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Weighted composition operators from \(F(p,q,s)\) to Bloch type spaces on the unit ball. (English) Zbl 1163.47021

Let \(B\) denote the complex \(n\)-dimensional unit ball. For \(p>0\), \(s>0\), \(q>-(n+1)\), with \(q+s>-1\), let \(F(p,q,s)\) denote the space of holomorphic functions \(f\) in \(B\) such that \[ \sup_{a\in B}\int_B| \nabla f(z)| ^p(1-| z| ^2)^qg^s(z,a)\,dv(z)<\infty, \] where \(dv\) is volume measure and \(g\) is Green’s function on \(B\). For \(\alpha>0\), let \(B^\alpha\) denote the space of holomorphic functions in \(B\) such that \[ \sup_{a\in B}(1-| a| ^2)^\alpha| \nabla f(a)| <\infty. \] The paper characterizes boundedness and compactness for weighted composition operators \(W_{\phi,\psi}\) between \(F(p,q,s)\) and \(B^\alpha\). Here, \(\phi\) is a holomorphic self-map of \(B\), \(\psi\) is a holomorphic function on \(B\), and \(W_{\phi,\psi}f=\psi f\circ\phi\).
Reviewer: Kehe Zhu (Albany)

MSC:

47B33 Linear composition operators
32A37 Other spaces of holomorphic functions of several complex variables (e.g., bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA))
46E15 Banach spaces of continuous, differentiable or analytic functions
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