Interpolation by periods in a planar domain

Abstract:

Let $\Omega \subset \mathbb {R}^2$ be a countably connected domain. With any closed differential form of degree $1$ in $\Omega$ with components in $L^2(\Omega )$ one associates the sequence of its periods around the holes in $\Omega$, that is around the bounded connected components of $\mathbb R^2\setminus \Omega$. For which $\Omega$ the collection of such period sequences coincides with $\ell ^2$? We give an answer in terms of metric properties of holes in $\Omega$.