TAILIEUCHUNG - Báo cáo toán học: "Weighted Composition Operators between Different Weighted Bergman Spaces in Polydiscs "

là polydiscs đơn vị Cn, φ (z) = (φ1 (z ),., φn (z)) là một holomorphic tự bản đồ của D và ψ (z) một chức năng holomorphic trên Dn. Các điều kiện cần và đủ được thiết lập cho ψCφ nhà điều hành thành phần trọng gây ra bởi φ (z) và ψ (z) được giới hạn hoặc nhỏ gọn giữa các không gian khác nhau trọng Bergman trong polydiscs. | Vietnam Journal of Mathematics 34 3 2006 255-264 Viet n a m J 0 u r n a I of MATHEMATICS VAST 2006 Weighted Composition Operators Between Different Weighted Bergman Spaces in Polydiscs Li Songxiao Department of Math. Shantou University 515063 Shantou Guangdong China and Department of Mathematics Jiaying University 514015 Meizhou Guangdong China Received April 04 2004 Revised September 18 2005 Abstract. Let Dn be the unit polydiscs of Cn z 1 z . pn z be a holomorphic self-map of Dn and Mz a holomorphic function on Dn. Necessary and sufficient conditions are established for the weighted composition operator . induced by ip z and Mz to be bounded or compact between different weighted Bergman spaces in polydiscs. 2000 Mathematics Subject Classification 47B38 32A36. Keywords Bergman space polydiscs weighted composition operator. 1. Introduction We adopt the notation described in 4-6 . Denote by Dn the unit polydisc in Cn by Tn the distinguished boundary of Dn by Apa Dnn the weighted Bergman spaces of order p with weights n n I 1 2 a 1. We use mn to denote the n-dimensional Lebesgue area measure on Tn normalized so that mn Tn 1. By ơn we shall denote the volume measure on Dn given by ơn Dn 1 and by ơn a we shall denote the weighted measure on Dn given by ơn a pn i 1 zj 2 aơn. If R is a rectangle on Tn then S R denote the corona associated to The author is partially supported by NNSF 10371051 and ZNSF 102025 . 256 Li Songxiao R. In particular if R I1 X I2 X X In c Tn with I being the intervals on Tn of length ỗ and centered at r 2 for i 1 - n then S R is given by S R S T1 X S I2 X X S In where S Ii reie D 1 - ỗi r 1 e0 e e0 ỗi . For a 1 0 p TO recall that the weighted Bergman space Apa Dn consists of all holomorphic functions on the polydisc satisfying the condition f ll f z p n 1 zi 2 adơn a X . a JDn i Denoted by H Dn the class of all holomorphic functions with domain Dn. Let p be a holomorphic self-map of Dn the composition operator Cv induced by p is defined .

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