I have been reading some semiclassical analysis and I have come across expressions boiling down to $(T_hf)(\xi) = O(h^K)$ for $K\in \mathbb{N}$ (or $K = \infty$) holding "uniformly" in $h$, $0 < h \leq 1$ quite often, where $T_h(.)$ is some linear operator, e.g. Fourier-transform or some differential operator, depending on the semi-classical parameter $h$, $f$ is just some nice function and $\xi\in \mathbb{R}^n$ for example. Here big oh of infinity, $O(h^\infty)$ means the following:
$$f(h) = O(h^\infty) \Longleftrightarrow \forall N\in\mathbb{N}:\exists C_N\geq 0:\forall h\in (0, 1]:|f(h)| \leq C_nh^N$$
An example of this sort of expression is e.g. in the proof of Zworski's Theorem 7.12 (Zworski: Semiclassical analysis), equation (7.4.17) or in Ivana Alexandrova's Semi-Classical Wavefront Set and Fourier Integral Operators, where she states that
$(x_0, \xi_0)$ does not belong to the essential support of $a_h$ if there exists an open neighborhood $U$ and $V$ for $x_0$ and $\xi_0$, respectively, such that for all $\alpha, \beta\in\mathbb{N}^n$ we have $$\partial_x^\alpha\partial_\xi^\beta a_h(x, \xi) = O(h^\infty) \text{ uniformly in $(x, \xi)\in U\times V$.}$$
So my question is that how the triple quantors $\forall \exists \forall $ holds "uniformly" w.r.t. some other parameter? Is the point that the constant $C_N$ cannot depend on the external $h$ parameter? Or is there more to the story?
Suppose $f = f(x_1,x_2,\dots;h)$. If we have an expression of the form $f = O(h^\infty)$, which holds uniformly in $x_1,x_2,\dots$, then it means $$ \text{for all $N\in\mathbb N$,}\ \exists C_N: \forall h\in(0,1],\ \forall x_1,x_2,\dots,\ |f(x_1,x_2,\dots;h)|\le C_Nh^N. $$ Contrast this with if I said $f(x_1,x_2,\dots;h) = O(h^\infty)$, but I don't specify that it holds uniformly in the parameters $x_1,x_2,\dots$: $$ \text{for all $N\in\mathbb N$,}\ \exists C_N(x_1,x_2,\dots): \forall h\in(0,1],\ |f(x_1,x_2,\dots;h)|\le C_N(x_1,x_2,\dots)h^N. $$ As you can see, the constant is allowed to vary with the parameters now.
You could also imagine mixed-type assumptions, where $f(x_1,x_2,\dots;h) = O(h^\infty)$, uniformly in $x_1$, but we say nothing of the other variables. Then we would interpret it formally as $$ \text{for all $N\in\mathbb N$,}\ \exists C_N(x_2,\dots): \forall h\in(0,1],\ \forall x_1,\ |f(x_1,x_2,\dots;h)|\le C_N(x_2,\dots)h^N. $$ The constant here has no dependence on the first variable $x_1$, but it may vary with respect to $x_2,\dots$.
Certainly in all cases, we never allow the constant to depend on $h$.