Extended Cesàro operators on Bergman spaces
Denote by \(H(B)\) the class of all holomorphic functions on the open unit ball \(B\) of \(\mathbb{C}^n\) and, for \(g\in H(B)\), let \(\Re g(z)=\sum_{j=1}^n z_j (\partial g/\partial z_j)\) denote the radial derivative. For \(g\in H(B)\), the extended Cesàro operator \(T_g\) with symbol \(g\) is the operator on \(H(B)\) defined by \[ T_g(f)(z)=\int_0^1 f(tz)\Re g(tz) \frac{dt}{t},\quad f\in H(B),\; z\in B. \] For a positive Lebesgue measurable function \(w\) defined on \(B\) and for \(0<p<\infty\), the weighted Bergman space \(L_{a,w}^p(B)\) is the space of all functions \(f\in H(B)\) such that \[ \int_B | f(z)| ^pw(z)dm(z)<\infty. \] The main result of the paper under review characterizes (for some class of weights \(w\)) all symbols \(g\) for which \(T_g\) is a bounded operator from the Bergman space \(L_{a,w}^p(B)\) to \(L_{a,w}^q(B)\) for \(0<p, q<\infty\). A similar characterization is given for compact operators.
scientific article; zbMATH DE number 2092936
scientific article