tailieunhanh - Báo cáo toán học: "A Carleson measure theorem for the Bergman space on the ball "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Một biện pháp Carleson định lý cho không gian Bergman trên quả bóng. | J. OPERATOR THEORY 7 1982 157-165 Copyright by INCREST. 1982 A CARLESON MEASURE THEOREM FOR THE BERGMAN SPACE ON THE BALL JOSEPH A. CIMA and WARREN R. WOGEN Let B be the closed unit disc in the complex plane c and let ỊẤ be a finite measure on B. For 0 e 0 2tt h e 0 1 let 2 0 A u e B i 1 - 7 z 1 e argz 0 h . The measure ii is called a Carleson measure if there is a constant c 0 such that Q O 0 Ch V 0 h . In 1 it is shown that n is a Carleson measure if and only if the injection mapping from the Hardy space H- into F 2 dju is bounded . there is a C 0 so that for hl-p t l z l2dp z c C V F. J J 2tĩ B 0 Several analogues of the Carleson theorem have subsequently been obtained. Hormander 6 proved a version of this result for the Hardy space of the unit ball B in C . Then Carleson 2 produced an example to show that the result is false for the Hardy space of the polydisc. In a more current paper Hastings 5 has proven a Carleson type theorem for the Bergman space of the polyđisc. Further Stegenga 9 has obtained analogous results for certain weighted Bergman spaces. In this paper we prove a Carleson theorem for the Bergman space of the unit ball in C . We also outline a short proof of Hormander s result 6 . As an application we consider the question of compactness of Toeplitz operators on Bergman spaces. After the results of this paper were obtained Professor Peter Duren pointed out the Hormander reference 6 . We wish to thank him for this reference. 158 JOSEPH A. C1MA and WARREN R. WOGEN 1. THE PRINCIPAL THEOREM Let B be the closed unit ball in cn Bl B and let be the boundary of B . If Í is a measure on B and p A 1 then Lfp dp is the lAịd ì closure of the poly normals in z - El .- . Further let m z denote volume measure on C . Fix n 1 and p n. Consider measures defined by the equations d v z 1 - zp dm z z e B d2 zx .- 1 - Ex e Bl. For 0 t 1 and I 6 F let W Jze B 1 - z . The notation Ộ denotes complex inner product in C . A measure Í on B is a 1 -Carleson measure if