Generalized interpolation in Bergman spaces and extremal functions (Q2721357)

For \(0< p\leq\infty\), \(\alpha> 0\), let \(B^{p,\alpha}\) be the Bergman space with the usual norm \(\|\cdot\|_{p, \alpha}\). The set of generalized free interpolating sequences \(\Lambda\subset \mathbb{D}= [|z|< 1]\) for the space \(B^{p,\alpha}\) is denoted by \(\text{Int}(B^{p, \alpha})\). In addition suppose that \(G_{\Lambda,\sigma}\) is a (generalized extremal function) solution of the problem NEWLINE\[NEWLINE\sup\{\text{Re }g(0): g|_\sigma\text{ constant, }g|_\Lambda= 0, \|g\|_{p, \alpha}= 1\}.NEWLINE\]NEWLINE Let \(\Lambda= \bigcup_{n\geq 1}\sigma_n\), where \(\sigma_n\subset \mathbb{D}\). The generalized Schuster-Seip condition, which is denoted by \((G_{\tau, \lambda})_{\tau, \lambda}\in (\text{SSG})\), takes place when there is a number \(\delta> 0\) such that \(\forall\tau\subset \mathbb{N}\), \(|\tau|< \infty\), and \(\lambda\in \sigma\) then \(G_{\tau, \lambda}(0)\geq \delta\), where \(\tau= \varphi_\lambda(\Lambda\setminus \sigma_\tau)\), \(\lambda= \varphi_\lambda(\sigma_\tau)\), and \(\varphi_\lambda= (\lambda- z)/(1- \overline\lambda z)\). One of the main results of the paper under review is the followingNEWLINENEWLINENEWLINETheorem: Let \(0< p<\infty\), \(\alpha> 0\). Suppose that \(\Lambda= \bigcup_{n\geq 1}\sigma_n\) has no accumulation points in \(\mathbb{D}\). Then \((\sigma_n)\in \text{Int}(B^{p, \alpha})\Rightarrow (G_{\tau,\lambda})_{\tau, \lambda}\in (\text{SSG})\).NEWLINENEWLINENEWLINEIt is proved that the condition (SSG) is also sufficient in the special case \(N= \sup_{n\geq 1}|\sigma_n|< \infty\).