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Let $\Omega$ be a bounded domain of $\mathbb{C}^n$ and $f$ be a holomorphic function defined on $\Omega$.

Is it possible that $L^2$-norm of $f$ is bounded but $f$ itself is unbounded?

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    My stomach feeling is no! In the sense that I don´t see how an holomorphic function on a bounded domain can be unbounded, a priori of any examination of the $L^2$-norm. EDIT: Now I see it...2012-02-09
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    @Student73 Yes it can, you may very well have singularities at the boundary of the domain. See theory of univalent functions, $H^p$-spaces, Bergman spaces, Dirichlet spaces etc... and my answer below.2012-02-10

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