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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