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

Abstract

Bergman-Privalov class A Nα(B) consists of all holomorphic functions on the unit ball B⊂ Cn such that ||f||A Nα:=∫ Bln(1+|f(z)|)d Vα(z) <∞, where α>-1, d Vα(z)=cα, n(1-|z| 2)αdV(z) (dV(z) is the normalized Lebesgue volume measure on B and cα, n is the normalization constant, that is, Vα(B)=1). Under a mild condition, we characterize surjective isometries (not necessarily linear) on A Nα(B), and prove that T is a surjective multiplicative isometry (not necessarily linear) on A Nα(B) if and only if it has the form Tf=f°ψ or Tf=f°ψ̄̄, for every f∈A Nα(B), where ψ is a unitary transformation of the unit ball. The corresponding results for the case of the Bergman-Privalov class on the unit polydisk Dn are also given. Our results extend and complement recent results by O. Hatori and Y. Iida.