Let $\varepsilon>0$, let $Y$ be a smooth coordinate manifold, and let $X=(-\varepsilon,\varepsilon)\times Y$ with coordinates $(t,y)$. Let $\mathcal{L}^1$ denote one-dimensional [Lebesgue measure](/page/Lebesgue%20Measure) on $\mathbb{R}$ and let $\mathcal{L}^N$ denote $N$-dimensional Lebesgue measure on $\mathbb{R}^N$. Let $U\subset X$ be open, let $\Gamma\subset \mathbb{R}^{N}_{0}$ be an open conic set, and let $\phi:U\times\Gamma\to\mathbb{R}$ be a smooth real nondegenerate phase function, positively homogeneous of degree $1$ in $\theta\in\Gamma$. Define

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

and

\begin{align*} \Lambda_{\phi}:=\{(x,\partial_x\phi(x,\theta)):(x,\theta)\in C_{\phi}\}\subset T^*U\setminus 0. \end{align*}

Let $W\subset U\times\Gamma$ be an open conic set such that the map $\kappa:C_{\phi}\cap W\to \Lambda_{\phi}$, defined by $\kappa(x,\theta)=(x,\partial_x\phi(x,\theta))$, is a diffeomorphism onto its image, denoted by $\Lambda$. Assume moreover that $C_{\phi}\cap W$ has an open conic neighbourhood $W_1\subset U\times\Gamma$ whose closure is contained in $W$, and that smooth homogeneous functions on $C_{\phi}\cap W$ extend to smooth homogeneous functions on $U\times\Gamma$ after multiplication by a degree-zero conic cutoff supported in $W$ and equal to $1$ near $C_{\phi}\cap W$.

Let $P:C^\infty(U)\to C^\infty(U)$ be a scalar differential operator of order $m$ with real homogeneous principal symbol $p_m:T^*U\setminus 0\to\mathbb{R}$. Assume that $p_m|_{\Lambda}=0$, that $dp_m\ne 0$ on $\Lambda$, that the Hamilton vector field $H_{p_m}$ is nonzero and tangent to $\Lambda$, and that $H_{p_m}t\ne 0$ on $\Lambda$. Define

\begin{align*} C_0:=(C_{\phi}\cap W)\cap\{t=0\}. \end{align*}

Assume further that the Hamilton flow of $H_{p_m}$ gives a conic flow-box diffeomorphism from an open interval in the flow parameter times $\Lambda\cap T^*_{\{t=0\}}U$ onto $\Lambda$; equivalently, every point of $\Lambda$ lies on a unique bicharacteristic segment of $H_{p_m}$ meeting $\Lambda\cap T^*_{\{t=0\}}U$ exactly once, with the induced lifted flow on $C_{\phi}\cap W$ respecting the conic homogeneity of the transport equations.

Fix the coordinate half-density convention for the phase $\phi$. Assume the associated local symbolic calculus has the following transport form on $C_{\phi}\cap W$: for every classical amplitude $a\in S^r_{\mathrm{cl}}(U\times\Gamma)$ with expansion $a\sim\sum_{j=0}^{\infty}a_{r-j}$, there is a classical amplitude $Q_\phi(a)\in S^{r+m}_{\mathrm{cl}}(U\times\Gamma)$ such that $P I_\phi(a)=I_\phi(Q_\phi(a))$ microlocally on the chosen conic patch; the leading homogeneous coefficient is $p_m(x,\partial_x\phi)a_r$; and, after the leading eikonal term vanishes on $C_\phi\cap W$, the coefficient of order $r+m-j$ restricted to $C_\phi\cap W$ is equivalent to a first-order transport equation $\mathcal{T}_{r-j}c_{r-j}=f_{r-j}$, where $c_{r-j}=a_{r-j}|_{C_\phi\cap W}$, $f_{r-j}$ is determined by the earlier restrictions $c_r,\dots,c_{r-j+1}$ and is homogeneous of degree $r-j+m-1$, and $\mathcal{T}_{\mu}c=Vc+\gamma_\mu c$ for each $\mu\in\mathbb{R}$, with $V$ the pullback of $H_{p_m}$ by $\kappa$ and $\gamma_\mu:C_\phi\cap W\to\mathbb{C}$ a smooth homogeneous coefficient. Assume also that, for this phase convention, the local wave-front criterion on the chosen conic patch is coefficientwise: $I_\phi(q)$ has empty wave front set on $\Lambda$ if the homogeneous coefficients of $q$ restrict to zero on $C_\phi\cap W$ at every order; conversely, microlocal smoothness on $\Lambda$ forces these restrictions to vanish at every order. Finally, assume the classical Borel summation procedure may be carried out with the above conic support and with a base cutoff giving local proper support in $U$.

Fix $r\in\mathbb{R}$. For each $j\in\mathbb{N}_0$, let $b_{r-j}:C_0\to\mathbb{C}$ be a smooth function homogeneous of degree $r-j$ in the conic variable inherited from $\theta$. Then there exists a classical amplitude $a\in S^r_{\mathrm{cl}}(U\times\Gamma)$ with local proper support in $U$ and conic support contained in $W$, with asymptotic expansion $a\sim\sum_{j=0}^{\infty}a_{r-j}$, where each $a_{r-j}$ is smooth and homogeneous of degree $r-j$ in $\theta$, such that $a_{r-j}|_{C_0}=b_{r-j}$ for every $j\in\mathbb{N}_0$. Moreover, for the oscillatory integral

\begin{align*} I_{\phi}(a)(x):=(2\pi)^{-N}\int_{\Gamma}e^{i\phi(x,\theta)}a(x,\theta)\,d\mathcal{L}^{N}(\theta), \end{align*}

the distribution $P I_{\phi}(a)$ is microlocally smooth on the chosen conic Lagrangian patch: $\operatorname{WF}(P I_{\phi}(a))\cap\Lambda=\varnothing$. For the fixed initial data $(b_{r-j})_{j\in\mathbb{N}_0}$, the restrictions $a_{r-j}|_{C_{\phi}\cap W}$ are uniquely determined by the recursive transport equations on $C_{\phi}\cap W$ along the lifted bicharacteristics of $H_{p_m}$. Equivalently, if $\widetilde a$ is another amplitude with the same properties and homogeneous coefficients $\widetilde a_{r-j}$, then $(a_{r-j}-\widetilde a_{r-j})|_{C_{\phi}\cap W}=0$ for every $j\in\mathbb{N}_0$.