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

Abstract EN

We introduce a new space ANlog,α(?) consisting of all holomorphic functions on the unit ball ?⊂ℂn such that ‖f‖ANlog,α:=∫?φe(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 ?, and cα,n is a normalization constant, that is, Vα(?)=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(?) 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(?) 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 ? 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(?).