Isometries of the Zygmund \(F\)-algebra (Q2845556)

Let \(\mathbb B\) and \(\mathbb S\) be the open unit ball and the unit sphere in the \(n\)-dimensional complex Euclidean space \(\mathbb C^n\), respectively, and \(d\sigma\) the normalized Lebesgue measure on \(\mathbb S\). Let \(\alpha >0\), \(t\geq 0\) and \(\varphi_\alpha (t) =t(\log (\gamma_\alpha +t))^\alpha\), where \(\gamma_\alpha=\max \{e,e^\alpha\}\). The Zygmund \(F\)-algebra \(N\log^\alpha N\) on \(\mathbb B\) is defined as a family of all holomorphic functions \(f\) defined on \(\mathbb B\) for which NEWLINE\[NEWLINE\sup_{0\leq r < 1} \int_{\mathbb S} \varphi_\alpha (\log (1+|f(r\zeta )|)\, d\sigma(\zeta) <\infty.NEWLINE\]NEWLINE In the paper under review the author characterizes the complex-linear isometries on \(N\log^\alpha N\). Let us mention, that \textit{O. Hatori} et al. [Abstr. Appl. Anal. 2012, Article ID 125987, 16 p. (2012; Zbl 1236.32002)] extended the results of the paper studying multiplicative isometries on the following \(F\)-algebras: the Smirnov class \(N_*(X)\), the Privalov class \(N^P(X)\), the Bergman-Privalov class \(AN_\alpha^p(X)\) and the Zygmund \(F\)-algebra \(N\log^\alpha N(X)\), where \(X\) is \(\mathbb B\) or the unit polydisk.