Carleson conditions in Bergman-Orlicz spaces
Let \(\varphi:[0,\infty)\to [0,\infty)\) be a continuous, increasing function with \(\varphi(0)=0\) and \(\lim_{t\to \infty}\varphi(t)=\infty\). Let \(\mathbb D\) denote the open unit disk in \(\mathbb C\). The Bergman-Orlicz space \(A^\varphi\) is the set of all holomorphic functions \(f\) on \(\mathbb D\) with \[ \int_{\mathbb D}\varphi(\lambda|f(z)|)dm(z)<\infty \] for some positive constant \(\lambda\) depending on \(f\). \(A^\varphi\) is a generalization of the classical Bergman space \(A^p\). The paper under review gives characterizations of (forward and reverse) Carleson conditions in terms of inequalities that involve functions in the Bergman-Orlicz space, which generalizes two important results from the theory of Bergman spaces to the context of Bergman-Orlicz spaces.
scientific article; zbMATH DE number 5795254