In Maciej Zworski's book $\textit{Semiclassical Analysis}$, an important step in proving $L^p$ bounds on quasimodes is deriving a microlocal oscillatory integral representation formula for families of operators $U(t)$ solving:

$$(hD_t+P(t))U(t)u=O_{L^2}(h^\infty)$$

$$U(0)=u$$

for appropriately microlocalised $u$.

I am failing to understand what Zworski means by his final statement before Theorem 10.4 in which he proceeds from having a $b(0,x,\eta)\in\mathcal{C}_c^\infty(\mathbb{R}^{2n})$ with $b(0,x,hD)u=u+O_{L^2}(h^\infty)$ to losing the error term. Is anyone with a copy of this book able to point out what I am missing?