The commutation formula for an $h^{-1}$-pseudodifferential operator with a rapidly oscillating exponential function in the complex phase case

This paper considers the action of the operator $a\bigl(x_1-ih\frac\partial{\partial x}\bigr)u\overset{\mathrm{def}}=\int a(x,h\xi)\times\exp i(x\xi)\widetilde u(\xi)\,d\xi$ on functions of the form $\exp(\frac{iS}h)\varphi(x)=u(x)$, where $\varphi\in C_0^\infty(\mathbf R^n)$ and $S\in C^\infty(\mathbf R^n)$. In particular, when $ S(x,h)=S(x)$, $\operatorname{im}S(x)\geqslant0$, one has
$$ a\biggl(x_1-ih\frac\partial{\partial x}\biggr)\exp\biggl(-\frac{iS}h\biggr)\varphi=\exp\biggl(\frac{iS}h\biggr)\sum_{j=0}^N h^jL_j\varphi+O(h^{N+1}). $$
It is proved that for $\operatorname{im}S\not\equiv0$ the differential operators $L_j$ can be obtained from the analogous differential operators for $\operatorname{im}S\equiv0$ by means of “almost analytic extension” with respect to the arguments $S',S'',\dots,S^{(k)}$.
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