TITLE: Hilbert Space - Norm of derivative
QUESTION [5 upvotes]: If $H$ is a Hilbert space of entire functions with weighted norm $||f||^{2}=\int_{R} |\frac{f(t)}{g(t)}|^{2}dt$ for some entire function $g$ (not necessary in $H$). Can we find any relation between the norm of $f$ and the norm of it's derivative? Something like:
$||f'||\leq C ||f||$ for some constant $C$. (Note: so far we don't know whether $f'$ belongs to $H$ or not).

REPLY [3 votes]: An explicit counterexample: fix some $f$ and define 
$$\tilde{f}(t) = e^{iKt}f(t) $$
for some large, unspeficied $K$. $\tilde{f}$ is clearly also in $H$. Then $$\tilde{f}'(t) = e^{iKt}f'(t) + Ke^{iKt}f(t)$$ An inequality of the form you seek will imply that 
$$K\|f\| - \|f'\| \leq \| f' + K f\| \leq C\|f\| $$
for all $K$, which is absurd.