# Weierstrass factorization with $L^2$ estimates?

Let $\Omega$ be a bounded domain in $\mathbb{C}$. Let $X$ be a discrete set of points whose boundary is in the boundary of $\Omega$. Can I find an $L^2$ holomorphic function which vanishes on $X$? Can I solve the problem in weighted $L^2$ spaces?

If there are counterexamples, are precise conditions on the set $X$ known to ensure the existence of an $L^2$ solution?

I have been learning about Hormander's approach to the $\bar{\partial}$-problem, and this seems like a natural question to ask from that perspective, but I have not been able to find any work done on this.

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You have to specify what you mean by $L^2$. Is this $L^2$ with respect to Lebesgue measure (area) in $D$? Whatever you mean by $L^2$, the answer is "no". The reason is Jensen's formula. It says that a function which has too many zeros must grow fast.
If you want to solve it in weighted $L^2$ space, then your weight must be related to the growth rate of the set $X$. If instead you want to fix the weight in $L^2$, the conditions on $X$ will come from the Jensen formula. If you are interested in $L^2$ without weight, look in the books about Bergman space. There you can find the exact conditions on $X$.