Isometries of weighted Bergman-Privalov spaces on the unit ball of \bm{C}ⁿ

Abstract

Let B denote the unit ball in \bm{C}ⁿ, and v the normalized Lebesgue measure on B. For α>-1, define dv_α(z)=Γ(n+α+1)/{Γ(n+1)Γ(α+1)}(1-|z|²)^αdv(z), z∈ B. Let H(B) denote the space of holomorphic functions in B. For p≥q 1, define
(\displaystyle AN)^{\bm{p}}(v_α)={f∈ H(B):\left//f\ ight//≡[∈t_B{log(1+|f|)}^pdv_α]^1/p<∞}.
(AN)^p(v_α) is an F-space with respect to the metric ρ(f, g)≡\left//f-g\ ight//. In this paper we prove that every linear isometry T of (AN)^p(v_α) into itself is of the form Tf=c(f\circψ) for all f∈(AN)^p(v_α), where c is a complex number with |c|=1 and ψ is a holomorphic self-map of B which is measure-preserving with respect to the measure v_α.