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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?

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  • $\begingroup$ The statement can be formalized as "$\forall B\subset C^{k+1}(X)$ a bounded subset, there exists $C$ such that for every $f \in B$ with $\mathrm{Im} f \geq 0$..." In particular the constant $C$ can depend on the choice of $B$. // When you have many quantifiers in your statement, and you ask about "necessary" conditions, you should specify which of the statements you want to remove from the hypotheses. This is especially the case since for every fixed $f$ the estimate is true. // Are you in particular looking for a sequence of $f_k$ such that the best corresponding $C_k$ diverges? $\endgroup$
    – Willie Wong
    Commented Feb 17, 2015 at 9:07
  • $\begingroup$ @WillieWong What I'm interested in is if $f(x,y)$ depends on parameter $y$ and is integrated over $x$. Then what I think the theorem says is if $f(x,y)$ (and its derivatives) is uniformly bounded, then $C$ is bounded and is not a function of $y$ (i.e. not $C(y)$). Then the question is what is the necessary condition for $C$ to be bounded.? $\endgroup$
    – teagut
    Commented Feb 17, 2015 at 15:04

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