Let \(\mathbb{D}\) denote the unit disk in \(\mathbb{C}\), and let \(\Gamma(\zeta)\) be the usual nontangential approach region with vertex \(\zeta\in\partial \mathbb{D}\), i.e. \(\Gamma(\zeta):= \{z\in \mathbb{D}:|\zeta- z|\leq 1-| z|^2\}\). Given a positive measure \(\mu\) on \(\mathbb{D}\), the author investigates the operator \[ Gf(\zeta):= \int_{\Gamma(\zeta)} | f(z)| {d\mu\over 1-| z|}, \] for \(f\in H^p\). Does there exist a constant \(C\) independent of \(f\) such that \[ \int_{\partial\mathbb{D}}\Biggl(\int_{\Gamma(\zeta)} | f(z)| {d\mu(z)\over 1-| z|}\Biggr)^p| d\zeta|\leq C\| f\|^p_p,\tag{1} \] i.e. is the operator \(G: H^p\to L^p\), \(f\mapsto Gf\) continuous? The author proves the following
Theorem: Let \(0<p<\infty\). Necessary and sufficient that (1) holds for all \(f\) in \(H^p\) is that \(\mu\) is a Carleson measure.