Principal Conormal Symbol Transformation Under Equivalent Stabilized Phases

Theorem

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Let $M$ be a smooth $n$-dimensional manifold, let $S \subset M$ be an embedded submanifold, and let \begin{align*} \Lambda := N^*S \setminus 0 \subset T^*M \setminus 0. \end{align*} Fix $\lambda_0 \in \Lambda$, and work microlocally in sufficiently small conic neighbourhoods of $\lambda_0$ in $\Lambda$ and of the corresponding critical points in the phase spaces. All phase functions, amplitudes, maps, half-densities, and identities below are restricted to these neighbourhoods. Let $N_1 \in \mathbb{N}$, and let \begin{align*} \phi_1: M \times \mathbb{R}^{N_1}_0 \to \mathbb{R} \end{align*} be a real smooth positively homogeneous nondegenerate conormal phase function of degree $1$ in $\theta \in \mathbb{R}^{N_1}_0$. Let \begin{align*} C_{\phi_1} := \{(x,\theta) : \partial_{\theta_j}\phi_1(x,\theta)=0 \text{ for every } 1 \leq j \leq N_1\} \end{align*} be its critical manifold, and assume that the map $\gamma_{\phi_1}: C_{\phi_1} \to \Lambda$ defined by \begin{align*} \gamma_{\phi_1}(x,\theta) := (x,d_x\phi_1(x,\theta)) \end{align*} is a local diffeomorphism onto its image. Use the oscillatory-integral normalization \begin{align*} I_{\phi_1}(a_1) := (2\pi)^{-N_1/2}\int_{\mathbb{R}^{N_1}} e^{i\phi_1(x,\theta)}a_1(x,\theta)\,d\mathcal{L}^{N_1}(\theta)\,|dx|^{1/2}. \end{align*} Let $r \in \mathbb{N}_0$, let $H \in \mathbb{R}^{r \times r}$ be a nonsingular symmetric matrix, let $\kappa: M \times \mathbb{R}^{N_1}_0 \to \mathbb{R}^{N_1}_0$ be a smooth map such that, for each relevant $x \in M$, the map $\eta \mapsto \kappa(x,\eta)$ is a positively homogeneous diffeomorphism of degree $1$ on the relevant conic set, and let \begin{align*} \rho: M \times \mathbb{R}^{N_1}_0 \to (0,\infty) \end{align*} be smooth and positively homogeneous of degree $1$ in $\eta$. Suppose that a second phase function \begin{align*} \phi_2: M \times \mathbb{R}^{N_1}_0 \times \mathbb{R}^r \to \mathbb{R} \end{align*} has the stabilized normal form \begin{align*} \phi_2(x,\eta,z)=\phi_1(x,\kappa(x,\eta))+\frac{1}{2}\rho(x,\eta)z^\top H z \end{align*} near its critical set. Let \begin{align*} C_{\phi_2}:=\{(x,\eta,0): \partial_{\eta_j}\phi_2(x,\eta,0)=0 \text{ for every } 1 \leq j \leq N_1,\ \partial_{z_k}\phi_2(x,\eta,0)=0 \text{ for every } 1 \leq k \leq r\}. \end{align*} Assume that the map $\gamma_{\phi_2}: C_{\phi_2} \to \Lambda$ defined by \begin{align*} \gamma_{\phi_2}(x,\eta,0) := (x,d_x\phi_2(x,\eta,0)) \end{align*} is a local diffeomorphism onto its image. Define the map $\beta: C_{\phi_2} \to C_{\phi_1}$ by \begin{align*} \beta(x,\eta,0) := (x,\kappa(x,\eta)). \end{align*} Assume that \begin{align*} \gamma_{\phi_1}\circ \beta=\gamma_{\phi_2}. \end{align*} Use the stabilized oscillatory-integral normalization with total phase-variable dimension $N_1+r$: \begin{align*} I_{\phi_2}(a_2):=(2\pi)^{-(N_1+r)/2}\int_{\mathbb{R}^{N_1}\times \mathbb{R}^r} e^{i\phi_2(x,\eta,z)}a_2(x,\eta,z)\,d\mathcal{L}^{N_1}(\eta)\,d\mathcal{L}^r(z)\,|dx|^{1/2}. \end{align*} Let $a_1$ and $a_2$ be classical amplitudes of the same conormal order for $I_{\phi_1}$ and $I_{\phi_2}$, respectively. Let $a_{1,0}$ denote the leading homogeneous term of $a_1$ restricted to $C_{\phi_1}$, and let $a_{2,0}$ denote the leading homogeneous term of $a_2$ in the homogeneous variable $\eta$, evaluated at $z=0$ and restricted to $C_{\phi_2}$. Fix smooth positive local half-densities $|dc_{\phi_1}|^{1/2}$ on $C_{\phi_1}$ and $|dc_{\phi_2}|^{1/2}$ on $C_{\phi_2}$. Define the positive smooth function $|J_\beta|^{1/2}: C_{\phi_2} \to (0,\infty)$ by \begin{align*} \beta^*|dc_{\phi_1}|^{1/2}=|J_\beta|^{1/2}|dc_{\phi_2}|^{1/2}. \end{align*} Let $F_H: C_{\phi_2} \to \mathbb{C}$ be defined by \begin{align*} F_H(x,\eta,0) := e^{i\pi\operatorname{sgn}(H)/4}\rho(x,\eta)^{-r/2}|\det H|^{-1/2}, \end{align*} where $\operatorname{sgn}(H)$ is the number of positive eigenvalues of $H$ minus the number of negative eigenvalues of $H$. With these conventions, the principal conormal symbol represented by $I_{\phi_1}(a_1)$ is \begin{align*} (\gamma_{\phi_1})_*\bigl(a_{1,0}|dc_{\phi_1}|^{1/2}\bigr), \end{align*} and the principal conormal symbol represented by $I_{\phi_2}(a_2)$ is \begin{align*} (\gamma_{\phi_2})_*\bigl(F_H a_{2,0}|dc_{\phi_2}|^{1/2}\bigr). \end{align*} These two principal conormal symbols are equal as homogeneous half-densities on $\Lambda$ if and only if, on $C_{\phi_2}$, \begin{align*} F_H a_{2,0}=|J_\beta|^{1/2}(a_{1,0}\circ\beta). \end{align*} Equivalently, on $C_{\phi_2}$, \begin{align*} a_{2,0}=e^{-i\pi\operatorname{sgn}(H)/4}\rho^{r/2}|\det H|^{1/2}|J_\beta|^{1/2}(a_{1,0}\circ\beta). \end{align*} In particular, if $I_{\phi_1}(a_1)$ and $I_{\phi_2}(a_2)$ agree microlocally modulo conormal distributions of one lower order near $\lambda_0$, then their principal conormal symbols agree. Conversely, equality of the displayed principal conormal symbols implies equality of the two microlocal conormal distributions modulo one lower conormal order near $\lambda_0$, but does not imply equality of the full distributions.

Discussion

No discussion available for this theorem.

Proof

[proofplan] We compare both principal symbols after pulling them back to the common critical manifold $C_{\phi_2}$. The extra nonhomogeneous variables $z$ in the stabilized phase are eliminated by stationary phase for the exact quadratic form $\frac{1}{2}\rho z^\top H z$, which produces the factor $F_H$. The remaining difference is only the change of critical-manifold half-density under $\beta$, and this contributes precisely $|J_\beta|^{1/2}$. Finally, the principal-symbol exact sequence for conormal distributions identifies equality of principal symbols with equality modulo one lower conormal order. [/proofplan] [step:Reduce equality of pushed-forward symbols to equality after pullback to $C_{\phi_2}$] Let \begin{align*} \sigma_1 := (\gamma_{\phi_1})_*\bigl(a_{1,0}|dc_{\phi_1}|^{1/2}\bigr) \end{align*} and \begin{align*} \sigma_2 := (\gamma_{\phi_2})_*\bigl(F_Ha_{2,0}|dc_{\phi_2}|^{1/2}\bigr) \end{align*} denote the two homogeneous half-densities on the same conic neighbourhood in $\Lambda$. Since $\gamma_{\phi_1}$ and $\gamma_{\phi_2}$ are local diffeomorphisms and satisfy $\gamma_{\phi_1}\circ\beta=\gamma_{\phi_2}$, the map $\beta$ is itself a local diffeomorphism, locally equal to $\gamma_{\phi_1}^{-1}\circ\gamma_{\phi_2}$. Therefore pullback of half-densities by $\beta$ is defined on the chosen neighbourhoods, and the half-densities $\sigma_1$ and $\sigma_2$ are equal if and only if their pullbacks by $\gamma_{\phi_2}$ are equal on $C_{\phi_2}$. For $\sigma_2$, the definition of pushforward by a local diffeomorphism gives \begin{align*} \gamma_{\phi_2}^*\sigma_2=F_Ha_{2,0}|dc_{\phi_2}|^{1/2}. \end{align*} For $\sigma_1$, using $\gamma_{\phi_2}=\gamma_{\phi_1}\circ\beta$ gives \begin{align*} \gamma_{\phi_2}^*\sigma_1=\beta^*\bigl(a_{1,0}|dc_{\phi_1}|^{1/2}\bigr). \end{align*} By the definition of pullback of a function times a half-density, \begin{align*} \beta^*\bigl(a_{1,0}|dc_{\phi_1}|^{1/2}\bigr)=(a_{1,0}\circ\beta)\beta^*|dc_{\phi_1}|^{1/2}. \end{align*} Using the defining convention \begin{align*} \beta^*|dc_{\phi_1}|^{1/2}=|J_\beta|^{1/2}|dc_{\phi_2}|^{1/2}, \end{align*} we obtain \begin{align*} \gamma_{\phi_2}^*\sigma_1=|J_\beta|^{1/2}(a_{1,0}\circ\beta)|dc_{\phi_2}|^{1/2}. \end{align*} Because $|dc_{\phi_2}|^{1/2}$ is nowhere vanishing, $\sigma_1=\sigma_2$ if and only if \begin{align*} F_Ha_{2,0}=|J_\beta|^{1/2}(a_{1,0}\circ\beta) \end{align*} on $C_{\phi_2}$. [/step] [step:Compute the leading stabilized contribution by stationary phase in the $z$ variables] Because the theorem is microlocal near $C_{\phi_2}$, choose the phase-space neighbourhood and the amplitude representative so that the $z$-support is contained in a small fixed neighbourhood of $0 \in \mathbb{R}^r$ on which $z=0$ is the only stationary point of the $z$-phase. On the complement of a smaller neighbourhood of $0$, the $z$-gradient of the phase is $\rho(x,\eta)Hz$, and its Euclidean norm is bounded from below by a positive multiple of $\rho(x,\eta)|z|$ on the chosen support. Repeated [integration by parts](/theorems/210) in $z$ with the standard nonstationary-phase operator therefore gives rapid decay in the homogeneous parameter $|\eta|$, so those contributions are microlocally smoothing near $\lambda_0$ and do not affect the principal conormal symbol. Fix $(x,\eta)$ in the relevant conic set and define the real quadratic map $q_{x,\eta}: \mathbb{R}^r \to \mathbb{R}$ by \begin{align*} q_{x,\eta}(z) := \frac{1}{2}\rho(x,\eta)z^\top H z. \end{align*} Since $\rho(x,\eta)>0$ and $H$ is nonsingular and symmetric, the Hessian matrix of $q_{x,\eta}$ at $z=0$ is $\rho(x,\eta)H$, which is nonsingular. Its signature is \begin{align*} \operatorname{sgn}(\rho(x,\eta)H)=\operatorname{sgn}(H), \end{align*} and its determinant satisfies \begin{align*} |\det(\rho(x,\eta)H)|^{-1/2}=\rho(x,\eta)^{-r/2}|\det H|^{-1/2}. \end{align*} After shrinking the conic neighbourhood if necessary, write $\eta=\tau\omega$ with $\tau>0$ and $\omega$ in a compact subset of the unit sphere. Since $\rho$ is smooth, positive, and homogeneous of degree $1$, there are constants $0<c<C<\infty$ such that \begin{align*} c\tau \leq \rho(x,\tau\omega) \leq C\tau \end{align*} on the chosen neighbourhood. Thus $\rho(x,\eta)$ is a uniform large parameter as $|\eta| \to \infty$ in the conic region. Applying the normalized [stationary phase theorem](/theorems/8198) [citetheorem:8198] in the nonhomogeneous variable $z$ to the quadratic phase $q_{x,\eta}$ gives the leading contribution \begin{align*} e^{i\pi\operatorname{sgn}(H)/4}\rho(x,\eta)^{-r/2}|\det H|^{-1/2}a_{2,0}(x,\eta,0). \end{align*} Thus the leading homogeneous term of the stabilized integral after eliminating $z$ is \begin{align*} F_H(x,\eta,0)a_{2,0}(x,\eta,0). \end{align*} All further stationary phase terms contain derivatives of the amplitude in the $z$ variables and inverse powers of the homogeneous parameter represented by $\rho(x,\eta)$; hence they lower the homogeneous order by one or more and contribute only to lower conormal order. [guided] The purpose of this step is to isolate the only effect of adding the stabilized variables $z$. Since the statement is microlocal near $C_{\phi_2}$, we first choose the phase-space neighbourhood and the amplitude representative so that the $z$-support lies in a small fixed neighbourhood of $0 \in \mathbb{R}^r$. On that neighbourhood, $z=0$ is the only stationary point of the $z$-phase; contributions from outside it are microlocally smoothing near $\lambda_0$ and cannot change the principal symbol. For each fixed pair $(x,\eta)$, define the real quadratic map $q_{x,\eta}: \mathbb{R}^r \to \mathbb{R}$ by \begin{align*} q_{x,\eta}(z) := \frac{1}{2}\rho(x,\eta)z^\top H z. \end{align*} This is a genuine nonhomogeneous stationary phase problem in the variable $z$; the variables $(x,\eta)$ are parameters. The critical point is $z=0$, because the derivative with respect to $z$ is the [linear map](/page/Linear%20Map) represented by $\rho(x,\eta)Hz$, and this vanishes exactly at $z=0$ since $\rho(x,\eta)>0$ and $H$ is invertible. The Hessian matrix at the critical point, meaning the Jacobian matrix of the gradient map $\nabla_z q_{x,\eta}:\mathbb{R}^r\to\mathbb{R}^r$ at $0$, is \begin{align*} \rho(x,\eta)H. \end{align*} The stationary phase theorem [citetheorem:8198] applies because this Hessian is invertible and the amplitude is classical in the homogeneous parameter $\eta$ while smooth with compactly localized support in the nonhomogeneous variable $z$. Multiplication by the positive scalar $\rho(x,\eta)$ does not change the numbers of positive and negative eigenvalues, so \begin{align*} \operatorname{sgn}(\rho(x,\eta)H)=\operatorname{sgn}(H). \end{align*} The determinant transforms by the elementary determinant identity for scalar multiplication of an $r \times r$ matrix: \begin{align*} \det(\rho(x,\eta)H)=\rho(x,\eta)^r\det H. \end{align*} Taking absolute values and the negative one-half power gives \begin{align*} |\det(\rho(x,\eta)H)|^{-1/2}=\rho(x,\eta)^{-r/2}|\det H|^{-1/2}. \end{align*} We also need the large parameter in stationary phase to be uniform on the conic neighbourhood. After shrinking that neighbourhood, write $\eta=\tau\omega$ with $\tau>0$ and $\omega$ in a compact subset of the unit sphere. Because $\rho$ is smooth, positive, and homogeneous of degree $1$, there are constants $0<c<C<\infty$ such that \begin{align*} c\tau \leq \rho(x,\tau\omega) \leq C\tau \end{align*} throughout the chosen neighbourhood. Hence $\rho(x,\eta)$ tends to infinity uniformly with $|\eta|$, which is exactly the large-parameter regime used in the stationary phase expansion. The normalized stationary phase formula therefore says that the leading term produced by integration in $z$ is \begin{align*} e^{i\pi\operatorname{sgn}(H)/4}\rho(x,\eta)^{-r/2}|\det H|^{-1/2}a_{2,0}(x,\eta,0). \end{align*} This is exactly \begin{align*} F_H(x,\eta,0)a_{2,0}(x,\eta,0). \end{align*} The use of the total normalization $(2\pi)^{-(N_1+r)/2}$ is essential here: stationary phase in $r$ variables cancels precisely the additional factor $(2\pi)^{-r/2}$, so no extra power of $2\pi$ remains in $F_H$. Finally, the next terms in stationary phase are obtained by applying differential operators in the $z$ variables and multiplying by negative powers of the large homogeneous scale. In this setting that scale is represented by the positive homogeneous function $\rho(x,\eta)$ of degree $1$ in $\eta$. Each subsequent term therefore has homogeneous degree at least one lower than the leading term. Hence those terms affect only lower conormal orders, while the principal conormal symbol receives exactly the factor $F_H$. [/guided] [/step] [step:Identify the principal symbol of the stabilized phase] By the definition of the principal conormal symbol associated to a nondegenerate homogeneous phase, using the nondegeneracy of $\phi_1$ and the local diffeomorphism property of $\gamma_{\phi_1}:C_{\phi_1}\to\Lambda$, the phase $\phi_1$ represents the principal symbol \begin{align*} (\gamma_{\phi_1})_*\bigl(a_{1,0}|dc_{\phi_1}|^{1/2}\bigr). \end{align*} For the stabilized phase, the preceding stationary phase computation first replaces the leading restricted amplitude $a_{2,0}$ on $C_{\phi_2}$ by $F_Ha_{2,0}$. The remaining homogeneous phase variables are $\eta$, the critical manifold is $C_{\phi_2}$, and the assumed local diffeomorphism $\gamma_{\phi_2}:C_{\phi_2}\to\Lambda$ identifies this critical manifold with the same conic Lagrangian. Therefore the principal symbol represented by the stabilized integral is \begin{align*} (\gamma_{\phi_2})_*\bigl(F_Ha_{2,0}|dc_{\phi_2}|^{1/2}\bigr). \end{align*} This uses the convention that $a_{2,0}$ is the leading homogeneous term in $\eta$ evaluated at the stationary value $z=0$. [/step] [step:Derive the displayed amplitude transformation formula] From the first step, equality of the two principal symbols is equivalent to \begin{align*} F_Ha_{2,0}=|J_\beta|^{1/2}(a_{1,0}\circ\beta) \end{align*} on $C_{\phi_2}$. Substituting the definition \begin{align*} F_H=e^{i\pi\operatorname{sgn}(H)/4}\rho^{-r/2}|\det H|^{-1/2} \end{align*} and multiplying both sides by \begin{align*} e^{-i\pi\operatorname{sgn}(H)/4}\rho^{r/2}|\det H|^{1/2} \end{align*} gives \begin{align*} a_{2,0}=e^{-i\pi\operatorname{sgn}(H)/4}\rho^{r/2}|\det H|^{1/2}|J_\beta|^{1/2}(a_{1,0}\circ\beta). \end{align*} This is the stated equivalent form. [/step] [step:Relate equality of principal symbols to equality modulo one lower conormal order] Since $S \subset M$ is an embedded submanifold, $\Lambda=N^*S\setminus 0$ is the conic Lagrangian associated to conormal distributions along $S$. We use the same classical conormal order and the same half-density normalization as in the oscillatory integrals displayed in the statement. The principal-symbol exact sequence for classical conormal distributions [citetheorem:8203], specialized to this conormal Lagrangian $\Lambda$, states that the principal symbol map has kernel equal to the conormal distributions of one lower order. Hence, if $I_{\phi_1}(a_1)$ and $I_{\phi_2}(a_2)$ agree microlocally modulo one lower conormal order near $\lambda_0$, their difference has zero principal symbol, so the two principal symbols are equal. Conversely, if the displayed principal symbols are equal, then the principal symbol of \begin{align*} I_{\phi_1}(a_1)-I_{\phi_2}(a_2) \end{align*} vanishes near $\lambda_0$. The same exact-sequence statement implies that this difference belongs microlocally to the conormal class of one lower order near $\lambda_0$. This proves equality modulo one lower conormal order. Since the principal symbol records only the leading homogeneous amplitude class, this conclusion does not determine the lower homogeneous terms of the amplitudes and therefore does not imply equality of the full distributions. [/step]

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