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.
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Báo cáo toán học: "Weighted Composition Operators between Different Weighted Bergman Spaces in Polydiscs " Vietnam Journal of Mathematics 34:3 (2006) 255–264 9LHWQD P-RXUQDO RI 0$7+(0$7, &6 ‹ 9$ 67 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, 2005Abstract. Let Dnbe the unit polydiscs of Cn , ϕ(z)=(ϕ1 (z),... ,ϕn (z)) be a holomor-phic self-map of D and ψ(z) a holomorphic function on Dn . Necessary and sufficient nconditions are established for the weighted composition operator ψCϕ induced by ϕ(z)and ψ(z) to be bounded or compact between different weighted Bergman spaces inpolydiscs.2000 Mathematics Subject Classification: 47B38, 32A36.Keywords: Bergman space, polydiscs, weighted composition operator.1. IntroductionWe adopt the notation described in [4-6]. Denote by Dn the unit polydisc inCn , by T n the distinguished boundary of Dn , by Ap (Dn ) the weighted Bergman α nspaces of order p with weights i=1 (1−|zi |2)α , α > −1. We use mn to denote then-dimensional Lebesgue area measure on T n , normalized so that mn (T n) = 1.By σn we shall denote the volume measure on Dn given by σn(Dn ) = 1, and nby σn,α we shall denote the weighted measure on Dn given by σn,α = i=1 (1 −|zi |2)α σn. If R is a rectangle on T n , then S (R) denote the corona associated to∗ The author is partially supported by NNSF(10371051) and ZNSF(102025).256 Li SongxiaoR. In particular, if R = I1 × I2 × · · · × In ⊂ T n , with Ii being the intervals on 0T n of length δi and centered at ei(θi +δi /2) for i = 1, · · ·, n, then S (R) is given byS (R) = S (I1 ) × S (I2 ) × · · · × S (In ), where S (Ii ) = {reiθ ∈ D : 1 − δi < r < 1, θ0 < θ < θi + δi }. i 0For α > −1, 0 < p < ∞, recall that the weighted Bergman space Ap (Dn ) αconsists of all holomorphic functions on the polydisc satisfying the condition n p |f ( z ) |p (1 − |zi|2 )αdσn,α < +∞. f = Apα Dn i=1 Denoted by H (Dn ) the class of all holomorphic functions with domain Dn .Let ϕ be a holomorphic self-map of Dn , the composition operator Cϕ induced byϕ is defined by (Cϕ f )(z ) = f (ϕ(z )) for z in Dn and f ∈ H (Dn ). If, in addition,ψ is a holomorphic function defined on Dn , the weighted composition operatorsψCϕ induced by ψ and ϕ is defined by (ψCϕ f )(z ) = ψ(z )f (ϕ(z )) n nfor z in D and f ∈ H (D ). It is interesting to characterize the composition operator on various analyticfunction spaces. The book [2] contains plenty of information. It is well knownthat composition operator is bounded on the Hardy space and the Bergman spacein the unit disc. This result does not carry over to the case of several complexvariables. Singh and Sharma has showed in [7] that not every holomorphicmap from Dn to Dn induces a composition operator on H p (Dn ). For example ...