Isometries of a Bergman-Privalov-type space on the unit ball

Abstract

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