$O_{\mathcal{S}'\rightarrow \mathcal{S}}(h^\infty)$

Question

What does the notation:

$$O_{\mathcal{S}'\rightarrow \mathcal{S}}(h^\infty)$$

mean in Zworski's semiclassical analysis?

(To be clear, I know what $O(h^\infty)$ means in the context of normed and semi-normed spaces.)

Answer 1

That notation would arise when you're looking at the asymptotics of a linear operator parametrized in $h$. In particular, for your case this means that the asymptotic formula has an error operator $E_h$, such that, for all $u \in \mathscr{S}'$, and for multi-indices $\alpha, \beta$, and integer $N > 0$, there exists $C_{\alpha, \beta, N,u} > 0$ such that $\| E_h u\|_{\alpha, \beta} \le C_{\alpha, \beta, N,u} h^N$ for sufficiently small $h$, where $\| \cdot \|_{\alpha, \beta}$ is the usual semi-norm for the Schwartz space.