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).