I am trying to understand theorem 7.7.1 in Hormander's Analysis of linear partial differential operators, vol.1.

Let $K \subset \mathbb{R}^n$ be a compact set, $X$ an open neighborhood of $K$ and $j, k$ non-negative integers. If $u \in C^k_0(K), f \in C^{k+1}(X)$ and $\text{Im} f \geq 0$ in $X$, then

$\omega^{j+k}\lvert\int u(x)(\text{Im} f(x))^je^{i \omega f(x)}dx\rvert \leq C\sum_{|\alpha|\leq k} \text{sup}|D^{\alpha}u|(|f'|^2+\text{Im}f)^{|\alpha|/2-k}, \quad \omega>0$

Here $C$ is bounded when $f$ stays in a bounded set in $C^{k+1}(X)$.

It's the "$f$ stays in a bounded set" part that is unclear. The way I understand it is that if $f$ depends on a parameter and $f$ and its derivatives are uniformly bounded then $C$ is bounded. Is that correct?

If it is, then I think this ($f$ and its $k+1$ derivatives are uniformly bounded) is only a sufficient condition for $C$ to be bounded. What is the necessary condition for $C$ to be bounded?