Groups of linear isometries of spaces \(M^q\) of holomorphic functions of several complex variables (Q2517993)

Let \(G\) be the unit ball or the unit polydisk in \(\mathbb C^n\) and \(\Gamma\) be the Bergman-Shilov boundary of \(G\). Let \(M^q\) be the class of all holomorphic functions \(f\) in \(G\) such that \[ \int \limits_{\Gamma} (\ln^+ \{\sup \limits_{0 \leq r < 1} |f (r \zeta)|)^q \sigma (d \zeta) < + \infty, \] where \(\sigma\) is an invariant probability measure on \(\Gamma\). The main result of this paper is the following Theorem. A mapping \(A: M^q \to M^q\) is a surjective linear isometry if and only if for all functions \(f \in M^q\) \[ A f (z) = \alpha f (\Phi (z)), z \in G, \] where \(\alpha \in \mathbb C, |\alpha| = 1,\) and \(\Phi\) is a biholomorphic automorhism of \(G\) leaving the point \(0\) fixed.