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

Let $0<p,q<\mathrm{\infty }$, $\mathrm{\Phi }$ be a generalized normal function and $L_{p,q}(\mathrm{\Phi })$ 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 $L_{p_1,p_2}(\mathrm{\Phi })$ to $L_{q_1,q_2}(\mathrm{\Phi })$ for $1\le p_i,q_i\le \mathrm{\infty }$ with $i\in \{1,2\}$. Moreover, we also establish the boundedness of Bergman-type operators $P_{s,t}$, where $s\in \mathbb{R}$ and $t>0$, on holomorphic radial-angular mixed space $H_{p,q}(\mathrm{\Phi })$ for all possible $0<p,q<\mathrm{\infty }$. As an application, we finally solve Gleason's problem on $H_{p,q}(\mathrm{\Phi })$ for all possible $0<p,q<\mathrm{\infty }$.