Weighted composition operators on the Zygmund space (Q437557)

Let \(H({\mathbb D})\) denote the space of holomorphic functions on the unit disk \({\mathbb D}\). For \(\alpha>0\), the Lipschitz space \(\Lambda^\alpha({\mathbb D})\) consists of those functions \(f\in H({\mathbb D})\) for which NEWLINE\[NEWLINE |f^{(J)}(z)|(1-|z|)^{J-\alpha} \leq C, \quad z\in{\mathbb D}, NEWLINE\]NEWLINE where \(f^{(J)}\) is the derivative of order \(J\) and \(J\) is an integer such that \(J>\alpha\).NEWLINENEWLINEGiven a function \(u\in H({\mathbb D})\) and a holomorphic mapping \(\varphi: {\mathbb D} \to {\mathbb D}\), the weighted composition operator \(uC_\varphi: H({\mathbb D}) \to H({\mathbb D})\) is defined by the formula NEWLINE\[NEWLINE (uC_\varphi f)(z) = u(z) f(\varphi(z)), \quad f\in H({\mathbb D}),\quad z\in {\mathbb D}. NEWLINE\]NEWLINENEWLINENEWLINEThe authors characterize the bounded and compact weighted composition operators \(uC_\varphi: \Lambda^\alpha({\mathbb D})\to \Lambda^\alpha({\mathbb D})\) for \(\alpha =1\).NEWLINENEWLINE Reviewers remark. Such characterizations were obtained for all \(\alpha>0\) by the reviewer, see [J. Math. Sci., New York 182, No.~5, 630--638 (2012); translation from Zap. Nauchn. Semin. POMI 389, 85--100 (2011; Zbl 1257.30065)].