

The amplitude may depend on both bases; Section 11 handles that dependence by a uniform localized Fourier series in $y$. The key new estimate is

An arbitrary positive phase does not satisfy (8.6). We prove it from the self-composition structure.

9. Why self-composition controls the phase gradient

Let $\phi(x,y,\theta)$ represent the localized $A$. Define

Let $\mathcal I$ be its ideal of derivatives in $(x,y,\theta,\tau)$. The projection isomorphism (8.3), together with sum-phase nondegeneracy, makes the Jacobian of these critical derivatives in the same variables invertible over $\mathbb C$. To see this, a vector in its kernel is a critical tangent with $\delta z=\delta\eta=0$; its composed canonical vector is zero by (8.3), and injectivity of the critical-map differential makes the entire vector zero. The positive critical-value/graph theorem therefore supplies generators

and

The degrees of $X,Y,\Theta,T$ are $0,0,1,1$, respectively. On a fixed unit-frequency patch the positive critical-value estimate is

This is non-strict positivity; no invertible imaginary Hessian is assumed.

Put $G=\operatorname{Im}\widehat\phi=\operatorname{Im}\psi$ and let primes denote real parts of the four graph functions. At $(X',Y',\Theta',T')$, each generator (9.2) has size at most $C\sqrt G$. Evaluating (9.3) there gives an error $O(G)$. Its imaginary part, using that $y\cdot\eta$ is real, gives

Each summand is nonnegative. The elementary nonnegative-function gradient inequality therefore bounds every imaginary phase gradient at either point by $C\sqrt G$.

The critical generators $\mathcal P_x$, $\mathcal P_\theta$, $\mathcal P_\tau$ give, at the same real points,

Here $\phi_\theta(X',z,T')$ denotes differentiation in its own third variable. The imaginary gradients are already controlled by (9.5), so (9.6) controls differences of the real $x$ gradients and of the real parameter gradients.

Write $\phi=\phi_1+i\phi_2$ and introduce the real map

Equations (9.5)–(9.6) imply

We now prove that $\partial_{(y,\theta)}f$, for fixed $x$, is injective at the mark. A real vector in its kernel has $d\phi_\theta=0$, $d\phi_x=0$, and real $d\phi_y$. Its critical-map image is the real canonical tangent $(0,0;dy,-d\phi_y)$. Condition (6.1) makes $dy=d\phi_y=0$. The critical-map differential is injective, so $d\theta=0$ as well.

Select an injective square minor of this derivative. The real inverse-function theorem applied to $(x,y,\theta)\mapsto(x,f_{\rm minor})$, after choosing a product box in the image of this local diffeomorphism and a smaller compact preimage, gives a uniformly bounded inverse derivative. For fixed $x$, the segment between the two minor-coordinate values lies in that box. The mean value theorem for the inverse gives

Apply this to (9.8):

Differentiate (9.3) with respect to the external variables $z,\eta$. Differentiating a product of two ideal generators gives a function in $\mathcal I$. Thus at the real graph point,

Also $\mathcal P_y=\phi_y(x,y,\theta)+\eta\in\mathcal I$, whence $\phi_y(X',Y',\Theta')+\eta=O(\sqrt G)$. Equation (9.9) and a uniform derivative bound compare this gradient with the one at $(X',z,T')$; conjugation changes it by another $O(\sqrt G)$ using (9.5). Subtracting the derivatives of $z\cdot\eta$ in (9.10) now gives

Finally restore homogeneity. At $\eta=r\widehat\eta$, $\psi_z$ scales by $r$, $\psi_\eta$ has degree zero and $G$ scales by $r$. Multiplying the two squared bounds by $r^{-1}$ and $r$, respectively, proves (8.6). Both weights are necessary.

10. Damping produces the endpoint symbol class

For a compact base patch define $S^0_{1/2,1/2}$ by

We prove the stronger estimate

A differentiated exponential is a finite sum of products $e^{i\psi}\prod_j\partial_z^{\alpha_j}\partial_\eta^{\beta_j}\psi$, where each factor has positive total derivative order and the multiindices sum to $(\alpha,\beta)$. If a factor has total order at least two, homogeneity on the fixed angular patch gives

The second inequality is exactly $|\alpha_j|+|\beta_j|\ge2$. A first-order factor instead uses (8.6):

A term with $k$ first-order factors is bounded by the right radial power times $G^{k/2}e^{-G}$. Since $\sup_{G\ge0}G^{k/2}e^{-G/2}<\infty$, it obeys (10.2). Summing the finite product expansion proves the estimate at every derivative order. Ordinary order-zero amplitudes, including extra compact base variables, can be multiplied into this estimate by Leibniz's rule.

The converse also holds for a smooth homogeneous $\psi$ on such a patch: $e^{i\psi}\in S^0_{1/2,1/2}$ implies (8.6). Fix a unit direction and write $g=G(z,\widehat\eta)$. The first derivative bounds at $t\widehat\eta$ give

For $0<g\le1$, choose $t=1/g$; both gradients are bounded by $C e\sqrt g$. If $g=0$, let $t\to\infty$, forcing both gradients to zero. If $g>1$, the gradients are uniformly bounded on the normalized compact patch and their squares are bounded by a constant times $g$. Homogeneity restores (8.6). This argument uses all radii; division by a finite-radius exponentially small quantity would not give the conclusion.

11. The exact endpoint proof and completion of sufficiency

The exact analytic input used here is proved in E0–E4 of the endpoint companion: a left symbol in $S^0_{1/2,1/2}$, with support in a fixed compact base set, is $L^2$ bounded by a finite number of its symbol seminorms. The proof uses the complete earlier AN-03 E23 packet estimate, a balanced dilation on each frequency shell, explicit summable off-diagonal Fourier kernels and the bounded overlap of neighboring shells. It includes the two-base amplitude actually occurring in (8.5). The broader AN-03 metric theorem B26 and conversion B8a were the original proof route; their role is recorded in the source comparison. The metric computations below identify the same endpoint class and its absence of a decreasing Planck factor.

Put $R=\langle\eta\rangle$ and

The symplectic dual is the same form: with the canonical symplectic matrix $J_0$, the matrix $J_0^t\operatorname{diag}(R,R^{-1})^{-1}J_0$ is $\operatorname{diag}(R,R^{-1})$, in every coordinate pair. Thus $g=g^\sigma$ and the Planck function is one. No lower-order gain from a Planck factor is asserted.

To check slow variation and temperateness, compare centers $X=(z,\eta)$, $Y=(z',\eta')$. Write $S=\langle\eta'\rangle$ and

The map $\eta\mapsto\langle\eta\rangle$ is 1-Lipschitz. If $d\le c^2$, with $c<1/2$, then $|R-S|\le c\sqrt S\le cS$. The ratios $R/S$ and $S/R$ are therefore uniformly bounded in this metric ball, proving slow variation.

Globally $R/S\le1+\sqrt{d/S}\le2(1+d)$. If $R\ge S/2$, then $S/R\le2$. Otherwise $|\eta-\eta'|\ge|S-R|>S/2$, so $d>S/4$; since $R\ge1$, $S/R\le S<4d$. Consequently

Self-duality makes this exactly symplectic temperateness. The constant weight one satisfies its weight conditions. The metric is even in frequency directions. Derivatives measured on $g$-unit vectors are bounded exactly when the Cartesian estimates (10.1) hold; a base derivative costs $R^{1/2}$ and a frequency derivative gains $R^{-1/2}$. Hence $S(1,g)=S^0_{1/2,1/2}$ with equivalent seminorms. The complete endpoint theorem E0–E4 applies directly to the compactly supported left symbols used below. No quantization change is needed.