MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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:



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?

share|cite|improve this question
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) – Willie Wong Jul 5 '13 at 11:46

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.