上半平面到其自身的调和同胚
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摘要
得到了实轴R上的保向同胚φ(x)在Beurling-Ahlfors延拓下是调和拟共形的充要条件.利用poisson积分具体给出了一个φ(x)延拓成上半平面到其自身的调和同胚.并且给出了这个调和同胚为拟共形的一个充分条件,得到了它的伸张估计.所得结果推广了Michalski的相关结果.
In this paper, the necessary and sufficient conditions for the extension of BeurlingAhlfors to be a harmonic quasiconformal mapping of the upper half-plane onto itself are obtained.Specifically, a kind of harmonic homeomorphism of the upper half-plane by applying poisson integral is considered. Furthermore, a sufficient condition for this harmonic homeomorphism to be quasiconformal is presented, and its dilatation is estimated. Some results of Michalski are generalized.
In this paper, the necessary and sufficient conditions for the extension of BeurlingAhlfors to be a harmonic quasiconformal mapping of the upper half-plane onto itself are obtained.Specifically, a kind of harmonic homeomorphism of the upper half-plane by applying poisson integral is considered. Furthermore, a sufficient condition for this harmonic homeomorphism to be quasiconformal is presented, and its dilatation is estimated. Some results of Michalski are generalized.
引文
[1] Ahlfors L V. Lecture on Quasionformal Mappings[M]. Princeton:Van Nostrand, 1966.
[2] Kalaj D, Pavlovic M. Boundary correspondence under quasiconformal harmonic diffeomorphisms of a half-plane[J]. Anna Acad Sci Fenn Math, 2005, 30:159-165.
[3] Axler S, Bourdon P, Ramey W. Harmonic Function Theory[M]. New York:Spring-Verlag,1992.
[4] Huang Xinzhong. Harmonic quasiconformal mapping on the upper half-plane[J]. Complex Var Elliptic Equ, 2013, 58(7):1005-1011.
[5] Chen Shaolin. John disks and K-quasiconformal harmonic mappings[J]. J Geome Anal,2015, 207(2):1-21.
[6]黄心中.单位圆盘上的调和拟共形同胚[J].数学年刊A辑,2008, 29(4):519-624.
[7] Shen Yuliang. Quasiconformal mappings and harmonic functions[J]. Adv Math, 1999, 4(4):347-357.
[8] Zhu Jianfeng. Quasiconformal harmonic mappings and the curvature of the boundary[J]. J Math Anal Appl, 2017, 446(2):1154-1166.
[9] Li Lilan. In.jectivity of sections of convex harmonic mappings and convolution theorems[J].Czech Math J, 2016, 66(2):331-350.
[10] Chen Huaihui, Gauthier P M, Hengartner W. Bloch constants for planar harmonic mappings[J]. Proc Amer Math Soc, 2000, 128(11):3231-3240.
[11]龙波涌,黄心中.Beurling-Ahlfors延拓的伸张函数[J].数学杂志,2006, 26(4):446-450.
[12] Liu Mingsheng. Estimates on Bloch constants for planar harmonic mappings[J].中国科学A辑,2009, 52(1):87-93.
[13] Michalski A. Sufficient conditions for quasiconformality of harmonic mappings of the upper half-plane onto itself[J]. Anna Umcs Math, 2008, 62(1):91-104.
[2] Kalaj D, Pavlovic M. Boundary correspondence under quasiconformal harmonic diffeomorphisms of a half-plane[J]. Anna Acad Sci Fenn Math, 2005, 30:159-165.
[3] Axler S, Bourdon P, Ramey W. Harmonic Function Theory[M]. New York:Spring-Verlag,1992.
[4] Huang Xinzhong. Harmonic quasiconformal mapping on the upper half-plane[J]. Complex Var Elliptic Equ, 2013, 58(7):1005-1011.
[5] Chen Shaolin. John disks and K-quasiconformal harmonic mappings[J]. J Geome Anal,2015, 207(2):1-21.
[6]黄心中.单位圆盘上的调和拟共形同胚[J].数学年刊A辑,2008, 29(4):519-624.
[7] Shen Yuliang. Quasiconformal mappings and harmonic functions[J]. Adv Math, 1999, 4(4):347-357.
[8] Zhu Jianfeng. Quasiconformal harmonic mappings and the curvature of the boundary[J]. J Math Anal Appl, 2017, 446(2):1154-1166.
[9] Li Lilan. In.jectivity of sections of convex harmonic mappings and convolution theorems[J].Czech Math J, 2016, 66(2):331-350.
[10] Chen Huaihui, Gauthier P M, Hengartner W. Bloch constants for planar harmonic mappings[J]. Proc Amer Math Soc, 2000, 128(11):3231-3240.
[11]龙波涌,黄心中.Beurling-Ahlfors延拓的伸张函数[J].数学杂志,2006, 26(4):446-450.
[12] Liu Mingsheng. Estimates on Bloch constants for planar harmonic mappings[J].中国科学A辑,2009, 52(1):87-93.
[13] Michalski A. Sufficient conditions for quasiconformality of harmonic mappings of the upper half-plane onto itself[J]. Anna Umcs Math, 2008, 62(1):91-104.