Symbol order reduction by a phase integration operator ($L^t:S_{1,0}^s\to S_{1,0}^{s-1}$)

For a homogeneous phase function, let $q=|{\nabla }_x\Phi {|}^2+|\theta {|}^2|{\nabla }_{\theta }\Phi {|}^2$ and $L=(iq{)}^{-1}({\nabla }_x\Phi \cdot {\nabla }_x+|\theta {|}^2{\nabla }_{\theta }\Phi \cdot {\nabla }_{\theta })$. Then $L{e}^{i\Phi }=e^{i\Phi }$. The formal transpose of a differential operator $L^t$ lowers the symbol class order by one: the $x$ coefficients have order $-1$, while the order-zero frequency coefficients accompany a frequency derivative.