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

Abstract

Bergman–Privalov class A N α ( B ) consists of all holomorphic functions on the unit ball B ⊂ C n such that ‖ f ‖ A N α : = ∫ B ln ( 1 + ∣ f ( z ) ∣ ) d V α ( z ) < ∞ , where α > − 1 , d V α ( z ) = c α , n ( 1 − ∣ z ∣ 2 ) α d V ( z ) ( d V ( z ) is the normalized Lebesgue volume measure on B and c α , n is the normalization constant, that is, V α ( B ) = 1 ). Under a mild condition, we characterize surjective isometries (not necessarily linear) on A N α ( B ) , and prove that T is a surjective multiplicative isometry (not necessarily linear) on A N α ( B ) if and only if it has the form T f = f ∘ ψ or T f = f ∘ ψ ¯ ¯ , for every f ∈ A N α ( 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 D n are also given. Our results extend and complement recent results by O. Hatori and Y. Iida.