函数的增长性质及其推广
摘要
本文得到右半平面中几类解析函数与调和函数的积分表示,利用Cartan估计和Hayman定理的方法,研究了它们的增长性质,并且把这些增长性质推广到了n维欧式半空间中.
在右半平面中,作者主要考虑了级小于2的调和函数当它在边界上连续时的积分表示和级小于2的解析函数当它在边界上奇异时的积分表达式,并得到了它们的增长性质,这一性质直接推广了Hayman定理。作者应用这种方法还得到了右半平面中一类级小于3的解析函数的积分表示,同时作者还得到右半平面中一类由积分表示的次调和函数的增长性质。在n维欧式半空间中,作者研究了一类次调和函数的增长性质,一类调和函数(由Piosson积分表示的Dirichlet问题的解)的渐进性质和另一类调和函数(由修正的Piosson积分表示的Dirichlet问题的解)的渐进性质。
在n维欧式半空间中,作者得到了调和函数的Carleman公式和具体的半球上调和函数的清楚表达式,并且由一个调和函数的上界得到了该调和函数的下界,当n=2时就是复平面中的结果。
这些结果推广了经典的复变函数理论,深刻揭示了复变函数和调和分析之间的联系与区别。
In these thesis, the author studied the integral representations of several classes of analytic and harmonic functions with new conditions, and gave the growth estimates by the ways of Cartan's estimate and Hayman's theorem, then the author generalized those estimate properties to n-dimension Euclidean half space.
On the right half plane, the author focused on the integral representions of harmonic functions that countious to the boundary and analytic functions that singular to the boundary with order less than 2, and proved their estimate properties, which generalized the Hayman theorem. The author gave the integral repressentation of a class of analytic functions with order less than 3, at the same time the author gave the growth properties of a class of subharmonic functions and functions whose order is finite in right half plane.
On the n dimensional Euclidean half space, the author studied the growth properties of subharmonic functions, the asymptotic properties of a class of harmonic functions(the solutions of half space Dirichlet problem represented by Poisson integral) and another class of harmonic functions (the modified solutions of half space Dirichlet problem represented by the modified Poisson integral).
On the n dimensional Euclidean half space, the author derived the Carleman formula of harmonic functions and the precise representation of harmonic function in the half sphere by using Hormander's theorem, the author also derivd a lower bound of harmonic function in the half space from the upper bound by using the representation of harmonic function in the half sphere and the carleman formula for the half space, which is the results of complex plane when n = 2.
These results generalize the theory of classic complex variables functions, illustrate the links and the difference between the complex variables functions and the harmonic analysis.
在右半平面中,作者主要考虑了级小于2的调和函数当它在边界上连续时的积分表示和级小于2的解析函数当它在边界上奇异时的积分表达式,并得到了它们的增长性质,这一性质直接推广了Hayman定理。作者应用这种方法还得到了右半平面中一类级小于3的解析函数的积分表示,同时作者还得到右半平面中一类由积分表示的次调和函数的增长性质。在n维欧式半空间中,作者研究了一类次调和函数的增长性质,一类调和函数(由Piosson积分表示的Dirichlet问题的解)的渐进性质和另一类调和函数(由修正的Piosson积分表示的Dirichlet问题的解)的渐进性质。
在n维欧式半空间中,作者得到了调和函数的Carleman公式和具体的半球上调和函数的清楚表达式,并且由一个调和函数的上界得到了该调和函数的下界,当n=2时就是复平面中的结果。
这些结果推广了经典的复变函数理论,深刻揭示了复变函数和调和分析之间的联系与区别。
In these thesis, the author studied the integral representations of several classes of analytic and harmonic functions with new conditions, and gave the growth estimates by the ways of Cartan's estimate and Hayman's theorem, then the author generalized those estimate properties to n-dimension Euclidean half space.
On the right half plane, the author focused on the integral representions of harmonic functions that countious to the boundary and analytic functions that singular to the boundary with order less than 2, and proved their estimate properties, which generalized the Hayman theorem. The author gave the integral repressentation of a class of analytic functions with order less than 3, at the same time the author gave the growth properties of a class of subharmonic functions and functions whose order is finite in right half plane.
On the n dimensional Euclidean half space, the author studied the growth properties of subharmonic functions, the asymptotic properties of a class of harmonic functions(the solutions of half space Dirichlet problem represented by Poisson integral) and another class of harmonic functions (the modified solutions of half space Dirichlet problem represented by the modified Poisson integral).
On the n dimensional Euclidean half space, the author derived the Carleman formula of harmonic functions and the precise representation of harmonic function in the half sphere by using Hormander's theorem, the author also derivd a lower bound of harmonic function in the half space from the upper bound by using the representation of harmonic function in the half sphere and the carleman formula for the half space, which is the results of complex plane when n = 2.
These results generalize the theory of classic complex variables functions, illustrate the links and the difference between the complex variables functions and the harmonic analysis.
引文
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[2]R.P.Boas.,Entire Functions,Academic Press Inc.,New York,1954.
[3]Levin,B.Ya.,Lections on Entire Functions.Translations of Math.Monographs.V.150.Am.Math.Soc.,Provedence,Rhode Island,1996.
[4]N.I.Akhiezer.,Lectures on integral transforms,Translations of Mathematical Monographs,vol.70,American Mathematical Society,Providence,RI,1998.
[5]Paul Koosis.,The Logarithmic Integral.I,Cambridge University Press,Cam-bridge,1998.
[6]B.Ya.Levin,Distribution of zeros of entire functions,revised ed.,Transla-tions of Mathematical Monographs,vol.5,American Mathematical Society,Providence,R.I.,1980.
[7]B.V.Vynnyts'kyi and V.L.Sharan.,On Zeros of one Class of Functions Analytic in a Half-plane,Ukrainian Mathematical Journal,Vol.55.,No.9,2003.
[8]Deng G.T.,Zeros of Analytic Functions in weighted Hardy Space(in Chi-nese)[J].Journal of Beijing Normal University(Natural Science Edition),2003,39(4):1.
[9]Sheldon Axler.,Paul Bourdon.,Wade Ramey.,Harmonic Function Theory,Springer-Verlag New York,1992.MR93f:31001,Zbi0447.31003.
[10]Yoshihiro Mizuta and Tetsu Shimomura.,Growth properties for modified Pois-son integrals in a half space,Pacific Journal of Mathematics,Vol.212,No.2,2003.
[11]M.Finkelstein and S.Scheinberg.,Kernels for solving problems of Dirichlet type in a half-plane,Adv.in Math.,18(1975),108-113.
[12]G.Szego.,Orthogonal polynomials,American Mathematical Society,Providence,1975.
[13]D.H.Armitage.,Representations of Harmonic Functions in Half-space.Proc.London Math.Soc.(3)38(1979)53-71.
[14]G.Folland,Lectures on partial differential equations,Princeton Univ.Press,Princeton,1976.
[15]D.Siegel and E.O.Talvila,Uniqueness for the n—dimensional half space Dirichlet problem,Pacific.J.Math.,175(1996),571-587,MR 98a:35020,Zbl 0865.35035.
[16]Elias.M.Stein and Guido Weiss,Fourier Analysis on Euclidean Space,Prince-ton University Press,1971.
[17]Lars Hormander.,Notions of Convexity.Birkh-user,Boston-Basel-Berlin,1994.
[18]Hoffman.K.,Banach Space of Analytic Function[M].Prentice-Hall:Engleuood Cliffs,1962.
[19]Walter Rudin.,Real and Complex Analysis,China Machine Press,Beijing,2004.
[20]Deng,G.T.,Analytic Functions in weighted Hardy Space(in Chinese)[J].Journal of Beijing Normal University(Natural Science Edition),2002,38(4):1
[21]D.Siegel.,The Dirichlet problem in a half-space and a new Phragmen-Lindelof Principle,Maximum principles and eigenvalue problems in partial differential equations(P.W.Schaefer,ed.),Longman,Harlow,1988,pp.208-217.
[22]Bengt Ove Turesson.,Nonlinear potential theory and weighted Sobolev spaces.Springer,2000.
[23]Yoshihiro Mizuta and Tetsu Shimomura.,Growth properties for modified Pois-son integrals in a half space,Pacific Journal of Mathematics,Vol.212,No.2,2003.
[24]G.T.Deng.,Incompleteness and closure of a linear apan of exponential system in a weighted Banach Space,Journal of Approximation Theory 125(2003)1-9.
[25]H.Groemer.,Geometric Applications of Fourier Series and Spherical Harmonics,Cambridge University Press,1996.
[26]Charles F.Dunkl and Yuan Xu.,Orthogonal Polynomials of Several Variables,Cambridge University Press,2001.
[27]John B.Garnett.,Bounded Analytic Functions,Academic Press,1981.
[28]T.M.Flett.,On the rate of growth of mean values of holomorphic and har-monic functions.Proc.London Math.Soc.,20(3)(1970),749-768,MR42 3286,Zbl 0211.39203.
[29]Miao.C.X.,Harmonic Analysis and the Apply in Partial Differential Equation,Science Press,Beijing,2004.
[30]Koosis P.,Introduction to H_p Spaces[M].Cambridge Unversity Press,Cam-bridge,1980.
[31]Tyn Myint.U.,Partial Differential Equation in Math and Physics,Shanghai Technology Press,Shanghai,1983.
[32]Hiroaki Aikawa,Marts Essen.,Potential Theory-Seledted Topics,Springer-Verlag Berlin Heidelberg,1996.
[33]A.M.Sedletskii.,Nonharrnonic Analysis,Journal Mathematical Science,Vol.116,No.5,2003.
[34]M.L.Cartwrigrt.,Integral Functions,Cambridge at the University Press,1956.
[35]Mats Andersson,Mikael Passare,Ragnar Sigurdsson.,Complex Convexity and Analytic Functionals,Birkhauser Verlag Basel.Boston.Berlin.
[36]Jie Xiao.,Holornorphic Q Classes,Springer 2001.
[37]A.M.Sedletskii.,Fourier Transforms and Approxirnations,Gordon and Breach Science Publishers,2000.
[38]E.M.Stein,Harmonic Analysis:Real-Variable Methods,Orthogonality,and Oscillatory Integrals,Princeton Univ.Press,Princeton,N.J.1993.1993.
[41]Frantisek Wolf.,The Poisson Intagral.A Study in the Uniqueness of Harmonic Functions.
[39]Paul Malliavin.,complex Analysis in the Aftermath of Paley-Wiener,Proceed-ings of Symposia in Applied Mathematics,Volume 52,1997.
[40]Marvin Rosenblum and James Rovnyak.,Hardy Class and Operator Theory,Dover Publications,Inc.Mineola,New York,1985.
[41]Paul R.Halmos.,Measure Theory,Springer-Verlag,1998.
[45]T.M.Flett.,On the Rate of Growth of Mean Values of Holomorphic and Harmonic Functions.Proc.London Math.Soc.(3)20(1970)749-68.
[46]U.Cegrell,S.Kolodziej.and N.Levenberg.,Two Problems on Potential Theory for Unbounded Sets.Math Scand.83(1998),265-276.
[47]Ikuko Miyarnoto and Hidenobu Yoshida.,Two Criterions of Wiener Type for Minimally Thin Sets and Rarefied Sets in a cone.J.Math.Soc.Japan.Vol.54,No.3,2002.
[48]D.Siegel and E.O.Talvila.,Uniqueness for n-dimensional Half Space Dirichlet Problem.Pacific Journal of Mathematics.Vol.175,No.2,1996.
[49]David Siegel and Erik Talvila.,Sharp Growth Estimates for Modified Poisson Integrals in a Half Space.Potential Analysis,15:333-360,2001.
[50]P.Sjogren.,Approach Regions for the Square Roots of the Poisson kernel and Bounded Functions.Bull.Austral.Math.Soc.,Vol.55,1997,521-527.
[51]Hidenobu Yoshida.,A Type of Uniqueness for the Dirichlet Problems on a Half-space with Continuous Data.Pacific Journal of Mathematics.Vol.172,No.2,1996.
[52]Juhani Riihentaus.,Removable Sets for Subharmonic Functions.Pacific Journal of Mathermatics.Vol.194,No.1,2000.
[53]Patrick Ahern and Carmen Cascante.,Exceptional Sets for Poisson Integrals of Potentials on the Unit Sphere in C~n,p≤1.Pacific Journal of Mathermat-ics.Vol.153,No.1,1992.
[54]Mark Pinkelstein and Stephen Scheinberg.,Kernels for Solving Problems of Dirichlet Type in a Half-plane.Advances in Mathematics 18,108-113(1975).
[55]S.J.Gardiner.,The Dirichlet and Neumann Problems for Harmonic Functions in Half-spaces.J.London Math.Soc.(2),24(1981)[502-512].
[56]A.Ancona,W.K.Hayman and J.-M.G.Wu.,The Growth of harmonic Functions Along a Path in Space.J.London Math.Soc.,vol.24,No.2,1981,513-524.
[57]U.Kuran.,Study of Superharmonic Functions in R~n×(0,+∞)by a Passage to R~n+3.J.London Math.Soc.,vol.20,No.3,1970,276-302.
[58]Ole Christensen.,An Introduction to Frames and Riesz Bases,Birkhauser,1996.
[59]B.V.Vynnytskyi.,V.L.Sharan.On Factorization of One Class of Functions Analytic in the Half-plane.Matematychui Studii.V.14.No.1.
[60]B.Ya.Levin.,Distribution of zeros of entire functions,revised ed.,Translations of mathematical Monographs,vol.5,American Mathematical Monographs,Providence,R.I.,1980.
[61]M.H.Protter and H.F.Weinberger,Maximum principles in differential equa-tions,Springer-Verlag,New York,1984.
[62]L.V.Alfors,Complex analysis:An introduction to the theory of analytic func-tions of one complex variable,third ed.,McGraw-Hill Book Co.,New York,1978.
[63]F.Ricci and E.Stein,Harmonic analysis on nilpotent groups and singular inte- grals Ⅰ:Oscillatory integral,J.Punct.Anal.,73(1987),179-194.
[64]闻国椿,共形映射和边植问题,高等教育出版社,北京,1985.
[65]张南岳陈怀惠,复变函数论选讲,北京大学出版社,北京,,1995.
[66]张鸣镛,位势论,北京大学出版社,北京,1998.
[67]龚升,简明复分析,北京大学出版社,北京,1996.
[68]沈燮昌,复变函数逼近论,科学出版社,北京,2003.
[69]吉田耕作,泛函分析,人民教育出版社,北京,1980.
[70]Erik Talvila,Growth estimates and Phragmen-Lindelof principles for half space problems,Department of Applied Mathematics of University of Waterloo,博士学位论文,1997.
[71]Secil Gergun,Representations of functions harmonic in the upper half-plane and their applications,Department of Mathematics and the Institute of Engineering and Science of Bilkent University,博士学位论文,2003.
[72]O,Akin,The integral representation of the positive solutions of the generalized Weinstein eqution on the quarter-space,SIAM J.Math.Anal.19,1988,1348-1354.
[73]F.Wolf,The Poisson integral.A study in the uniqueness of harmonic functions,Acta.Math.74,1941,65-100.
[74]R.T.Seeley,Spherical harmonics,Amer.Math.Monthly 73 Supplement,1996.
[75]H.Yoshida,A type of uniqueness for the Dirichlet problem on a half-space with continuous data,Pacific J.Math.148,1991,369-395.
[76]H.Yoshida,A boundedness criterion for subharmonic functions,J.London Math.Soc.Vol.24,No.2,1981,148-160.
[77]B.Muckenhoupt and E.M.Stein,Classical expansions and their relation to con-jugate harmonic functions,Trans.A.M.S.118,1965,17-92.
[78]R.M.Mcleod,The generalized Riemann integral,Mathematical Association of America,Providence,RI,1980.
[79]Y.Mizuta,On the behavior of harmonic functions near a hyperplane,Analysis 2,1982,203-218.
[80]Lu S.Z.and Wang.N.Y.,Real Analysis,Beingjing Normal University Press,Beijing,1997.
[2]R.P.Boas.,Entire Functions,Academic Press Inc.,New York,1954.
[3]Levin,B.Ya.,Lections on Entire Functions.Translations of Math.Monographs.V.150.Am.Math.Soc.,Provedence,Rhode Island,1996.
[4]N.I.Akhiezer.,Lectures on integral transforms,Translations of Mathematical Monographs,vol.70,American Mathematical Society,Providence,RI,1998.
[5]Paul Koosis.,The Logarithmic Integral.I,Cambridge University Press,Cam-bridge,1998.
[6]B.Ya.Levin,Distribution of zeros of entire functions,revised ed.,Transla-tions of Mathematical Monographs,vol.5,American Mathematical Society,Providence,R.I.,1980.
[7]B.V.Vynnyts'kyi and V.L.Sharan.,On Zeros of one Class of Functions Analytic in a Half-plane,Ukrainian Mathematical Journal,Vol.55.,No.9,2003.
[8]Deng G.T.,Zeros of Analytic Functions in weighted Hardy Space(in Chi-nese)[J].Journal of Beijing Normal University(Natural Science Edition),2003,39(4):1.
[9]Sheldon Axler.,Paul Bourdon.,Wade Ramey.,Harmonic Function Theory,Springer-Verlag New York,1992.MR93f:31001,Zbi0447.31003.
[10]Yoshihiro Mizuta and Tetsu Shimomura.,Growth properties for modified Pois-son integrals in a half space,Pacific Journal of Mathematics,Vol.212,No.2,2003.
[11]M.Finkelstein and S.Scheinberg.,Kernels for solving problems of Dirichlet type in a half-plane,Adv.in Math.,18(1975),108-113.
[12]G.Szego.,Orthogonal polynomials,American Mathematical Society,Providence,1975.
[13]D.H.Armitage.,Representations of Harmonic Functions in Half-space.Proc.London Math.Soc.(3)38(1979)53-71.
[14]G.Folland,Lectures on partial differential equations,Princeton Univ.Press,Princeton,1976.
[15]D.Siegel and E.O.Talvila,Uniqueness for the n—dimensional half space Dirichlet problem,Pacific.J.Math.,175(1996),571-587,MR 98a:35020,Zbl 0865.35035.
[16]Elias.M.Stein and Guido Weiss,Fourier Analysis on Euclidean Space,Prince-ton University Press,1971.
[17]Lars Hormander.,Notions of Convexity.Birkh-user,Boston-Basel-Berlin,1994.
[18]Hoffman.K.,Banach Space of Analytic Function[M].Prentice-Hall:Engleuood Cliffs,1962.
[19]Walter Rudin.,Real and Complex Analysis,China Machine Press,Beijing,2004.
[20]Deng,G.T.,Analytic Functions in weighted Hardy Space(in Chinese)[J].Journal of Beijing Normal University(Natural Science Edition),2002,38(4):1
[21]D.Siegel.,The Dirichlet problem in a half-space and a new Phragmen-Lindelof Principle,Maximum principles and eigenvalue problems in partial differential equations(P.W.Schaefer,ed.),Longman,Harlow,1988,pp.208-217.
[22]Bengt Ove Turesson.,Nonlinear potential theory and weighted Sobolev spaces.Springer,2000.
[23]Yoshihiro Mizuta and Tetsu Shimomura.,Growth properties for modified Pois-son integrals in a half space,Pacific Journal of Mathematics,Vol.212,No.2,2003.
[24]G.T.Deng.,Incompleteness and closure of a linear apan of exponential system in a weighted Banach Space,Journal of Approximation Theory 125(2003)1-9.
[25]H.Groemer.,Geometric Applications of Fourier Series and Spherical Harmonics,Cambridge University Press,1996.
[26]Charles F.Dunkl and Yuan Xu.,Orthogonal Polynomials of Several Variables,Cambridge University Press,2001.
[27]John B.Garnett.,Bounded Analytic Functions,Academic Press,1981.
[28]T.M.Flett.,On the rate of growth of mean values of holomorphic and har-monic functions.Proc.London Math.Soc.,20(3)(1970),749-768,MR42 3286,Zbl 0211.39203.
[29]Miao.C.X.,Harmonic Analysis and the Apply in Partial Differential Equation,Science Press,Beijing,2004.
[30]Koosis P.,Introduction to H_p Spaces[M].Cambridge Unversity Press,Cam-bridge,1980.
[31]Tyn Myint.U.,Partial Differential Equation in Math and Physics,Shanghai Technology Press,Shanghai,1983.
[32]Hiroaki Aikawa,Marts Essen.,Potential Theory-Seledted Topics,Springer-Verlag Berlin Heidelberg,1996.
[33]A.M.Sedletskii.,Nonharrnonic Analysis,Journal Mathematical Science,Vol.116,No.5,2003.
[34]M.L.Cartwrigrt.,Integral Functions,Cambridge at the University Press,1956.
[35]Mats Andersson,Mikael Passare,Ragnar Sigurdsson.,Complex Convexity and Analytic Functionals,Birkhauser Verlag Basel.Boston.Berlin.
[36]Jie Xiao.,Holornorphic Q Classes,Springer 2001.
[37]A.M.Sedletskii.,Fourier Transforms and Approxirnations,Gordon and Breach Science Publishers,2000.
[38]E.M.Stein,Harmonic Analysis:Real-Variable Methods,Orthogonality,and Oscillatory Integrals,Princeton Univ.Press,Princeton,N.J.1993.1993.
[41]Frantisek Wolf.,The Poisson Intagral.A Study in the Uniqueness of Harmonic Functions.
[39]Paul Malliavin.,complex Analysis in the Aftermath of Paley-Wiener,Proceed-ings of Symposia in Applied Mathematics,Volume 52,1997.
[40]Marvin Rosenblum and James Rovnyak.,Hardy Class and Operator Theory,Dover Publications,Inc.Mineola,New York,1985.
[41]Paul R.Halmos.,Measure Theory,Springer-Verlag,1998.
[45]T.M.Flett.,On the Rate of Growth of Mean Values of Holomorphic and Harmonic Functions.Proc.London Math.Soc.(3)20(1970)749-68.
[46]U.Cegrell,S.Kolodziej.and N.Levenberg.,Two Problems on Potential Theory for Unbounded Sets.Math Scand.83(1998),265-276.
[47]Ikuko Miyarnoto and Hidenobu Yoshida.,Two Criterions of Wiener Type for Minimally Thin Sets and Rarefied Sets in a cone.J.Math.Soc.Japan.Vol.54,No.3,2002.
[48]D.Siegel and E.O.Talvila.,Uniqueness for n-dimensional Half Space Dirichlet Problem.Pacific Journal of Mathematics.Vol.175,No.2,1996.
[49]David Siegel and Erik Talvila.,Sharp Growth Estimates for Modified Poisson Integrals in a Half Space.Potential Analysis,15:333-360,2001.
[50]P.Sjogren.,Approach Regions for the Square Roots of the Poisson kernel and Bounded Functions.Bull.Austral.Math.Soc.,Vol.55,1997,521-527.
[51]Hidenobu Yoshida.,A Type of Uniqueness for the Dirichlet Problems on a Half-space with Continuous Data.Pacific Journal of Mathematics.Vol.172,No.2,1996.
[52]Juhani Riihentaus.,Removable Sets for Subharmonic Functions.Pacific Journal of Mathermatics.Vol.194,No.1,2000.
[53]Patrick Ahern and Carmen Cascante.,Exceptional Sets for Poisson Integrals of Potentials on the Unit Sphere in C~n,p≤1.Pacific Journal of Mathermat-ics.Vol.153,No.1,1992.
[54]Mark Pinkelstein and Stephen Scheinberg.,Kernels for Solving Problems of Dirichlet Type in a Half-plane.Advances in Mathematics 18,108-113(1975).
[55]S.J.Gardiner.,The Dirichlet and Neumann Problems for Harmonic Functions in Half-spaces.J.London Math.Soc.(2),24(1981)[502-512].
[56]A.Ancona,W.K.Hayman and J.-M.G.Wu.,The Growth of harmonic Functions Along a Path in Space.J.London Math.Soc.,vol.24,No.2,1981,513-524.
[57]U.Kuran.,Study of Superharmonic Functions in R~n×(0,+∞)by a Passage to R~n+3.J.London Math.Soc.,vol.20,No.3,1970,276-302.
[58]Ole Christensen.,An Introduction to Frames and Riesz Bases,Birkhauser,1996.
[59]B.V.Vynnytskyi.,V.L.Sharan.On Factorization of One Class of Functions Analytic in the Half-plane.Matematychui Studii.V.14.No.1.
[60]B.Ya.Levin.,Distribution of zeros of entire functions,revised ed.,Translations of mathematical Monographs,vol.5,American Mathematical Monographs,Providence,R.I.,1980.
[61]M.H.Protter and H.F.Weinberger,Maximum principles in differential equa-tions,Springer-Verlag,New York,1984.
[62]L.V.Alfors,Complex analysis:An introduction to the theory of analytic func-tions of one complex variable,third ed.,McGraw-Hill Book Co.,New York,1978.
[63]F.Ricci and E.Stein,Harmonic analysis on nilpotent groups and singular inte- grals Ⅰ:Oscillatory integral,J.Punct.Anal.,73(1987),179-194.
[64]闻国椿,共形映射和边植问题,高等教育出版社,北京,1985.
[65]张南岳陈怀惠,复变函数论选讲,北京大学出版社,北京,,1995.
[66]张鸣镛,位势论,北京大学出版社,北京,1998.
[67]龚升,简明复分析,北京大学出版社,北京,1996.
[68]沈燮昌,复变函数逼近论,科学出版社,北京,2003.
[69]吉田耕作,泛函分析,人民教育出版社,北京,1980.
[70]Erik Talvila,Growth estimates and Phragmen-Lindelof principles for half space problems,Department of Applied Mathematics of University of Waterloo,博士学位论文,1997.
[71]Secil Gergun,Representations of functions harmonic in the upper half-plane and their applications,Department of Mathematics and the Institute of Engineering and Science of Bilkent University,博士学位论文,2003.
[72]O,Akin,The integral representation of the positive solutions of the generalized Weinstein eqution on the quarter-space,SIAM J.Math.Anal.19,1988,1348-1354.
[73]F.Wolf,The Poisson integral.A study in the uniqueness of harmonic functions,Acta.Math.74,1941,65-100.
[74]R.T.Seeley,Spherical harmonics,Amer.Math.Monthly 73 Supplement,1996.
[75]H.Yoshida,A type of uniqueness for the Dirichlet problem on a half-space with continuous data,Pacific J.Math.148,1991,369-395.
[76]H.Yoshida,A boundedness criterion for subharmonic functions,J.London Math.Soc.Vol.24,No.2,1981,148-160.
[77]B.Muckenhoupt and E.M.Stein,Classical expansions and their relation to con-jugate harmonic functions,Trans.A.M.S.118,1965,17-92.
[78]R.M.Mcleod,The generalized Riemann integral,Mathematical Association of America,Providence,RI,1980.
[79]Y.Mizuta,On the behavior of harmonic functions near a hyperplane,Analysis 2,1982,203-218.
[80]Lu S.Z.and Wang.N.Y.,Real Analysis,Beingjing Normal University Press,Beijing,1997.