Let $U \subset \mathbb{R}^n$ be open, let $N \in \mathbb{N}$, and let $\phi \in C^\infty(U \times \mathbb{R}^N_0; \mathbb{R})$ be a real-valued phase function such that
for every $(x,\theta) \in U \times \mathbb{R}^N_0$.
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Let $a \in S^m(U \times \mathbb{R}^N)$ be a classical symbol whose $x$-support is proper over $U$, and let $u \in \mathcal{D}'(U)$ be the oscillatory integral distribution defined, in the usual homogeneous oscillatory-integral sense, by
Let $\operatorname{esssupp}(a) \subset U \times \mathbb{R}^N_0$ denote the conic essential support of the full symbol modulo $S^{-\infty}$, and define the critical set
is a conic immersed Lagrangian submanifold of $T^*U \setminus 0$, and the preceding inclusion is the wave front set estimate for this Lagrangian parametrization restricted to $\operatorname{esssupp}(a)$.