Schur's test, Bergman-type operators and Gleason's problem on radial-angular mixed spaces

Abstract

<abstract><p>Let $ 0 &lt; p, q &lt; \infty $, $ \Phi $ be a generalized normal function and $ L_{p, q}(\Phi) $ the radial-angular mixed space. In this paper, we first generalize the classical Schur's test to radial-angular mixed spaces setting and then find the sufficient and necessary condition for the boundedness of integral operators from $ L_{p_1, p_2}(\Phi) $ to $ L_{q_1, q_2}(\Phi) $ for $ 1\leq p_i, q_i\leq \infty $ with $ i\in\{1, 2\} $. Moreover, we also establish the boundedness of Bergman-type operators $ P_{s, t} $, where $ s\in {\mathbb R} $ and $ t &gt; 0 $, on holomorphic radial-angular mixed space $ H_{p, q}(\Phi) $ for all possible $ 0 &lt; p, q &lt; \infty $. As an application, we finally solve Gleason's problem on $ H_{p, q}(\Phi) $ for all possible $ 0 &lt; p, q &lt; \infty $.</p></abstract>