The Lindelöf principle in ℂn

Peter Dovbush

Open Mathematics, Tome 11 (2013), p. 1763-1773 / Harvested from The Polish Digital Mathematics Library

Résumé

Let D be a bounded domain in ℂn. A holomorphic function f: D → ℂ is called normal function if f satisfies a Lipschitz condition with respect to the Kobayashi metric on D and the spherical metric on the Riemann sphere ̅ℂ. We formulate and prove a few Lindelöf principles in the function theory of several complex variables.

Détail

Publié le : 2013-01-01

Zbl 1278.32003

EUDML-ID : urn:eudml:doc:269191

Bibliographie

[1] Abate M., The Lindelöf principle and the angular derivative in strongly convex domains, J. Anal. Math., 1990, 54, 189–228 http://dx.doi.org/10.1007/BF02796148[Crossref] | Zbl 0694.32015

[2] Abate M., Angular derivatives in strongly pseudoconvex domains, In: Several Complex Variables and Complex Geometry, 2, Santa Cruz, 1989, Proc. Sympos. Pure Math., 52(2), American Mathematical Society, Providence, 1991, 23–40 http://dx.doi.org/10.1090/pspum/052.2/1128532[Crossref]

[3] Abate M., The Julia-Wolff-Carathéodory theorem in polydisks, J. Anal. Math., 1998, 74, 275–306 http://dx.doi.org/10.1007/BF02819453[Crossref] | Zbl 0912.32005

[4] Abate M., Angular derivatives in several complex variables, In: Real Methods in Complex and CR Geometry, Lecture Notes in Math., 1848, Springer, Berlin, 2004, 1–47 http://dx.doi.org/10.1007/978-3-540-44487-9_1[Crossref]

[5] Abate M., Tauraso R., The Lindelöf principle and angular derivatives in convex domains of finite type, J. Aust. Math. Soc., 2002, 73(2), 221–250 http://dx.doi.org/10.1017/S1446788700008818[Crossref] | Zbl 1113.32301

[6] Aladro G., Application of the Kobayashi metric to normal functions of several complex variables, Utilitas Math., 1987, 31, 13–24 | Zbl 0585.32027

[7] Aladro G., Krantz S.G., A criterion for normality in ℂn, J. Math. Anal. Appl., 1991, 161(1), 1–8 http://dx.doi.org/10.1016/0022-247X(91)90356-5[Crossref]

[8] Bagemihl F., Seidel W., Sequential and continuous limits of meromorphic functions, Ann. Acad. Sci. Fenn. Ser. A I Math., 1960, 280, 1–17 | Zbl 0095.05801

[9] Bayne R.E., Kwack M.H., A Lindelöf property for uniformly normal families, Missouri J. Math. Sci., 2010, 22(2), 130–138 | Zbl 1208.30033

[10] Cameron R.H., Storvick D.A., A Lindelöf theorem and analytic continuation for functions of several variables, with an application to the Feynman integral, In: Entire Functions and Related Parts of Analysis, LaJolla, 1966, American Mathematical Society, Providence, 1968, 149–156 http://dx.doi.org/10.1090/pspum/011/0237815[Crossref]

[11] Cima J.A., Krantz S.G., The Lindelöf principle and normal functions of several complex variables, Duke Math. J., 1983, 50(1), 303–328 http://dx.doi.org/10.1215/S0012-7094-83-05014-7[Crossref] | Zbl 0522.32003

[12] Čirka E.M., The Lindelöf and Fatou theorems in ℂn, Mat. Sb. (N.S.), 1973, 92(134), 622–644 (in Russian) | Zbl 0285.32005

[13] Dovbush P.V., Normal functions of several complex variables, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1981, 36(1), 38–42 (in Russian) | Zbl 0471.32002

[14] Dovbush P.V., Lindelöf’s theorem in ℂn, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1981, 36(6), 33–36 (in Russian)

[15] Dovbush P.V., Boundary behavior of normal holomorphic functions of several complex variables, Dokl. Akad. Nauk SSSR, 1982, 263(1), 14–17 (in Russian) | Zbl 0531.32003

[16] Dovbush P.V., Lindelöf’s theorem in ℂn, Ukrainian Math. J., 1988, 40(6), 673–676 http://dx.doi.org/10.1007/BF01057192[Crossref] | Zbl 0679.32012

[17] Dovbush P.V., Bloch functions on complex Banach manifolds, Math. Proc. R. Ir. Acad., 2008, 108(1), 27–32 http://dx.doi.org/10.3318/PRIA.2008.108.1.27[Crossref] | Zbl 1170.32001

[18] Dovbush P.V., On normal and non-normal holomorphic functions on complex Banach manifolds, Ann. Sc. Norm. Super. Pisa Cl. Sci., 2009, 8(1), 1–15 | Zbl 1183.32004

[19] Dovbush P.V., Boundary behaviour of Bloch functions and normal functions, Complex Var. Elliptic Equ., 2010, 55(1–3), 157–166 | Zbl 1192.32006

[20] Dovbush P.V., The Lindelöf principle for holomorphic functions of infinitely many variables, Complex Var. Elliptic Equ., 2011, 56(1–4), 315–323 | Zbl 1256.32002

[21] Dovbush P.V., On the Lindelöf-Gehring-Lohwater theorem, Complex Var. Elliptic Equ., 2011, 56(5), 417–421 http://dx.doi.org/10.1080/17476931003628240[Crossref] | Zbl 1219.32001

[22] Frosini C., Busemann functions and the Julia-Wolff-Carathéodory theorem for polydiscs, Adv. Geom., 2010, 10(3), 435–463 http://dx.doi.org/10.1515/advgeom.2010.016[Crossref][WoS]

[23] Funahashi K., Normal holomorphic mappings and classical theorems of function theory, Nagoya Math. J., 1984, 94, 89–104 | Zbl 0533.32013

[24] Garnett J.B., Marshall D.E., Harmonic Measure, New Math. Monogr., 2, Cambridge University Press, Cambridge, 2008

[25] Gauthier P., A criterion for normalcy, Nagoya Math. J., 1968, 32, 277–282 | Zbl 0157.39802

[26] Gavrilov V.I., Dovbush P.V., Normal functions, Math. Montisnigri, 2001, 14, 5–61 (in Russian)

[27] Gehring F.W., Lohwater A.J., On the Lindelöf theorem, Math. Nachr., 1958, 19, 165–170 http://dx.doi.org/10.1002/mana.19580190111[Crossref] | Zbl 0089.05303

[28] Hahn K.T., Inequality between the Bergman metric and Carathéodory differential metric, Proc. Amer. Math. Soc., 1978, 68(2), 193–194 | Zbl 0376.32020

[29] Hahn K.T., Asymptotic behavior of normal mappings of several complex variables, Canad. J. Math., 1984, 36(4), 718–746 http://dx.doi.org/10.4153/CJM-1984-041-9[Crossref] | Zbl 0564.32015

[30] Hahn K.T., Higher-dimensional generalizations of some classical theorems on normal meromorphic functions, Complex Variables Theory Appl., 1986, 6(2–4), 109–121 http://dx.doi.org/10.1080/17476938608814163[Crossref]

[31] Hahn K.T., Nontangential limit theorems for normal mappings, Pacific J. Math., 1988, 135(1), 57–64 http://dx.doi.org/10.2140/pjm.1988.135.57[Crossref] | Zbl 0618.32004

[32] Järvi P., An extension theorem for normal functions, Proc. Amer. Math. Soc., 1988, 103(4), 1171–1174 http://dx.doi.org/10.2307/2047105[Crossref] | Zbl 0659.32022

[33] Joseph J.E., Kwack M.H., Some classical theorems and families of normal maps in several complex variables, Complex Variables Theory Appl., 1996, 29(4), 343–362 http://dx.doi.org/10.1080/17476939608814902[Crossref]

[34] Kobayashi S., Invariant distances on complex manifolds and holomorphic mappings, J. Math. Soc. Japan, 1967, 19(4), 460–480 http://dx.doi.org/10.2969/jmsj/01940460[Crossref] | Zbl 0158.33201

[35] Korányi A., Harmonic functions on Hermitian hyperbolic space, Trans. Amer. Math. Soc., 1969, 135, 507–516 [Crossref] | Zbl 0174.38801

[36] Krantz S.G., The Lindelöf principle in several complex variables, J. Math. Anal. Appl., 2007, 326(2), 1190–1198 http://dx.doi.org/10.1016/j.jmaa.2006.03.059[Crossref]

[37] Kwack M.H., Families of Normal Maps in Several Variables and Classical Theorems in Complex Analysis, Lecture Notes Ser., 33, Seoul National University, Seoul, 1996

[38] Lehto O., Virtanen K.I., Boundary behaviour and normal meromorphic functions, Acta Math., 1957, 97(1–4), 47–65 http://dx.doi.org/10.1007/BF02392392[Crossref] | Zbl 0077.07702

[39] Lindelöf E., Sur un Principe Général de l’Analyse et ses Applications á la Théorie de la Représentation Conforme, Acta Soc. Sci. Fennicae, 46(4), Suomen Tiedeseura, Helsinki, 1915 | Zbl 45.0665.02

[40] Montel P., Sur les familles de fonctions analytiques qui admettent des valeurs exceptionelles dans un domaine, Ann. Sci. École Norm. Sup., 1912, 29, 487–535 | Zbl 43.0509.05

[41] Pommerenke Ch., Univalent Functions, Studia Mathematica/Mathematische Lehrbücher, 25, Vandenhoeck & Ruprecht, Göttingen, 1975

[42] Sagan H., Space-Filling Curves, Universitext, Springer, New York, 1994 http://dx.doi.org/10.1007/978-1-4612-0871-6[Crossref] | Zbl 0806.01019

[43] Schiff J.L., Normal Families, Universitext, Springer, New York, 1993 http://dx.doi.org/10.1007/978-1-4612-0907-2[Crossref]

[44] Stein E.M., Boundary Behavior of Holomorphic Functions of Several Complex Variables, Math. Notes, 11, Princeton University Press, Princeton, 1972 | Zbl 0242.32005

[45] Whyburn G.T., Analytic Topology, Amer. Math. Soc. Colloq. Publ., 28, American Mathematical Society, New York, 1942

[46] Zaidenberg M.G., Schottky-Landau growth estimates for s-normal families of holomorphic mappings, Math. Ann., 1992, 293(1), 123–141 http://dx.doi.org/10.1007/BF01444708[Crossref] | Zbl 0766.32005

[47] Zavyalov B.I., Drozhzhinov Yu.N., On a multidimensional analogue of Lindelöf’s theorem, Dokl. Akad. Nauk SSSR, 1982, 262(2), 269–270 (in Russian)