Weighted composition operators between weighted Bergman spaces and weighted Banach spaces of holomorphic functions (Q949069)

Let \(w,v\) be weights on the open unit disc \(\mathbb{D}\) of the complex plane. The weighted Bergman space \(A_{w,p}\) is the vector space of analytic functions on \(\mathbb{D}\) such that \(\| f\| _{w,p}^p=\int_\mathbb{D} | f(z)| ^pw(z)\,dA(z)\) is finite, where \(dA\) denotes the normalized area measure. Let \(H^\infty_v\) denote the vector space of analytic functions on \(\mathbb{D}\) such that \(\| f\| _{v}=\sup_\mathbb{D} v(z) | f(z)|\,dA(z)\) is finite. \textit{A.\,K.\thinspace Sharma} and \textit{S.\,D.\thinspace Sharma} [Commun.\ Korean Math.\ Soc.\ 21, No.\,3, 465--474 (2006; Zbl 1160.47308)] characterized the boundedness and compactness of weighted composition operators \(\psi C_\phi:A_{w,p}\to H^\infty_v\) for weights of the standard type \((1-| z| ^2)^\alpha\). Inspired by this work, the author extends such characterizations for \(w=| u|\), with \(u\) a non-vanishing analytic function.