Weighted composition operators between Zygmund type spaces and their essential norms (Q1949279)

Let \(H(\mathbb{D})\) denote the space of holomorphic functions on the unit disk \(\mathbb{D}\). For \(\alpha\in\mathbb{R}\), the space \(\Lambda^\alpha(\mathbb{D})\) consists of those functions \(f\in H(\mathbb{D})\) for which \[ |f^{(J)}(z)|(1-|z|)^{J-\alpha} \leq C, \quad z\in{\mathbb D}, \] where \(f^{(J)}\) is the derivative of order \(J\) and \(J\) is an integer such that \(J>\alpha\). In fact, \(\Lambda^\alpha(\mathbb{D})\), \(\alpha<0\), is a growth space, \(\Lambda^0(\mathbb{D})\) is the Bloch space, and \(\Lambda^\alpha(\mathbb{D})\), \(\alpha>0\), is the Lipschitz space of order \(\alpha\). In particular, \(\Lambda^1(\mathbb{D})\) is the Zygmund space. Given 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 \[ (uC_\varphi f)(z) = u(z) f(\varphi(z)), \quad f\in H({\mathbb D}),\quad z\in {\mathbb D}. \] For \(\alpha, \beta<2\), the authors characterize the bounded weighted composition operators \(uC_\varphi: \Lambda^\alpha({\mathbb D})\to \Lambda^\beta({\mathbb D})\). Also, they estimate the essential norms of the operators under consideration. Reviewer's remark. The bounded and compact weighted composition operators \(uC_\varphi: \Lambda^\alpha({\mathbb D})\to \Lambda^\beta({\mathbb D})\) were earlier characterized for all \(\alpha, \beta\in\mathbb{R}\) by the reviewer [J. Math. Sci., New York 182, No. 5, 630--638 (2012); translation from Zap. Nauchn. Semin. POMI 389, 85--100 (2011; Zbl 1257.30065)].