Criterion of surjectivity of the Cauchy transform operator on a Bergman space (Q2782552)

Let \(G\subset\mathbb{C}\) a bounded Jordan domain and \(B_2(G)\) the Bergman space. The Cauchy transform of a continuous linear functional \(S\) is denoted by \(\widetilde S=S({1\over z-\rho})\), \(\;\zeta\in \mathbb{C}\setminus \overline G\). Further let \(B^1_2(\mathbb{C}\setminus\overline G)=\) NEWLINE\[NEWLINE\left\{\gamma\in H( \mathbb{C} \setminus\overline G),\gamma (\infty)=0,\;\|\gamma \|^2_{B^1_2 (\mathbb{C} \setminus \overline G)}=\int_{\mathbb{C} \setminus\overline G}\bigl|\gamma' (\xi)\bigr |^2d \gamma(\xi)\right\}.NEWLINE\]NEWLINE It is proved that in a bounded Jordan domain the Cauchy transform is a surjective continuous operator from \(B^*_2(G)\) to \(B^1_2(\mathbb{C} \setminus\overline G)\) if and only if \(G\) is a quasidisk. The definition of a quasidisk is given by the authors. Furthermore, it is shown that the surjectivity of the Cauchy transform is equivalent to the surjectivity of the singular integral operator NEWLINE\[NEWLINE(\pi_{2,G^.} (z)=\left\{\begin{matrix} V.P.\int_\mathbb{C} {1\over(z-\xi)^2} dv(z), & \xi\in\mathbb{C} \setminus\overline G\\ 0, & \xi\in G\end{matrix}.\right.NEWLINE\]NEWLINENEWLINENEWLINEFor the entire collection see [Zbl 0980.00031].