A new characterization for Carleson measures and some applications
For \(s\in{\mathbb R}\), a non-negative measure \(\mu\) on \({\mathbb D}\) is called \(s\)-Carleson if \(\sup\limits_{I\subset\partial {\mathbb D}}\frac{\mu(S(I))}{|I|^s}<\infty\), and it is said to be compact \(s\)-Carleson if \(\lim\limits_{|I|\to 0}\frac{\mu(S(I))}{|I|^s}\), where \(S(I)\) is the Carleson square based on an arc \(I\subset\partial{\mathbb D}\). Let \(\Gamma(\zeta)=\{z\in{\mathbb D}:|\zeta-z|<2(1-|z|)\}\), where \(\zeta\in\partial{\mathbb D}\). For \(a\in{\mathbb D}\), \(\tau>0\), \(s\in{\mathbb R}\), and \(0<p\leq\infty\), put \({\mathcal A}_\mu(a,\tau,s,p)=\left\|\int_{\Gamma(\cdot)}\frac{(1-|a|^2)^\tau}{|1-\overline{a}z|^{\tau+s}}\frac{d\mu(z)}{1-|z|}\right\|_{L^p}\). Natural questions arise: whether an \(s\)-Carleson measure (resp., a compact \(s\)-Carleson measure) can be characterized by \(\sup\limits_{a\in{\mathbb D}}{\mathcal A}_\mu(a,\tau,s,p)<\infty\) (resp., by \({\mathcal A}_\mu(a,\tau,s,p)\to 0\) as \(|a|\to 1_-\))? The paper is devoted to answer these questions and to provide some applications to boundedness and compactness of Volterra type operators.
scientific article; zbMATH DE number 5982597