Zero sets and uniqueness sets with one cluster point for the Dirichlet space
Let \((z_n)\) be a sequence of points on the open unit disk \(\mathbb{D}\). In a classical paper, \textit{H. Shapiro} and \textit{A. Shields} [Math. Z. 80, 217--229 (1962; Zbl 0115.06301)] have shown that if \(\sum(-\log(1-|z_n|))^{-1}\) is finite, then \((z_n)\) is a zero sequence for the Dirichlet space, thus improving a previous result of L. Carleson. The present authors prove the following (Theorem 2.4). Let \(\varphi:[0,1]\to[0,\infty)\) be an increasing function such that \(\int_0^1\varphi(t)t^{-2}dt<\infty\), and suppose that there exists a non-zero function \(f\) in the Dirichlet space such that \(|f(\zeta)|^2\leq\varphi(|1-\zeta|/2)\) for almost every \(\zeta\) on the unit circle. Let \((z_n)\) be a Blaschke sequence on \(\mathbb{D}\) such that \(\sum\varphi(|1-z_n|)\) is finite. Then \((z_n)\) is a zero sequence for the Dirichlet space. The proof uses the theorem by Shapiro and Shields mentioned above and some preliminary steps. The authors use their results to construct families of zero sets for the Dirichlet space by giving suitable functions \(\varphi\). One of the examples is \(\varphi(t)=\exp(-\beta t^{-\alpha})\) where \(0<\alpha<1\) and \(\beta>0\).
scientific article; zbMATH DE number 5577087
scientific article