Authors: Stević, Stevo 
Affiliations: Mathematical Institute of the Serbian Academy of Sciences and Arts 
Title: On some isometries on the Bergman-Privalov class on the unit ball
Journal: Nonlinear Analysis, Theory, Methods and Applications
Volume: 75
Issue: 4
First page: 2448
Last page: 2454
Issue Date: 1-Mar-2012
Rank: M21
ISSN: 0362-546X
DOI: 10.1016/j.na.2011.10.042
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.
Keywords: Bergman-Privalov class | Multiplicative isometry | Nonlinear isometry
Publisher: Elsevier

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