A nullset for normal functions in several variables. (English) Zbl 0755.32006
The main result in this paper is the following: Let \(E\) be a closed subset of a domain \(\Omega\) in \(\mathbb{C}^ n\), \(f:\Omega/E\to C^*\) be meromorphic. If \(f\) is normal and \(E\) is an analytic subvariety or of locally finite \((2n-2)\)-dimensional Hausdorff measure in \(\Omega\) satisfying certain geometric conditions then \(f\) can be extended to a meromorphic function \(f^*:\Omega\to C^*\).
In the case of a sub-variety, a sufficient geometric condition is that the singularities of \(E\) are normal crossings.
Two examples, which give natural exceptional sets in the considering setting, of meromorphic functions with values in \(C^*\) which satisfy the specified conditions, but which are such that the extension result cannot be applied, are discussed. Incidentally a new proof for the following Theorem (of Parreau): “\(f\) be in the Nevanlinna class and \(E\) be polar in \(R^ 2\) then \(f\) has meromorphic extension \(f^*\) to \(\Omega\)” is given.
In the case of a sub-variety, a sufficient geometric condition is that the singularities of \(E\) are normal crossings.
Two examples, which give natural exceptional sets in the considering setting, of meromorphic functions with values in \(C^*\) which satisfy the specified conditions, but which are such that the extension result cannot be applied, are discussed. Incidentally a new proof for the following Theorem (of Parreau): “\(f\) be in the Nevanlinna class and \(E\) be polar in \(R^ 2\) then \(f\) has meromorphic extension \(f^*\) to \(\Omega\)” is given.
Reviewer: I.H.Nagaraja Rao (Waltair)
MSC:
| 32A20 | Meromorphic functions of several complex variables |
| 32A17 | Special families of functions of several complex variables |
| 32H25 | Picard-type theorems and generalizations for several complex variables |
| 30D45 | Normal functions of one complex variable, normal families |
References:
| [1] | Lars V. Ahlfors, Conformal invariants: topics in geometric function theory, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. McGraw-Hill Series in Higher Mathematics. · Zbl 0272.30012 |
| [2] | C. Carathéodory, Theory of functions, vol. 2, Chelsea, New York, 1960. · JFM 54.0359.01 |
| [3] | Urban Cegrell, Removable singularity sets for analytic functions having modulus with bounded Laplace mass, Proc. Amer. Math. Soc. 88 (1983), no. 2, 283 – 286. · Zbl 0514.32011 |
| [4] | Joseph A. Cima and Ian R. Graham, On the extension of holomorphic functions with growth conditions across analytic subvarieties, Michigan Math. J. 28 (1981), no. 2, 241 – 256. · Zbl 0438.32008 |
| [5] | Joseph A. Cima and Ian Graham, Removable singularities for Bloch and BMO functions, Illinois J. Math. 27 (1983), no. 4, 691 – 703. · Zbl 0522.32009 |
| [6] | Joseph A. Cima and Steven G. Krantz, The Lindelöf principle and normal functions of several complex variables, Duke Math. J. 50 (1983), no. 1, 303 – 328. · Zbl 0522.32003 · doi:10.1215/S0012-7094-83-05014-7 |
| [7] | Roy O. Davies and Henryk Fast, Lebesgue density influences Hausdorff measure; large sets surface-like from many directions, Mathematika 25 (1978), no. 1, 116 – 119. · Zbl 0378.28012 · doi:10.1112/S0025579300009335 |
| [8] | H. Federer, Geometric measure theory, Springer-Verlag, Berlin, 1969. · Zbl 0176.00801 |
| [9] | Dennis A. Hejhal, Linear extremal problems for analytic functions, Acta. Math. 128 (1972), no. 1-2, 91 – 122. · Zbl 0226.30019 · doi:10.1007/BF02392161 |
| [10] | M. Hervé, Several complex variables, 2nd ed., Oxford University Press, New Delhi, 1987. · Zbl 0646.32001 |
| [11] | Jaakko Hyvönen and Juhani Riihentaus, On the extension in the Hardy classes and in the Nevanlinna class, Bull. Soc. Math. France 112 (1984), no. 4, 469 – 480 (English, with French summary). · Zbl 0587.32011 |
| [12] | Jaakko Hyvönen and Juhani Riihentaus, Removable singularities for holomorphic functions with locally finite Riesz mass, J. London Math. Soc. (2) 35 (1987), no. 2, 296 – 302. · Zbl 0622.32014 · doi:10.1112/jlms/s2-35.2.296 |
| [13] | Pentti Järvi, An extension theorem for normal functions, Proc. Amer. Math. Soc. 103 (1988), no. 4, 1171 – 1174. · Zbl 0659.32022 |
| [14] | Peter Kiernan, Extensions of holomorphic maps, Trans. Amer. Math. Soc. 172 (1972), 347 – 355. · Zbl 0255.32014 |
| [15] | Olli Lehto and K. I. Virtanen, Boundary behaviour and normal meromorphic functions, Acta Math. 97 (1957), 47 – 65. · Zbl 0077.07702 · doi:10.1007/BF02392392 |
| [16] | Kiyoshi Noshiro, Cluster sets, Ergebnisse der Mathematik und ihrer Grenzgebiete. N. F., Heft 28, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1960. · Zbl 0090.28801 |
| [17] | M. Parreau, Sur les moyennes des fonctions harmoniques et analytiques et la classification des surfaces de Riemann, Ann. Inst. Fourier Grenoble 3 (1951), 103 – 197 (1952) (French). · Zbl 0047.32004 |
| [18] | Juhani Riihentaus, Removable singularities of analytic functions of several complex variables, Math. Z. 158 (1978), no. 1, 45 – 54. · Zbl 0351.32014 · doi:10.1007/BF01214564 |
| [19] | Juhani Riihentaus, Removable singularities in the Nevanlinna class and in the Hardy classes, Proc. Amer. Math. Soc. 102 (1988), no. 3, 546 – 550. · Zbl 0647.32016 |
| [20] | Donald Sarason, Function theory on the unit circle, Virginia Polytechnic Institute and State University, Department of Mathematics, Blacksburg, Va., 1978. Notes for lectures given at a Conference at Virginia Polytechnic Institute and State University, Blacksburg, Va., June 19 – 23, 1978. · Zbl 0398.30027 |
| [21] | Bernard Shiffman, On the removal of singularities of analytic sets, Michigan Math. J. 15 (1968), 111 – 120. · Zbl 0165.40503 |
| [22] | J. Väisälä, Lectures on \( n\)-dimensional quasiconformal mappings, Springer-Verlag, Berlin, 1971. · Zbl 0221.30031 |
| [23] | S. A. Vinogradov and V. P. Havin, Free interpolation in \( {{\text{H}}^\infty }\) and in other certain classes of functions, I, J. Soviet Math. 9 (1978), 137-171. · Zbl 0398.41002 |
| [24] | Shinji Yamashita, Functions of uniformly bounded characteristic on Riemann surfaces, Trans. Amer. Math. Soc. 288 (1985), no. 1, 395 – 412. · Zbl 0563.30029 |
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