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.
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