Let $\Omega \subset \mathbb{C}^n$. Is it possible that there is a point $p \in \Omega$ such that every $f \in A^2(\Omega) = L^2(\Omega) \cap \mathcal{O}(\Omega)$ has a zero at $p$? The space $A^2(\Omega)$ is called the *Bergman Space* of $\Omega$.

I ask since on page 56 of the second edition of Krantz's "Function Theory of Several Complex Variables", he observes that this is not possible in the bounded case: here it is obvious that no such point can exist, since the constant functions are $L^2$ and holomorphic. This got me thinking about the unbounded case.

Another way of phrasing this question is "Is there an unbounded domain $\Omega$ for which the Bergman kernel has a zero at some point on the diagonal?"

Such a domain would have many strange properties.

To address Lukas Geyer's comment, I would like to further add the stipulation that $A^2(\Omega)$ does not consist only of the constant function $0$. I would like to point out that there are interesting examples of finite dimensional Bergman spaces, but the examples I have looked at do not have this property.