Groups of linear isometries of spaces M q of holomorphic functions of several complex variables

Abstract

C n = {z = (z1, z2, . . . , zn) | zk ∈ C, 1 ≤ k ≤ n} be the standard n-dimensional complex coordinate space. ByG we denote the ball Bn = {z ∈ C | |z1| + |z2| + · · ·+ |zn| < 1} or the polydisk U = {z ∈ C | |z1| < 1, . . . , |zn| < 1} in the space Cn, and by Γ we denote the Bergman–Shilov boundary of G: ifG = Bn, then we take Γ = Sn = {z ∈ C | |z1| + · · ·+ |zn| = 1}, and ifG = Un, then Γ = T n = {z ∈ C | |z1| = · · · = |zn| = 1}. On Γ, there exists a natural invariant probability measure σ(dζ), ζ ∈ Γ, which coincides with the normed Lebesgue measure on the sphere Sn if Γ = Sn and is the direct product of normed Lebesgue measures on the unit circles that compose the torus T n in the Cartesian product if Γ = T n. For n = 1, we have G = Bn = U = U = {z ∈ C | |z| < 1}, Γ = Sn = T n = T = {z ∈ C | |z| = 1} and σ(dζ) = |dζ|/2π, ζ ∈ T . For an arbitrary nondecreasing functions φ(t) ≥ 0 convex on [−∞,+∞), the classes φ(N) are defined (see [1, p. 91 (Russian transl.)] and [2, p. 46 (Russian transl.)]) as sets of all functions f holomorphic in the domainG and satisfying the condition