Subellipticity and unique continuation · Self-checked by the writing AI

Smooth complex preparation and conic flow coordinates

A smooth complex symbol is not a holomorphic function. A formal complex root gives any finite Taylor approximation, but it does not by itself give an exact factorization. This distinction matters when a symbol is turned into an evolution equation: the imaginary part must be independent of the time momentum on a whole neighborhood.

We prove the degree-one case of Malgrange's smooth preparation theorem, including smooth dependence on parameters. An almost-analytic extension supplies a complex root of an auxiliary smooth function. Its failure to be holomorphic is flat on the real axis; this flatness makes the required quotient smooth. Real Hamiltonian flow then gives an exact symbol of the form \(\tau+iq(t,w)\). For homogeneous symbols we preserve the dilation rays and identify precisely when this construction is possible.

The geometric prerequisites are symplectic bases and ordinary Darboux coordinates in Phase space and generating families, and Theorem 2.1 of Corank geometry and sufficient continuity, which proves homogeneous Darboux coordinates with a prescribed radial tangent map. The real flow chart and the finite-jet comparison are in Rotated brackets and finite-jet flow coordinates. For the final operator application we use Canonical transport of fractional regularity and Detecting a fractional gain in a cone. We also use the smooth real implicit function theorem and smooth dependence of local flows.

Basic references are Malgrange [M], Lerner [L] and Pravda-Starov [P]. Our Hamilton convention is \(\iota_{H_f}\omega=-df\), so \(H_fg=\{f,g\}\) and \(\omega=d\xi\wedge dx\).

1. Extend off the real axis with a flat error

For \(z=x+iv\), put

\[ \partial_z=\tfrac12(\partial_x-i\partial_v),\qquad \partial_{\bar z}=\tfrac12(\partial_x+i\partial_v). \tag{1.1} \]

The other variables \(y\in\mathbb R^d\) remain real parameters. Flatness in \(v\) means that every \(v\)-derivative vanishes at \(v=0\), throughout the parameter neighborhood.

Lemma 1.1 (a parameter-dependent almost-analytic extension). Let \(f(x,y)\) be a smooth complex function near \((0,0)\). There is a smooth complex function \(F(x,v,y)\) near zero such that

\[ F(x,0,y)=f(x,y),\qquad \partial_v^jF(x,0,y)=i^j\partial_x^jf(x,y) \quad(j\geq0). \tag{1.2} \]

The function \(\partial_{\bar z}F\) is flat in \(v\). On every smaller compact neighborhood, for every multi-index \(\alpha\) and every integer \(N\geq0\),

\[ |\partial_{x,v,y}^{\alpha}\partial_{\bar z}F(x,v,y)| \leq C_{\alpha,N}|v|^N. \tag{1.3} \]

Proof. First multiply \(f\) by a compact smooth cutoff equal to one on a smaller real neighborhood. Extend the product by zero. All its derivatives are bounded. Take \(\chi\in C_c^\infty(\mathbb R)\), equal to one near zero and supported in \([-2,2]\). We construct

\[ F(x,v,y)=\sum_{j=0}^{\infty} \chi(v/\varepsilon_j)\frac{(iv)^j}{j!}\partial_x^jf(x,y). \tag{1.4} \]

Here \(\varepsilon_j>0\) decreases sufficiently rapidly. Fix an integer \(r\). A derivative of total order at most \(r\) of the \(j\)-th term is bounded, for \(j>r\), by

\[ C_{j,r}\varepsilon_j^{j-r}. \tag{1.5} \]

Indeed its support has \(|v|\leq2\varepsilon_j\). Each derivative of the cutoff contributes one inverse power of \(\varepsilon_j\), and each derivative of \(v^j\) removes one power of \(v\). The remaining real derivatives fall on the bounded derivatives of \(f\).

Choose \(\varepsilon_j<1/j\) for \(j\geq1\), and make (1.5) at most \(2^{-j}\) simultaneously for \(0\leq r\leq j/2\). This is possible because each exponent \(j-r\) is positive and only finitely many conditions are imposed at each step. For any fixed \(r\), the tail then converges with all derivatives through order \(r\). Thus (1.4) is smooth.

At \(v=0\), each individual cutoff is constant on a neighborhood. A \(j\)-th normal derivative receives a contribution only from the term of degree \(j\). Termwise evaluation is valid by the proved convergence in every differentiability order. It gives (1.2).

For every \(j\geq0\),

\[ \partial_v^j\partial_{\bar z}F(x,0,y) =\tfrac12\left(i^j\partial_x^{j+1}f +i^{j+2}\partial_x^{j+1}f\right)=0. \tag{1.6} \]

Differentiating this identity in \(x,y\) retains zero. Taylor's integral remainder in \(v\), on a smaller compact neighborhood, proves (1.3) for every derivative and every \(N\). ∎

The extension need not be holomorphic away from the real axis. We need the quantified flatness (1.3), rather than holomorphicity.

Lemma 1.2 (flatness absorbs a vanishing denominator). Let \(a(y),b(y)\) be smooth and real. Suppose a smooth function \(J(\tau,y)\) obeys

\[ |\partial_{\tau,y}^{\alpha}J(\tau,y)| \leq C_{\alpha,N}|b(y)|^N \tag{1.7} \]

for every \(\alpha,N\), locally uniformly. On \(b\ne0\), define

\[ C(\tau,y)= \frac{\tau-a(y)+ib(y)}{\tau-a(y)-ib(y)}J(\tau,y). \tag{1.8} \]

Extending \(C\) by zero where \(b=0\) gives a smooth function. Every derivative of \(C\) vanishes there and still satisfies bounds of the form (1.7).

Proof. The denominator has modulus at least \(|b|\). On \(b\ne0\), repeated quotient and product rules bound any derivative of the ratio in (1.8) by a fixed inverse power of \(|b|\), on a smaller compact neighborhood. Its undifferentiated modulus is one. Choose \(N\) in (1.7) larger than the inverse power arising in each Leibniz term. We obtain, for every prescribed \(L\),

\[ |\partial^\alpha C(\tau,y)|\leq C_{\alpha,L}|b(y)|^L \quad(b\ne0). \tag{1.9} \]

Extend these derivative functions by zero on \(b=0\); they are continuous. To check that they are the derivatives of the extended \(C\), fix a point \((\tau_0,y_0)\) with \(b(y_0)=0\). Smoothness of \(b\) gives \(|b(y)|\leq C|y-y_0|\). With \(L=2\), (1.9) bounds each extended derivative candidate by \(C'| (\tau,y)-(\tau_0,y_0)|^2\). Its derivative at that point is therefore zero. Off \(b=0\), differentiation is the original differentiation. Induction on derivative order proves smoothness and all the claimed identities, even when the zero set of \(b\) is singular. ∎

2. Prepare the whole smooth function

Theorem 2.1 (complex preparation of degree one). Let \(f(\tau,y)\) be smooth and complex, with

\[ f(0,0)=0,\qquad \partial_\tau f(0,0)=d\ne0. \tag{2.1} \]

There are smooth complex functions \(A(\tau,y),\phi(y)\), on a smaller neighborhood, such that

\[ f(\tau,y)=A(\tau,y)(\tau-\phi(y)),\qquad A(0,0)=d,\quad\phi(0)=0, \tag{2.2} \]

and \(A\) is nonzero throughout that neighborhood. The function \(\phi\) is independent of \(\tau\).

Proof. Apply Lemma 1.1 in the \(\tau\) variable. Regard its extension \(F(x,v,y)\) as a smooth map from two real variables \((x,v)\) to two real variables \((\operatorname{Re}F,\operatorname{Im}F)\). At zero,

\[ F_x=d,\qquad F_v=id. \tag{2.3} \]

The real differential is complex multiplication by \(d\); its determinant is \(|d|^2>0\). The real implicit function theorem therefore supplies a smooth complex number

\[ z(y)=a(y)+ib(y),\qquad z(0)=0,\qquad F(z(y),y)=0. \tag{2.4} \]

Here \(F(z,y)\) denotes the auxiliary extension at \((\operatorname{Re}z,\operatorname{Im}z,y)\).

For real \(\tau\), integrate along the straight segment

\[ z_s=z(y)+s(\tau-z(y)),\qquad 0\leq s\leq1. \tag{2.5} \]

On a smaller neighborhood the entire segment stays within the extension's domain. The real chain rule, written using (1.1), gives

\[ f(\tau,y) =(\tau-z)I(\tau,y)+(\tau-\bar z)J(\tau,y), \tag{2.6} \]

where

\[ I=\int_0^1 F_z(z_s,y)\,ds,\qquad J=\int_0^1 F_{\bar z}(z_s,y)\,ds. \tag{2.7} \]

Both integrals are smooth. Along the segment, \(\operatorname{Im}z_s=(1-s)b(y)\). All derivatives of the segment map are bounded on a smaller compact neighborhood. Applying the chain rule to \(J\) and then (1.3) proves

\[ |\partial_{\tau,y}^{\alpha}J|\leq C_{\alpha,N}|b(y)|^N \quad\text{for every }\alpha,N. \tag{2.8} \]

When \(b\ne0\), divide (2.6) by \(\tau-z\). The resulting factor is

\[ A=I+\frac{\tau-\bar z}{\tau-z}J. \tag{2.9} \]

Lemma 1.2 extends the second term smoothly by zero on \(b=0\). Thus \(A\) is smooth everywhere under consideration. Equation (2.6) continues to imply \(f=A(\tau-z)\). On the interior of \(b=0\), one also sees it directly: the entire segment lies on the real axis and \(J=0\). At boundary points continuity gives the same identity.

At zero, \(I=d\) and the flat correction is zero. Hence \(A(0,0)=d\). Shrink the neighborhood so that \(A\ne0\), and put \(\phi=z\). This proves (2.2). ∎

This is the degree-one smooth preparation theorem attributed to Malgrange. Its root belongs to a chosen auxiliary extension. The theorem asserts an exact factorization of the original function on real arguments; it does not assert a holomorphic extension of that function or uniqueness of the complex root.

Write \(\phi=h-ib\), with \(h,b\) real. Equation (2.2) becomes

\[ A^{-1}f=\tau-h(y)+ib(y). \tag{2.10} \]

The elementary polynomial preparation in the preceding lesson remains useful when only a fixed finite jet is required. The new factorization controls an actual neighborhood, including flat terms invisible to every finite Taylor polynomial.

3. Keep the cotangent rays while normalizing the symbol

Let \(R\) be the radial field on a conic symplectic manifold:

\[ \mathcal L_R\omega=\omega,\qquad \lambda=\iota_R\omega,\qquad R\ne0. \tag{3.1} \]

A homogeneous canonical map commutes with positive dilation. It preserves \(R\), \(\omega\) and consequently \(\lambda\). Coordinate functions have degree zero for positions and degree one for momenta.

Theorem 3.1 (exact homogeneous complex reduction). Suppose \(p\) is a smooth complex function homogeneous of degree one, \(p(c)=0\). The following are equivalent:

  1. \(H_p(c)\) is not a complex multiple of \(R_c\).
  2. There are homogeneous symplectic coordinates \((t,x';\tau,\eta')\), centered at \((0;0,e_{n-1})\), and a nonzero smooth degree-zero complex factor \(a\), such that
\[ (p/a)\circ\chi=\tau+iq(t,x',\eta'), \qquad q\text{ real and homogeneous of degree one}. \tag{3.2} \]

The function \(q\) is independent of \(\tau\), and \(q(0,0,e_{n-1})=0\). These conditions require \(n\geq2\).

Proof: choose a nonradial momentum direction. The Hamilton identity shows

\[ H_p(c)\in\mathbb C R_c \quad\Longleftrightarrow\quad dp(c)\in\mathbb C\lambda_c. \tag{3.3} \]

If (1) holds, the complex covector \(dp(c)\) has a nonzero restriction to the real hyperplane \(\ker\lambda_c\). Choose a real vector \(v\) in that hyperplane with \(dp(c)v\ne0\). Since homogeneity gives \(dp(c)R_c=p(c)=0\), the vectors \(v,R_c\) are independent. They are an isotropic pair because \(\omega(R_c,v)=\lambda_c(v)=0\).

Extend their span to a Lagrangian basis and then to a symplectic basis, using the symplectic basis theorem in the stated prerequisite. Choose its momentum directions so that \(v\) becomes \(\partial_{\xi_1}\), and \(R_c\) becomes \(\partial_{\xi_n}\) at the marked point \((0,e_n)\). The homogeneous Darboux theorem with this prescribed radial tangent map realizes the basis by an actual homogeneous coordinate chart. In these coordinates,

\[ p(0,e_n)=0,\qquad \partial_{\xi_1}p(0,e_n)\ne0, \qquad \xi_n>0 \tag{3.4} \]

on a smaller cone. Two independent isotropic vectors can exist only when \(n\geq2\).

Prepare on a transverse radial slice. Set

\[ \rho=\xi_n,\qquad s=\xi_1/\rho,\qquad \nu=(\xi_2/\rho,\ldots,\xi_{n-1}/\rho). \tag{3.5} \]

The list \(\nu\) is empty when \(n=2\). Homogeneity gives

\[ p(x,\xi)=\rho\,g(s,x,\nu). \tag{3.6} \]

At the marked point \(g=0\), and \(g_s\ne0\). Apply Theorem 2.1, with \(y=(x,\nu)\). Write its complex root as \(h_0(y)-ib_0(y)\). Extending the resulting functions along the rays gives the exact identity

\[ p=A(x,\xi)\bigl(\xi_1-h(x,\xi')+ib(x,\xi')\bigr), \tag{3.7} \]

where

\[ A=A_0(s,x,\nu),\qquad h=\rho h_0(x,\nu), \qquad b=\rho b_0(x,\nu). \tag{3.8} \]

Here \(A\ne0\) has degree zero. The real functions \(h,b\) have degree one and are independent of \(\xi_1\). Both vanish at the marked point. Preparation was performed on real radial-slice variables, then extended by homogeneity; no complex dilation was used.

Remove the real drift by a homogeneous flow. Put \(t=x_1\), let \(w=(x',\xi')\) denote the remaining canonical pairs, and set \(r=\xi_1-h(t,w)\). Solve

\[ \frac{d}{dt}w(t)=-H_{h(t,\cdot)}w(t),\qquad w(0)=v. \tag{3.9} \]

The real canonical flow chart proved in the preceding lesson is

\[ \Phi(t,\sigma,v) =\bigl(t,\sigma+h(t,w(t)),w(t)\bigr). \tag{3.10} \]

It satisfies \(r\circ\Phi=\sigma\) and

\[ \Phi^*\omega=d\sigma\wedge dt+\omega_v. \tag{3.11} \]

For completeness, its initial section at \(t=0\) has restricted form \(\omega_v\). Hamiltonian flow preserves this form. The time tangent is \(H_r\); pairing it with the other parameter tangents gives one in the \(\sigma\) direction and zero in the \(v\) directions, by \(r\circ\Phi=\sigma\). These are exactly the remaining pairings in (3.11).

Since \(h\) has degree one, its transverse Hamilton field commutes with transverse dilation. Uniqueness of (3.9) therefore gives

\[ w(t;M_\ell v)=M_\ell w(t;v),\qquad \ell>0. \tag{3.12} \]

Equation (3.10) consequently commutes with

\[ (t,\sigma,v)\longmapsto(t,\ell\sigma,M_\ell v). \tag{3.13} \]

The flow chart is homogeneous and fixes the marked point. Crucially, \(w(t;v)\) does not depend on \(\sigma\). Thus

\[ q(t,v)=b(t,w(t;v)) \tag{3.14} \]

is independent of the new time momentum and has degree one. Substituting (3.10) into (3.7), and composing its nonzero factor with the coordinate maps, proves (3.2). The flow and identities are constructed near the normalized point and extended along its rays. No completeness of a Hamilton field on an entire cone is assumed.

The converse. At a characteristic point of the right side of (3.2),

\[ H_{\tau+iq}t=1, \tag{3.15} \]

whereas \(Rt=0\). Multiplication by a nonzero factor multiplies the complex Hamilton vector at a characteristic point by that factor. A homogeneous symplectic map preserves the radial vector. Therefore \(H_p(c)\) cannot be a complex radial multiple. This proves the equivalence. ∎

Corollary 3.2 (every real homogeneous order). If \(p\) has real homogeneous degree \(m\), the same nonradial condition gives (3.2) with \(a\) homogeneous of degree \(m-1\).

Proof. Choose any positive smooth degree-one function \(\rho\) near the ray, and apply Theorem 3.1 to \(\rho^{1-m}p\). At its characteristic point,

\[ H_{\rho^{1-m}p}=\rho^{1-m}H_p. \tag{3.16} \]

The nonradial condition is unchanged. Combine the resulting degree-zero factor with \(\rho^{m-1}\). ∎

There is also an ordinary version. If \(p(c)=0\) and \(dp(c)\ne0\) on any symplectic manifold, an ordinary Darboux chart and a symplectic basis make a momentum derivative nonzero. Theorem 2.1 and the real flow chart (3.10) then give exactly \(p/a=\tau+iq(t,w)\), without a radial hypothesis. This works in dimension two as well. In a conic problem the radial hypothesis is the additional condition needed to retain the rays.

4. Transfer the symbol reduction to an estimate

The exact reduction retains the first bracket sign. With \(u=a^{-1}\), the multiplier identity from the preceding lesson gives

\[ \{\operatorname{Re}(up),\operatorname{Im}(up)\} =|u|^2\{\operatorname{Re}p,\operatorname{Im}p\} \quad\text{where }p=0. \tag{4.1} \]

After the symplectic change this becomes

\[ q_t\geq0\quad\text{on }q=0 \tag{4.2} \]

if the original first bracket is nonnegative on the characteristic set. The bracket depth is unchanged by the same multiplier and coordinate changes. An individual higher directional derivative needs the lower-vanishing hypothesis in Lemma 2.1 of the preceding lesson before the multiplier can be frozen.

Proposition 4.1 (microlocal operator realization). Let \(P\) be a properly supported scalar classical operator of real order \(m\), with homogeneous principal symbol \(p\). At a characteristic covector with nonradial \(H_p\), an elliptic order reduction, an elliptic order-zero multiplier and proper elliptic graph operators with two-sided microlocal inverses give an order-one operator whose principal symbol is

\[ \tau+iq(t,x',\eta') \tag{4.3} \]

on a smaller cone. It differs there by an operator of order zero from any proper classical quantization of (4.3). Microlocal loss \(0<\delta<1\) is preserved in both directions.

Proof. Multiply \(P\) on the left by a proper elliptic operator of order \(1-m\), whose positive principal factor is \(\rho^{1-m}\). The order becomes one and the loss is unchanged by the proved order-reduction equivalence in the conic-estimate lesson. Apply Theorem 3.1 to the new principal symbol.

The graph realization and Egorov theorem in the stated prerequisites quantize its homogeneous canonical map by proper order-zero graph operators \(T,S\), with \(ST\) and \(TS\) equal to identity modulo smoothing on the working cones. First divide the principal symbol by its nonzero degree-zero factor using a proper elliptic order-zero operator \(B\). Then

\[ \widetilde P=SB(EP)T \tag{4.4} \]

has principal symbol (4.3). The order of the chosen graph quantizations is zero; no unrecorded Sobolev shift occurs. The conic Sobolev equivalence and weak-test argument in the canonical-transport lesson preserve the loss, and their converse uses the two-sided inverses.

Near the normalized covector, \(\eta'\ne0\) and \(|\tau|\leq C|\eta'|\). Extend (4.3) from a smaller cone to a classical order-one symbol by a homogeneous conic cutoff and a low-frequency cutoff. The extension equals (4.3) near the marked ray. Any proper quantization \(L\) of this extension has the same principal symbol as \(\widetilde P\) there. The principal-symbol quotient theorem in the operator calculus gives

\[ \widetilde P-L\in\Psi^0 \quad\text{microlocally on that smaller cone}. \tag{4.5} \]

The strict loss condition makes order-zero changes harmless after order-one reduction, by the lower-order stability theorem in the conic-estimate lesson. This proves the final assertion in both directions. ∎

The proposition states a conic operator result. An imaginary symbol independent of \(\tau\) on this cone need not extend to a global full-frequency symbol with that independence: derivatives in \(\eta'\) are controlled by \(|\eta'|\), which may be much smaller than \(|\tau|\) elsewhere. The conic cutoff in the proof keeps exactly the region where these scales are comparable.

At a characteristic covector where a strict fractional gain is possible, the radial exclusion theorem in Weighted packets and the limit of the gain supplies the nonradial hypothesis of Theorem 3.1. Thus the exact homogeneous reduction is available for every such candidate. The reduction supplies coordinates and a principal-symbol model; deciding the full estimate still requires the sign obstruction and the finite-type analysis.

5. Exercises with complete solutions

Exercise 1 — an exact prepared factor, 8 points. Consider

\[ f(\tau,y)=(2+i+\tau)(\tau-y^2+iy^3). \]

Find \(A,\phi\) in (2.2), compute \(d\), and determine the real zero set near zero.

Solution. Take \(A=2+i+\tau\), \(\phi=y^2-iy^3\). Then \(d=2+i\), and \(A\ne0\) on a neighborhood. A real zero requires \(y^3=0\) and \(\tau=y^2\), hence \(y=\tau=0\). The factorization is an identity on real arguments.

Exercise 2 — preparation need not choose a unique complex root, 10 points. Set \(f(\tau,y)=\tau+iy\), and let

\[ e(y)=\begin{cases}e^{-1/y^2},&y\ne0,\\0,&y=0.\end{cases} \]

Compare the root \(\phi_0=-iy\) with \(\phi_1=e(y)-iy\). Prove that both give smooth nonzero prepared factors near zero.

Solution. The first factor is one. For \(y\ne0\), the second is

\[ A_1=\frac{\tau+iy}{\tau-e(y)+iy} =1+\frac{e(y)}{\tau-e(y)+iy}. \tag{5.1} \]

The denominator has modulus at least \(|y|\). Every derivative of \(e\) is bounded by every prescribed positive power of \(|y|\). Differentiation of the correction in (5.1) costs only a finite inverse power of \(|y|\). The zero-extension argument of Lemma 1.2 therefore makes that correction smooth and flat on \(y=0\); define \(A_1=1\) there. Its value at zero is one, so it is nonzero nearby. Multiplication verifies \(f=A_1(\tau-\phi_1)\) everywhere by continuity. The two roots differ by a nonzero flat function. Neither result is a uniqueness assertion for the real implicit function theorem applied to the same auxiliary extension.

Exercise 3 — an exact homogeneous coordinate map, 10 points. Near \((t,x_2;\xi_1,\xi_2)=(0,0;0,1)\), take

\[ p=(1+i\xi_1/\xi_2) \bigl(\xi_1-t^2\xi_2+i t^3\xi_2\bigr). \]

Find the full coordinate map (3.10), verify preservation of the canonical one-form, and identify the reduced imaginary symbol and its first bracket sign.

Solution. The factor is nonzero near the marked ray, and \(h=t^2\xi_2\), \(b=t^3\xi_2\). Since \(H_h=t^2\partial_{x_2}\) on the transverse pair, (3.9) gives

\[ x_2=z-t^3/3,\qquad \xi_2=\eta, \qquad \xi_1=\sigma+t^2\eta. \tag{5.2} \]

Directly,

\[ \xi_1dt+\xi_2dx_2 =(\sigma+t^2\eta)dt+\eta(dz-t^2dt) =\sigma dt+\eta dz. \tag{5.3} \]

Thus the map preserves the one-form, its derivative, and the dilation action. After dividing by the original factor and applying the map, the symbol is \(\sigma+i t^3\eta\). Its imaginary part is independent of \(\sigma\), and its first bracket is \(3t^2\eta\geq0\) in \(\eta>0\). At the marked point the third time derivative is \(6\), and the type is three.

Exercise 4 — a radial characteristic field, 8 points. On \(T^*\mathbb R^2\), consider \(p=x_2\xi_2\) at \(c=(0,0;0,1)\). Compute \(dp,H_p\) there. Explain why an ordinary real momentum coordinate is possible and the homogeneous reduction (3.2) is impossible.

Solution. At \(c\), \(dp=dx_2=\lambda_c\) and

\[ H_p=x_2\partial_{x_2}-\xi_2\partial_{\xi_2} =-R_c. \]

The differential is nonzero, so ordinary real function-preserving Darboux coordinates can take \(p\) as a momentum. But the field is radial. The converse in Theorem 3.1 rules out a homogeneous reduction to (3.2). A homogeneous time coordinate would have \(Rt=0\), whereas the normalized symbol would satisfy \(H_{p/a}t=1\); at this characteristic point \(H_{p/a}\) is still radial, a contradiction.

Exercise 5 — the first two derivatives of the complex root, 10 points. In Theorem 2.1 let \(y\) be one real parameter. Write

\[ e=f_y(0),\quad u=f_{\tau\tau}(0),\quad v=f_{\tau y}(0),\quad w=f_{yy}(0). \]

Determine \(\phi'(0)\) and \(\phi''(0)\). Explain why these formulas are valid although \(f\) was given only for real \(\tau\).

Solution. Differentiate the exact prepared identity at zero. Since \(\tau-\phi=0\) there,

\[ \phi'(0)=-e/d, \qquad A_\tau(0)=u/2, \qquad A_y(0)=v+(u/2)\phi'(0). \]

The second \(y\)-derivative gives

\[ \phi''(0)=-\frac{w+2v\phi'(0)+u\phi'(0)^2}{d}. \tag{5.4} \]

These are identities among derivatives of smooth functions on their real domains. Equivalently they follow by formal polynomial substitution into the second Taylor polynomial, as in the finite preparation lesson. No evaluation of the original \(f\) at a nonreal argument is required. Flat changes of root, such as Exercise 2, do not alter either derivative.

Exercise 6 — changing order while preserving the loss, 10 points. Let an operator of order \(m=5/2\) have principal symbol, near the ray \((0;0,1)\),

\[ p=\xi_2^{3/2}(\xi_1+i t^4\xi_2). \]

Give its order-one reduction and the necessary lower bound on loss. State exactly what the coordinate and packet results establish, and identify the missing analytic conclusion.

Solution. On \(\xi_2>0\), choose a proper elliptic multiplier of order \(-3/2\) with principal factor \(\xi_2^{-3/2}\) in the working cone. The reduced principal symbol is \(\xi_1+i t^4\xi_2\). Its Hamilton field at the marked characteristic point is \(\partial_t\), so it is nonradial; no further coordinate change is needed.

Use \(k=4\), with position and momentum weights \(m_1=1,\mu_1=4\) and \(m_2=\mu_2=5/2\). In the shifted germ \(\xi_2=1+\eta_2\), the terms \(\xi_1\) and \(i t^4\) have weight four, and \(i t^4\eta_2\) has weight \(13/2\). The weighted packet theorem therefore requires

\[ \delta\geq4/5 \]

for any strict-loss estimate. The equation can gain at most \(1/5\) after order-one reduction, and the original order gives output Sobolev order at most \(5/2-4/5=17/10\) from an \(L^2\) equation norm. These results prove order equivalence, the exact principal form and a necessary bound. Sufficiency for the displayed model uses the favorable even-order estimate in Finite type and the sign of the symbol, together with the proved lower-order stability when passing to a microlocal estimate. A general variable transverse symbol requires further analytic estimates; the coordinate theorem and necessary packet bound do not supply them.

References

Written by GPT-6.1 Sol (OpenAI), at Ultra reasoning effort, September 2026. Self-checked by the writing AI. Public domain (CC0).