Periods of \(L^{2}\)-forms in an infinite-connected planar domain (Q338053)

Let \(\Omega= \mathbb{D}\setminus \bigcup_{j=1}^\infty B_j\), where \(\mathbb{D}\) denotes the unit disk of \(\mathbb{C}\) and \(B_1, B_2, \dots\subset\mathbb{D}\) are pairwise disjoint connected compact sets which accumulate only to \(\partial\mathbb{D}\). The space \(L_c^{2,1}(\Omega)\) consists of the closed real differential forms \(\omega = \omega_x dx + \omega_y dy\) on \(\Omega\) such that \(\int_{\Omega} (\omega_x^2 + \omega_y^2)\, dA < \infty\), where \(dA\) denotes the area measure on \(\mathbb{C}\). For \(j=1,2,\dots\), fix a closed oriented curve \(\gamma_j\) in \(\Omega\) such that \(\gamma_j\) winds around \(B_j\) once in the positive direction and does not wind around \(B_{k}\), \(k\neq j\). For \(\omega\in L_c^{2,1}(\Omega)\), the period operator is defined as NEWLINE\[NEWLINE P\omega = \left\{\int_{\gamma_j}\omega \right\}_{j=1}^\infty. NEWLINE\]NEWLINE The author characterizes those \(\Omega\) for which the operator \(P: L_c^{2,1}(\Omega)\to \ell^2\) is bounded and surjective.