Theorems Principal Conormal Symbol Transformation Under Equivalent Stabilized Phases Attributions

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Statement

Let $M$ be a smooth $n$-dimensional manifold, let $S \subset M$ be an embedded submanifold, and let

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\begin{align*} \Lambda := N^*S \setminus 0 \subset T^*M \setminus 0. \end{align*}

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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.

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Let $N_1 \in \mathbb{N}$, and let

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\begin{align*} \phi_1: M \times \mathbb{R}^{N_1}_0 \to \mathbb{R} \end{align*}

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be a real smooth positively homogeneous nondegenerate conormal phase function of degree $1$ in $\theta \in \mathbb{R}^{N_1}_0$. Let

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\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*}

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be its critical manifold, and assume that the map $\gamma_{\phi_1}: C_{\phi_1} \to \Lambda$ defined by

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\begin{align*} \gamma_{\phi_1}(x,\theta) := (x,d_x\phi_1(x,\theta)) \end{align*}

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is a local diffeomorphism onto its image. Use the oscillatory-integral normalization

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\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*}

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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

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\begin{align*} \rho: M \times \mathbb{R}^{N_1}_0 \to (0,\infty) \end{align*}

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be smooth and positively homogeneous of degree $1$ in $\eta$. Suppose that a second phase function

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\begin{align*} \phi_2: M \times \mathbb{R}^{N_1}_0 \times \mathbb{R}^r \to \mathbb{R} \end{align*}

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has the stabilized normal form

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\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*}

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near its critical set. Let

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\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*}

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Assume that the map $\gamma_{\phi_2}: C_{\phi_2} \to \Lambda$ defined by

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\begin{align*} \gamma_{\phi_2}(x,\eta,0) := (x,d_x\phi_2(x,\eta,0)) \end{align*}

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is a local diffeomorphism onto its image. Define the map $\beta: C_{\phi_2} \to C_{\phi_1}$ by

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\begin{align*} \beta(x,\eta,0) := (x,\kappa(x,\eta)). \end{align*}

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Assume that

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\begin{align*} \gamma_{\phi_1}\circ \beta=\gamma_{\phi_2}. \end{align*}

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Use the stabilized oscillatory-integral normalization with total phase-variable dimension $N_1+r$:

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\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*}

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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 $C_{\phi_2}$.

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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

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\begin{align*} \beta^*|dc_{\phi_1}|^{1/2}=|J_\beta|^{1/2}|dc_{\phi_2}|^{1/2}. \end{align*}

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Let $F_H: C_{\phi_2} \to \mathbb{C}$ be defined by

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\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*}

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where $\operatorname{sgn}(H)$ is the number of positive eigenvalues of $H$ minus the number of negative eigenvalues of $H$.

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With these conventions, the principal conormal symbol represented by $I_{\phi_1}(a_1)$ is

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\begin{align*} (\gamma_{\phi_1})_*\bigl(a_{1,0}|dc_{\phi_1}|^{1/2}\bigr), \end{align*}

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and the principal conormal symbol represented by $I_{\phi_2}(a_2)$ is

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\begin{align*} (\gamma_{\phi_2})_*\bigl(F_H a_{2,0}|dc_{\phi_2}|^{1/2}\bigr). \end{align*}

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These two principal conormal symbols are equal as homogeneous half-densities on $\Lambda$ if and only if, on $C_{\phi_2}$,

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\begin{align*} F_H a_{2,0}=|J_\beta|^{1/2}(a_{1,0}\circ\beta). \end{align*}

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Equivalently, on $C_{\phi_2}$,

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\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*}

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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.

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