The authors characterize the Dirichlet space of a multiply connected domain \(\Sigma\) in the sphere \(\overline{\mathbb C}=\mathbb C\cup\{\infty\}\) bounded by non-intersecting quasicircles \(\Gamma_1,\dots,\Gamma_n\). Assume that \(\infty\in\Sigma\). Let \(\Omega_k^+\) and \(\Omega_k^-\) denote the bounded and unbounded components of the complements of \(\Gamma_k\) in \(\overline{\mathbb C}\), \(\Sigma=\cap_{k=1}^n\Omega_k\). Let \(f=(f_1,\dots,f_n)\) be an \(n\)-tuple of quasiconformally extendible conformal maps \(f_k:\mathbb D^+\to\Omega_k^+\), where \(\mathbb D^+=\{z:|z|<1\}\). Denote the Dirichlet space of \(\Sigma\) by \[\mathcal D_{\infty}(\Sigma)=\left\{h:\Sigma\to\mathbb C: h\;\text{holomorphic},h(\infty)=0,\int_{\Sigma}|h'|^2dA<\infty\right\}.\] For the Cauchy projections \(P(\Omega_k^-)\) of functions on \(\Gamma_j\) into \(\mathcal D_{\infty}(\mathbb D^-)\), define for \(\mathcal C_{f_k^{-1}}h_k=h_k\circ f_k^{-1}|_{\Gamma_k}\), \[\mathbf{I}_f:=\overset n\bigoplus\mathcal D_{\infty}(\mathbb D^-)\to\mathcal D_{\infty}(\Sigma)\] \[(h_1,\dots,h_n)\mapsto\sum_{k=1}^nP(\Omega_k^-)\mathcal C_{f_k^{-1}}h_k.\] The first main result is given in the following theorem.
Theorem 3.9. Let \(\Sigma\), \(\Gamma_k\), \(\Omega_k^+\) and \(\Omega_k^-\) be defined as above. Let \(f=(f_1,\dots,f_n)\) where \(f_k:\mathbb D^+\to\Omega_k^+\) is a conformal map for \(k=1,\dots,n\). Then the map \(\mathbf{I}_f\) is a bounded isomorphism.
Also, the authors introduce the generalized Grunsky operator \(\mathbf{Gr}(f)\) corresponding to the \(n\)-tuple \((f_1,\dots,f_n)\) and derive integral expressions for its blocks. Finally, they show that \[\mathbf{W}=\{(h\circ f_1|_{\mathbb S^1},\dots,h\circ f_n|_{\mathbb S^1}):h\in\mathcal D(\Sigma)\}\subseteq\overset n\bigoplus\mathcal H(\mathbb S^1)\] is the graph of this Grunsky operator. Here \(\mathbb S^1=\{z:|z|=1\}\) and \(\mathcal H(\mathbb S^1)\) is the \(H^{1/2}\) Sobolev space on \(\mathbb S^1\).