Hartogs type theorems for CR \(L^{2}\) functions on coverings of strongly pseudoconvex manifolds. (English) Zbl 1142.32004
Let \(M'@>{\pi}>>M\) be a holomorphic covering of a pseudoconvex complex manifold \(M\). Let \(D=\pi^{-1}(\widetilde{D})\), for a \(\widetilde{D}\Subset{M}\).
The author proves an analog of Bochner’s extension theorem, showing that, under suitable assumptions, the \(L^2\) CR functions defined on the boundary \(\partial{D}\) extend holomorphically to \(D\). The CR functions \(f\) on \(\partial{D}\) to be extended are required to be locally Lipschitz continuous and globally \(L^2\) on \(\partial{D}\).
Reviewer: Mauro Nacinovich (Roma)
MSC:
| 32A40 | Boundary behavior of holomorphic functions of several complex variables |
| 32V25 | Extension of functions and other analytic objects from CR manifolds |
References:
| [1] | S. Bochner, Analytic and meromorphic continuation by means of Green’s formula , Ann. of Math., 44 (1943), 652–673. JSTOR: · Zbl 0060.24206 · doi:10.2307/1969103 |
| [2] | A. Brudnyi, Representation of holomorphic functions on coverings of pseudoconvex domains in Stein manifolds via integral formulas on these domains , J. Funct. Anal., 231 (2006), 418–437. · Zbl 1090.32001 · doi:10.1016/j.jfa.2005.06.004 |
| [3] | A. Brudnyi, Holomorphic functions of slow growth on coverings of pseudoconvex domains in Stein manifolds , Compositio Math., 142 (2006), 1018–1038. · Zbl 1120.32020 · doi:10.1112/S0010437X06002156 |
| [4] | A. Brudnyi, Hartogs type theorems on coverings of Stein manifolds , Internat. J. Math., 17 (2006), no. 3, 339–349. · Zbl 1092.32022 · doi:10.1142/S0129167X06003515 |
| [5] | A. Brudnyi, On holomorphic \(L^2\)-functions on coverings of strongly pseudoconvex manifolds , Publications of RIMS, Kyoto University, 43 (2007), no. 4, 963–976. · Zbl 1157.32028 · doi:10.2977/prims/1201012376 |
| [6] | H. Cartan, Sur les fonctions de plusieurs variables complexes. Les espaces analytiques , Proc. Intern. Congress Mathematicians Edinbourgh 1958, Cambridge Univ. Press, 1960, pp. 33–52. · Zbl 0117.04602 |
| [7] | J.-P. Demailly, Estimations \(L^2\) pour l’opérateur \(\overline\partial\) d’un fibré vectoriel holomorphe semi-positif au-dessus d’une variété kahlérienne complète , Ann. Sci. Ecole Norm. Sup. (4), 15 (3) (1982), 457–511. · Zbl 0507.32021 |
| [8] | H. Federer, Geometric measure theory, Springer-Verlag, New York, 1969. · Zbl 0176.00801 |
| [9] | M. Gromov, G. Henkin and M. Shubin, Holomorphic \(L^2\) functions on coverings of pseudoconvex manifolds , GAFA, Vol. 8 (1998), 552–585. · Zbl 0926.32011 · doi:10.1007/s000390050066 |
| [10] | C. Grant and P. Milman, Metrics for singular analytic spaces , Pacific J. Math., 168 (1995), no. 1, 61–156. · Zbl 0822.32004 · doi:10.2140/pjm.1995.168.61 |
| [11] | H. Grauert and R. Remmert, Theorie der Steinschen Räume, Springer-Verlag, Berlin, 1977. |
| [12] | R. Harvey and H. B. Lawson, On boundaries of complex analytic varieties, I , Ann. of Math. (2), 102 (1975), no. 2, 223–290. JSTOR: · Zbl 0317.32017 · doi:10.2307/1971032 |
| [13] | G. Henkin, The method of integral representations in complex analysis , Several complex variables, I, Introduction to complex analysis, A translation of Sovremennye problemy matematiki. Fundamentalńye napravleniya, Tom 7, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekn. Inform., Moscow 1985. Encyclopaedia of Mathematical Sciences, 7, Springer-Verlag, Berlin, 1990. |
| [14] | J. J. Kohn and H. Rossi, On the extension of holomorphic functions from the boundary of a complex manifold , Ann. of Math. (2), 81 (1965), 451–472. JSTOR: · Zbl 0166.33802 · doi:10.2307/1970624 |
| [15] | F. Lárusson, An extension theorem for holomorphic functions of slow growth on covering spaces of projective manifolds , J. Geom. Anal., 5 (1995), no. 2, 281–291. · Zbl 0827.32011 · doi:10.1007/BF02921678 |
| [16] | E. McShane, Extension of range functions , Bull. Amer. Math. Soc., 40 (1934), no. 12, 837–842. · Zbl 0010.34606 · doi:10.1090/S0002-9904-1934-05978-0 |
| [17] | R. Narasimhan, Imbedding of holomorphically complete complex spaces , Amer. J. Math., 82 (1960), no. 4, 917–934. JSTOR: · Zbl 0104.05402 · doi:10.2307/2372949 |
| [18] | T. Ohsawa, Complete Kähler manifolds and function theory of several complex variables , Sugaku Expositions, 1 (1) (1988), 75–93. · Zbl 0692.32001 |
| [19] | R. Remmert, Sur les espaces analytiques holomorphiquement séparables et holomorphiquement convexes , C. R. Acad. Sci. Paris, 243 (1956), 118–121. · Zbl 0070.30401 |
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