mp.mathematical physics - Explicit bounds for the asymptotics of oscillatory integrals

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 $Ksmallsetminus { x_0}$.
For positive $hbar$, define:
$$ I_hbar = int_{mathbb R^n} e^{ivarphi(x)/hbar} , a(x),dx $$
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:
$$ left| I_hbar - hbar^{n/2} ,e^{ivarphi(x_0)/hbar} sum_{k=0}^{N-1} A_{2k}(x,partial) , a(x_0),hbar^kright| leq C_N, hbar^{N + frac n 2} sum_{|alpha| leq 2N + n+1} sup_K | partial^alpha a|$$
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.

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