Weighted composition operators from the weighted Bergman space to the weighted Hardy space on the unit ball
Let \(0<p<\infty\), \(\alpha>-1\), and \(\beta\geq 0\). The weighted Hardy space \(H_\beta^p(\mathbb{B})\) consists of all holomorphic functions on the unit ball \(\mathbb{B}\) of \(\mathbb{C}^n\) such that \(\| f\|^p_{H_\beta^p}=\sup_{0<r<1}(1-r)^\beta\int_{\partial\mathbb{B}}|f(r\zeta)|^p\,d\sigma(\zeta)<\infty\), where \(d\sigma\) is the normalized Lebesgue measure on the boundary \(\partial\mathbb{B}\) of \(\mathbb{B}\). The weighted Bergman space \(A_\alpha^p(\mathbb{B})\) consists of all holomorphic functions on \(\mathbb{B}\) such that \(\| f\|_{A_\alpha^p}^p=\int_{\mathbb{B}}|f(z)|^pc_\alpha(1-|z|^2)^\alpha\,dV(z)<\infty\), where \(dV\) is the normalized Lebesgue measure on \(\mathbb{B}\) and \(c_\alpha\) is a constant chosen such that \(\|1\|_{A_\alpha^p}=1\). For \(\beta=0\), \(H_\beta^p(\mathbb{B})\) coincides with the Hardy space \(H^p(\mathbb{B})\). For \(\alpha\geq 0\), \(H_\alpha^p(\mathbb{B})\) is a closed subspace of \(A_\alpha^p(\mathbb{B})\). The authors establish criteria for boundedness and compactness of a weighted composition operator \(uC_\phi: A_\alpha^p(\mathbb{B})\to H_\beta^q(\mathbb{B})\) defined by \(uC_\phi f=u(f\circ\phi)\), where \(\phi\) is a holomorphic self-map of \(\mathbb{B}\) and \(u\) is a holomorphic function on \(\mathbb{B}\).
scientific article; zbMATH DE number 5668003
scientific article