In the proof of proposition 1.3.2 in the book on Fourier Integral Operators by Duistermaat, it is claimed that $\langle e^{-i\tau \psi}\varphi, u\rangle = \langle\mathcal{F}^{-1}(e^{-i\tau\psi}), \mathcal{F}(\varphi u)\rangle$. Here $\psi \in C^\infty(\mathbb{R}^n\times\mathbb{R}^p)$, and $\varphi \in C^\infty_0(U_0)$ for some neighborhood $U_0$. They technically also define a function $\varphi'$ which is equal to 1 on the support of $\varphi$ and then claim $$\langle e^{-i\tau \psi}\varphi, u\rangle = \langle\mathcal{F}^{-1}(e^{-i\tau\psi}\varphi'), \mathcal{F}(\varphi u)\rangle,$$ but I believe this should not be relevant to this first step. They use the standard inner product
$$\langle f, g\rangle = \int f(x)g(x)\text{d}x$$
with real functions.
It seems to me that the given statement cannot be true because \begin{align} \langle e^{-i\tau \psi}\varphi, u\rangle &= \langle e^{-i\tau \psi}\varphi'\varphi, u\rangle \\ &= \langle e^{-i\tau \psi}\varphi', \varphi u\rangle \\ &= \langle \mathcal{F}(e^{-i\tau \psi}\varphi'), \mathcal{F}(\varphi u)\rangle \\ &= \langle \mathcal{P}\circ \mathcal{F}^{-1}(e^{-i\tau \psi}\varphi'), \mathcal{F}(\varphi u)\rangle \\ &\neq \langle\mathcal{F}^{-1}(e^{-i\tau\psi}\varphi'), \mathcal{F}(\varphi u)\rangle. \end{align}
Unfortunately, if I continue their derivation with my altnerative equality I cannot proof the proposition. Does anyone know where I went wrong?