tailieunhanh - Báo cáo toán học: "Some Results on the Relation Between Pluripolarity of Graphs and Holomorphicity"

ông nhằm mục đích của bài viết này là để cho một số kết quả về mối quan hệ giữa pluripolarity của đồ thị và holomorphicity các chức năng phức tạp có giá trị. Đặc biệt, chúng tôi điều tra mối quan hệ giữa pluripolarity của đồ thị và holomorphicity của Hartogs loại và loại Forelli. | Vietnam Journal of Mathematics 34 3 2006 275-284 Viet n a m J 0 u r n a I of MATHEMATICS VAST 2006 Some Results on the Relation Between Pluripolarity of Graphs and Holomorphicity Le Mau Hai and Phung Van Manh Deparment of Mathematics Hanoi University of Education 136 Xuan Thuy Rd. Hanoi Vietnam Received February 14 2005 Revised April 13 2006 Abstract. The aim of this paper is to give some results on the relation between pluripolarity of graphs and holomorphicity of complex-valued functions. In particular we investigate the relation between the pluripolarity of graphs and the holomorphicity of Hartogs type and of Forelli type. 2000 Mathematics Subject Classification 32H02 32H25 32U05. Keywords Pluripolar sets Holomorphic graphs of Hartogs type holomorphic graphs of Forelli type. 1. Introduction In 2003 Shcherbina proved the following result which solves a problem posed several years ago by Chirka see 4 Let D be a domain in Cn and f D C be a continuous function. The graph r f of f is pluripolar subset of Cn 1 if and only if f is holomorphic see 10 . Next Shcherbina extented the above result to the multi-functions. Namely he proved that if a1 z am z are continuous on D such that the set E z w e D X C wm ai z wm-1 . am z 0 is a pluripolar set in Cn 1 then a1 z am z are holomorphic on D see 11 . In this paper we give some results of type of Shcherbina on the relation between pluriporlarity of graphs and holomorphicity of complex-valued functions defined on a domain in Cn. In particular we establish the relation between the 276 Le Mau Hai and Phung Van Manh pluripolarity of graphs and the holomorphicity of Hartogs type and of Forelli type see Theorems and below . The paper is organized as follows. Beside the introduction the paper contains three sections. In the second one we recall some well-known facts in the pluripotential theory. The third section is devoted to present holomorphic graphs of Hartogs type. The fourth one deals with holomorphic graphs of Forelli