Let $X$ be a smooth $n$-dimensional manifold, and let $\Lambda \subset T^*X \setminus 0$ be a smooth conic Lagrangian submanifold. For $m \in \mathbb{R}$, let $I^m(X,\Lambda;\Omega_X^{1/2})$ denote the space of classical Lagrangian half-density distributions associated to $\Lambda$, defined locally by nondegenerate conic phase functions $\phi:U\times\Gamma\to\mathbb{R}$, where $U\subset X$ is a coordinate chart domain, $\Gamma\subset\mathbb{R}^N\setminus 0$ is an open conic set, $N\in\mathbb{N}$ is the number of auxiliary variables, and the amplitudes are classical symbols satisfying the order convention

\begin{align*} a \in S_{\mathrm{cl}}^{m+n/4-N/2}(U\times\Gamma). \end{align*}

Here $\mathcal{L}^N$ denotes $N$-dimensional [Lebesgue measure](/page/Lebesgue%20Measure) on $\mathbb{R}^N$, used in the auxiliary-variable oscillatory integral. All local oscillatory integral representatives are understood with the stated local conic support, properness, and conic cutoff regularization conventions.

For such a phase, write

\begin{align*} C_\phi=\{(p,\theta)\in U\times\Gamma:\partial_{\theta_j}\phi(p,\theta)=0\text{ for all }j\in\{1,\dots,N\}\} \end{align*}

and let

\begin{align*} \kappa_\phi:C_\phi\to\Lambda_\phi\subset\Lambda \end{align*}

be the associated smooth conic parametrization. For $\mu=m+n/4-N/2$, the local leading amplitude is taken in the quotient $\mathcal{Q}_\phi^\mu$ of homogeneous degree-$\mu$ functions on the chosen conic support by the null-symbol relation generated by homogeneous leading terms whose restriction to $C_\phi$ is zero, equivalently by finite sums $\sum_{j=1}^N q_j\partial_{\theta_j}\phi$ with each coefficient $q_j$ homogeneous of degree $\mu$, modulo nonstationary terms and symbols one order lower. The symbolic reduction convention is that such null symbols define oscillatory integrals in $I^{m-1}(X,\Lambda;\Omega_X^{1/2})$, after [integration by parts](/theorems/210) in the auxiliary variables, and conversely every vanishing leading class is reducible in this way after shrinking the conic support.

Let $\mathcal{M}_\Lambda$ be the Maslov line bundle of $\Lambda$, and let $\Omega_\Lambda^{1/2}$ be the half-density bundle on $\Lambda$. Let

\begin{align*} S_{\mathrm{hom}}^m(\Lambda;\mathcal{M}_\Lambda\otimes\Omega_\Lambda^{1/2}) \end{align*}

be the [vector space](/page/Vector%20Space) of smooth sections of $\mathcal{M}_\Lambda\otimes\Omega_\Lambda^{1/2}$ over $\Lambda$ that are homogeneous of degree $m$ under the positive fiber dilations on $T^*X\setminus 0$, with the same local conic support convention as the amplitudes defining $I^m(X,\Lambda;\Omega_X^{1/2})$.

The local stationary-phase normalization assigns to each nondegenerate conic phase $\phi$ an isomorphism

\begin{align*} \tau_\phi:\mathcal{Q}_\phi^\mu\to S_{\mathrm{hom}}^m(\Lambda_\phi;\mathcal{M}_\Lambda\otimes\Omega_\Lambda^{1/2}) \end{align*}

obtained by restricting the leading amplitude class to $C_\phi$, transporting it by $\kappa_\phi$, and multiplying by the stationary-phase half-density and Maslov factors. In local reductions, the Maslov factor is computed from the signature of the nondegenerate Hessian block of the phase in the auxiliary variables normal to the critical set being eliminated. These maps satisfy the standard transition law under coordinate changes on $X$, homogeneous changes of auxiliary variables, and stabilization by nondegenerate quadratic forms.

Then the principal symbol map

\begin{align*} \sigma_m:I^m(X,\Lambda;\Omega_X^{1/2})\to S_{\mathrm{hom}}^m(\Lambda;\mathcal{M}_\Lambda\otimes\Omega_\Lambda^{1/2}) \end{align*}

obtained locally by applying $\tau_\phi$ to the leading homogeneous amplitude class is a well-defined surjective [linear map](/page/Linear%20Map), and

\begin{align*} \ker \sigma_m=I^{m-1}(X,\Lambda;\Omega_X^{1/2}). \end{align*}

Equivalently, the sequence

\begin{align*} 0\to I^{m-1}(X,\Lambda;\Omega_X^{1/2})\to I^m(X,\Lambda;\Omega_X^{1/2})\xrightarrow{\sigma_m} S_{\mathrm{hom}}^m(\Lambda;\mathcal{M}_\Lambda\otimes\Omega_\Lambda^{1/2})\to 0 \end{align*}

is exact.