Isometries of the Zygmund $F$-algebra

In his monograph A. Zygmund introduced the space $N{\log }^{\alpha }N$
$(\alpha >0)$ of holomorphic functions on the unit ball that satisfy

$$\sup _{0\le r <1} \int _{\mathbb{S}} {\varphi }_{\alpha }(\log (1+\vert f(r\zeta )\vert))d\sigma (\zeta ) < \infty ,$$

where ${\varphi }_{\alpha }(t)= t\{\log ({\gamma }_{\alpha }+t)\}^{\alpha }$ for $t \in [0, \infty )$ and ${\gamma }_{\alpha }=\max \{e, e^{\alpha }\}$. In 2002, O.M. Eminyan provided some basic properties of $N{\log }^{\alpha }N$. In this paper we will characterize injective and surjective linear isometries of $N{\log }^{\alpha }N$. As an application, we will consider isometrically equivalent composition operators or multiplication operators on $N{\log }^{\alpha }N$.