Isometries of a Bergman-Privalov-Type Space on the Unit Ball

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)