Piotr Jucha
On the dimension of the Bergman space
It is known that the space of square integrable holomorphic functions (the Bergman space) of a planar domain is either trivial or infinite dimensional. On the other hand, there are non-pseudoconvex domains in $\mathbb C^n$ for $n>1$ which have Bergman spaces of finite dimension. The problem whether there exist such pseudoconvex domains is open.
We solve the problem for some Hartogs domains using Hormander's $L^2$-techniques. In particular, the dimension of the Bergman space of a domain $\{(z,w): |w|<e^{-u(z)}\}$ is fully characterized by the properties of a subharmonic function $u$.