Recall the following theorem (c.f. LC Evans, M Zworski, "Lectures on semiclassical analysis", Theorem 3.15, depending on the version):

**Theorem:** Let $\varphi: \mathbb R^n \to \mathbb R$ be smooth and $a: \mathbb R^n \to \mathbb R$ smooth with compact support $K$. Suppose that there exists $x_0 \in K$ with $\partial \varphi(x_0) = 0$ and $\det \partial^2 \varphi(x_0) \neq 0$, and suppose that $\partial \varphi \neq 0$ on $K\smallsetminus \{ x_0\}$.
For positive $\hbar$, define:
$$ I_\hbar = \int_{\mathbb R^n} e^{i\varphi(x)/\hbar} \, a(x)\,dx $$
Then for $k=0,1,\dots$, there exists differential operators $A_{2k}(x,\partial)$ of order $\leq 2k$, and constants $C_N$, all depending on $\varphi$, such that for each $N$ we have:
$$ \left| I_\hbar - \hbar^{n/2} \,e^{i\varphi(x_0)/\hbar} \sum_{k=0}^{N-1} A_{2k}(x,\partial) \, a(x_0)\,\hbar^k\right| \leq C_N\, \hbar^{N + \frac n 2} \sum_{|\alpha| \leq 2N + n+1} \sup_K | \partial^\alpha a|$$
where $\partial^\alpha$ is shorthand for some product of $\frac{\partial}{\partial x^i}$s.

**My question:** I know how to give the operators $A_{2k}$ explicitly; they depend only on the Taylor expansion of $\varphi$ at $x_0$, and are succinctly described combinatorially by ``Feynman diagrams''. What I would like to know is how explicitly the $C_N$ can be given? For example, can $C_N$ be taken to depend on the maximum values of some finite list (depending on $N$, of course) of derivatives of $\varphi$?

The reason I'm asking is that the above theorem gives $I_\hbar$ to an accuracy of $O(\hbar^\infty)$, but I would like to vary $\varphi$ and study $I_\hbar$ in some limit, and to know that my $O(\hbar^\infty)$ estimates still hold, I need to swap some limits, which requires more explicit description of the estimates.

As with any post, feel free to re-tag as appropriate.

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