数学物理学报(英文版) ›› 2018, Vol. 38 ›› Issue (3): 926-934.doi: 10.1016/S0252-9602(18)30793-8

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A BOUNDARY SCHWARZ LEMMA FOR PLURIHARMONIC MAPPINGS FROM THE UNIT POLYDISK TO THE UNIT BALL

李灵, 李红裔, 赵迪   

  1. LMIB, School of Mathematics and Systems Science, Beihang University, Beijing 100191, China
  • 收稿日期:2017-05-16 出版日期:2018-06-25 发布日期:2018-06-25
  • 作者简介:Ling LI,E-mail:normalee 2013@163.com;Hongyi LI,E-mail:hylbuaa@163.com;Di ZHAO,E-mail:zdhyl2010@163.com
  • 基金资助:

    Supported by the Natural and Science Foundation of China (61379001, 61771001).

A BOUNDARY SCHWARZ LEMMA FOR PLURIHARMONIC MAPPINGS FROM THE UNIT POLYDISK TO THE UNIT BALL

Ling LI, Hongyi LI, Di ZHAO   

  1. LMIB, School of Mathematics and Systems Science, Beihang University, Beijing 100191, China
  • Received:2017-05-16 Online:2018-06-25 Published:2018-06-25
  • Supported by:

    Supported by the Natural and Science Foundation of China (61379001, 61771001).

摘要:

In this article, we present a Schwarz lemma at the boundary for pluriharmonic mappings from the unit polydisk to the unit ball, which generalizes classical Schwarz lemma for bounded harmonic functions to higher dimensions. It is proved that if the pluriharmonic mapping fP(Dn, BN) is C1+α at z0Er∂Dn with f(0)=0 and f(z0)=w0∂BN for any n, N ≥ 1, then there exist a nonnegative vector λf=(λ1,0, …, λr, 0, …,0)TR2n satisfying λi ≥ 1/22n-1 for 1 ≤ i ≤ r such that
(Df(z'0))T w'0=diag(λf)z'0,
where z'0 and w'0 are real versions of z0 and w0, respectively.

关键词: Boundary Schwarz lemma, pluriharmonic mapping, unit polydisk, unit ball

Abstract:

In this article, we present a Schwarz lemma at the boundary for pluriharmonic mappings from the unit polydisk to the unit ball, which generalizes classical Schwarz lemma for bounded harmonic functions to higher dimensions. It is proved that if the pluriharmonic mapping fP(Dn, BN) is C1+α at z0Er∂Dn with f(0)=0 and f(z0)=w0∂BN for any n, N ≥ 1, then there exist a nonnegative vector λf=(λ1,0, …, λr, 0, …,0)TR2n satisfying λi ≥ 1/22n-1 for 1 ≤ i ≤ r such that
(Df(z'0))T w'0=diag(λf)z'0,
where z'0 and w'0 are real versions of z0 and w0, respectively.

Key words: Boundary Schwarz lemma, pluriharmonic mapping, unit polydisk, unit ball