Theorems Equivalence of Nondegenerate Phase Parametrisations Attributions

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Statement

Let $X$ be a smooth Hausdorff second-countable manifold of dimension $n$, let $U \subset X$ be a coordinate neighbourhood with coordinate map identifying $T^*U$ with $U \times \mathbb{R}^n$, and let $\Lambda_0 \subset T^*U \setminus 0$ be an open conic subset of an embedded conic Lagrangian submanifold. Let $\mathbb{R}^N_0:=\mathbb{R}^N\setminus\{0\}$ for every $N\in\mathbb{N}$.

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Let $N_\phi,N_\psi \in \mathbb{N}$, let $\Theta_\phi \subset \mathbb{R}^{N_\phi}_0$ and $\Theta_\psi \subset \mathbb{R}^{N_\psi}_0$ be open conic subsets, and let

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\begin{align*} \phi:U\times\Theta_\phi\to\mathbb{R} \end{align*}

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and

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\begin{align*} \psi:U\times\Theta_\psi\to\mathbb{R} \end{align*}

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be smooth real-valued phase functions, positively homogeneous of degree $1$ in the fibre variables. Assume that $\phi$ and $\psi$ are nondegenerate homogeneous phase functions in the sense that their fibre-critical sets

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\begin{align*} C_\phi:=\{(x,\theta)\in U\times\Theta_\phi:\nabla_\theta\phi(x,\theta)=0\} \end{align*}

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and

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\begin{align*} C_\psi:=\{(x,\eta)\in U\times\Theta_\psi:\nabla_\eta\psi(x,\eta)=0\} \end{align*}

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are smooth conic submanifolds and the differentials of the fibre-gradient maps have maximal rank transverse to these critical sets. Assume also that the maps

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\begin{align*} \lambda_\phi:C_\phi\to\Lambda_0 \end{align*}

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and

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\begin{align*} \lambda_\psi:C_\psi\to\Lambda_0 \end{align*}

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defined by $\lambda_\phi(x,\theta)=(x,\nabla_x\phi(x,\theta))$ and $\lambda_\psi(x,\eta)=(x,\nabla_x\psi(x,\eta))$ are conic diffeomorphisms. By Euler's identity, $\phi|_{C_\phi}=0$ and $\psi|_{C_\psi}=0$.

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Assume the local stable homogeneous Morse normal-form branches for $\phi$ and $\psi$ have been chosen so that, after adding explicitly homogeneous quadratic stabilizing variables, the stabilized phases have the same fibre dimension, matching transverse Hessian branch data, and are related on connected conic neighbourhoods of their critical sets by a smooth fibre-preserving conic diffeomorphism with no additive base term. These chosen branches determine the constant Maslov factor below.

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For $\mu\in\mathbb{R}$ and $j\in\{\phi,\psi\}$, let $S^\mu_{\mathrm{cl}}(U\times\Theta_j)$ denote the space of properly supported classical symbols of order $\mu$ in the fibre variable, and let $S^{-\infty}(U\times\Theta_j)$ denote the residual symbols. The oscillatory-integral maps

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\begin{align*} I_\phi:S^\mu_{\mathrm{cl}}(U\times\Theta_\phi)\to\mathcal{D}'(U;\Omega_U^{1/2}) \end{align*}

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and

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\begin{align*} I_\psi:S^\nu_{\mathrm{cl}}(U\times\Theta_\psi)\to\mathcal{D}'(U;\Omega_U^{1/2}) \end{align*}

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are defined in the standard homogeneous oscillatory sense by

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\begin{align*} I_\phi(a)(x):=(2\pi)^{-N_\phi/2}\int_{\Theta_\phi}e^{i\phi(x,\theta)}a(x,\theta)\,d\mathcal{L}^{N_\phi}(\theta)\,|dx|^{1/2} \end{align*}

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and analogously for $I_\psi(b)$ with phase $\psi$, fibre variable $\eta$, and measure $d\mathcal{L}^{N_\psi}(\eta)$.

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After replacing $U$, $\Theta_\phi$, and $\Theta_\psi$ by smaller conic neighbourhoods of the inverse images of $\Lambda_0$, for every $\mu\in\mathbb{R}$ there exists a linear isomorphism

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\begin{align*} T_{\phi\to\psi}:S^\mu_{\mathrm{cl}}(U\times\Theta_\phi)/S^{-\infty}(U\times\Theta_\phi)\to S^{\mu+(N_\phi-N_\psi)/2}_{\mathrm{cl}}(U\times\Theta_\psi)/S^{-\infty}(U\times\Theta_\psi) \end{align*}

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such that, for every properly supported $a\in S^\mu_{\mathrm{cl}}(U\times\Theta_\phi)$, one has $I_\phi(a)-I_\psi(T_{\phi\to\psi}a)\in C^\infty(U;\Omega_U^{1/2})$. Moreover, the induced map on principal symbols is the canonical identification of the associated half-density principal symbols on $\Lambda_0$, including the constant Maslov factor determined by the chosen local branches and by the normalization $(2\pi)^{-N/2}$ in the oscillatory integral convention above.

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