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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?

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    $\begingroup$ Let $B(t)u = b(t,x,hD)u$. The operator $B(0)u = u + O(h^\infty)$. We write that as $(B(0) - \mathrm{Id}) = O(h^\infty)$. Now, if I read correctly: note that we already have the propagator $F(t)$ which acts well on "error terms". So $$ U(t) = B(t) - F(t)(B(0) - \mathrm{Id}) $$ should solve (10.2.2) $\endgroup$
    – Willie Wong
    Commented Jul 5, 2013 at 11:46

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