On some isometries on the Bergman-Privalov class on the unit ball

Bergman-Privalov class AN(alpha)(B) consists of all holomorphic functions on the unit ball B subset of C-n such that parallel to f parallel to(AN alpha) := integral(B) ln(1+ vertical bar f(z)vertical bar) dV(alpha)(z) < infinity, where alpha > -1, dV(alpha)(z) = c(alpha, n)(1-vertical bar z vertical bar(2))(alpha)dV(z) (dV(z) is the normalized Lebesgue volume measure on B and c(alpha, n) is the normalization constant, that is, V-alpha(B) = 1). Under a mild condition, we characterize surjective isometries (not necessarily linear) on AN(alpha)(B), and prove that T is a surjective multiplicative isometry (not necessarily linear) on AN(alpha)(B) if and only if it has the form Tf = f o psi or Tf = f o psi, for every f is an element of AN(alpha)(B), where. is a unitary transformation of the unit ball. The corresponding results for the case of the Bergman-Privalov class on the unit polydisk D-n are also given. Our results extend and complement recent results by O. Hatori and Y. Iida. (C) 2011 Elsevier Ltd. All rights reserved.