Let \(\mathbb{D}\) be the open unit disk in the complex plane. The Hardy space \(H^p=H^p(\mathbb{D})\) for positive \(p\), consists of analytic functions on the open unit disk such that \[ \|f\|_{H^p}=\left (\sup_{0\le r<1}\frac{1}{2\pi}\int_0^{2\pi}|f(re^{i\theta})|^pd\theta<\infty\right )^{1/p}<\infty. \] Similarly, the Bergman space \(A^p=A^p(\mathbb{D})\) is the space of analytic functions for which \[ \|f\|_{A^p}=\left(\frac{1}{\pi}\int_{\mathbb{D}}|f(z)|^pdA(z)\right)^{1/p}<\infty, \] where \(dA(z)=dxdy\) is the area measure in the complex plane. Let \(\{z_k\}\) be a sequence in \(\mathbb{D}\) satisfying the condition \(\sum_k (1-|z_k|)<\infty\). If \(B\) is the Blaschke function associated to this zero sequence, and \(f\) is a function in the Hardy space whose zero set is precisely \(\{z_k\}\), then \(f/B\) belongs to \(H^p\) and \(\|f/B\|_{H^p}=\|f\|_{H^p}\). Blaschke functions are simple forms of a wider class of functions called inner functions. In the Hardy space, inner functions are solutions of the extremal problem \[ \sup\{\operatorname{Re}f(0):\|f\|_{H^p}\le 1, f(z_k)=0,\, k=1,2,\dots\}, \] where \(\{z_k\}\) is a zero sequence in \(H^p\), that is, there is at least one function \(f\in H^p\) that vanishes precisely on \(\{z_k\}\). Now, consider a zero sequence in the Bergman space \(A^p\). A theory of zero-divisors was developed by pioneering work of H. Hedenmalm [J. Reine Angew. Math. 422, 45–68 (1991; Zbl 0734.30040)] for \(p=2\), and subsequently by P. Duren et al. [Pac. J. Math. 157, No. 1, 37–56 (1993; Zbl 0739.30029)], for general \(p\). It turned out that if \(G\) is an inner function for the Bergman space, then \(\|f/G\|_{A^p}\le \|f\|_{A^p}\), which means that zero divisors are no longer isometric, but rather contractive.
The paper under review aims to explore the notion of inner function and zero divisor for Hardy and Bergman spaces defined on multiply connected domains, rather than simply connected domains. Given a space of analytic functions \(\mathcal{X}\) on a multiply connected domain \(\Omega\), and a zero set \(\mathcal{Z}\) for \(\mathcal{X}\), the authors aim to answer the following questions: Does there exist a function \(G\in \mathcal{X}\) that divides out zeros, that is to say, if \(f\in\mathcal{X}\) vanishes on \(\mathcal{Z}\), do we have the estimate \(\|f/G\|_{\mathcal{X}}\le \|f\|_{\mathcal{X}}\)? If this does not hold, can we hope in finding a constant \(C\) such that \(\|f/G\|_{\mathcal{X}}\le C \|f\|_{\mathcal{X}}\)? In this latter case, the authors call \(G\) a quasi-contractive divisor. It is shown in Theorem 2 that the Hardy space \(H^2(\Omega)\) supports quasi-contractive divisors, and so does the space \(H^p(\Omega)\) for other values of \(p\). The similar question for the Bergman space \(A^2(\Omega)\) remains unanswered. The paper focuses on an account on unsuccessful attempts to tackle this problem, and explains that there are good reasons to believe that the problem might be challenging. In the end, some open questions are raised.