Interpolation by periods in planar domain

Abstract

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