Let $D$ be a unit disc (I am actually interested in a much more general setting, but let's start with explicit examples). Let $E$ be an open subset of $D$. Consider the functional on the space of all holomorphic functions on $D$ defined by $\|f\|^2_E=\int_{E}|f(z)|^{2}dA(z)$. I need to find out for which $E$ this norm defines a Banach space of holomorphic functions. I've reduced this problem to the following: does there exist a sequence $f_1,f_2,...$ of holomorphic functions on $D$ and a sequence of points $z_1,z_2,...$, converging in $D\backslash E$, such that $\|f\|_E\le 1$, but $|f_n(z_n)|\to\infty$?

Off-course, if $D=E$, then the space is the classical Bergman Space, which is a Banach Space. The situation is similar in the case when $D\backslash E$ is a compact, i.e. $E$ "covers" the boundary.

I have very limited knowledge of these kind of matters, but I believe, that the answer depends on how $D\backslash E$ "intersects" with the unit circle. For example, it would be interesting to consider $D\backslash E$ to be a horde in $D$, solid triangle with a vertex on $\partial D$ or a disc of a smaller radius touching $\partial D$. My guess is that the first two give rise to a Banach space, while the last one doesn't.

Thank you.