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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).

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I think not. In a small neighborhood $U$, $g$ will be almost constant, so your new norm will look very much like the standard $L^2$ norm. And of course for the standard norm one can find functions with $||f||$ arbitrarily small and $||f'||$ arbitrarily large.

What makes you think this should hold?

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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.

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