Copyright © 2009 Stevo Stević and Sei-Ichiro Ueki. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We introduce a new space ANlog⁡,α(𝔹) consisting of all holomorphic functions on the unit ball 𝔹⊂ℂn such that ‖f‖ANlog⁡,α:=∫𝔹φe(ln⁡(1+|f(z)|))dVα(z)<∞, where α>−1, dVα(z)=cα,n(1−|z|2)αdV(z) (dV(z) is the normalized Lebesgue volume measure on 𝔹, and cα,n is a normalization constant, that is, Vα(𝔹)=1), and φe(t)=tln⁡(e+t) for t∈[0,∞). Some basic properties of this space are presented. Among other results we proved that ANlog⁡,α(𝔹) with the metric d(f,g)=‖f−g‖ANlog⁡,α is an F-algebra with respect to pointwise addition and multiplication. We also prove that every linear isometry T of ANlog⁡,α(𝔹) into itself has the form Tf=c(f∘ψ) for some c∈ℂ such that |c|=1 and some ψ which is a holomorphic self-map of 𝔹 satisfying a measure-preserving property with respect to the measure dVα. As a consequence of this result we obtain a complete characterization of all linear bijective isometries of ANlog⁡,α(𝔹).