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

Bergman–Privalov class ANα(B)ANα(B) consists of all holomorphic functions on the unit ball B⊂CnB⊂Cn such that ‖f‖ANα:=∫Bln(1+∣f(z)∣)dVα(z)<∞,‖f‖ANα:=∫Bln(1+∣f(z)∣)dVα(z)<∞, where α>−1α>−1, dVα(z)=cα,n(1−∣z∣2)αdV(z)dVα(z)=cα,n(1−∣z∣2)αdV(z) (dV(z)dV(z) is the normalized Lebesgue volume measure on BB and cα,ncα,n is the normalization constant, that is, Vα(B)=1Vα(B)=1). Under a mild condition, we characterize surjective isometries (not necessarily linear) on ANα(B)ANα(B), and prove that TT is a surjective multiplicative isometry (not necessarily linear) on ANα(B)ANα(B) if and only if it has the form Tf=f∘ψTf=f∘ψ or Tf=f∘ψ¯¯, for every f∈ANα(B)f∈ANα(B), where ψψ is a unitary transformation of the unit ball. The corresponding results for the case of the Bergman–Privalov class on the unit polydisk DnDn are also given. Our results extend and complement recent results by O. Hatori and Y. Iida.