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Hartogs theorem for forms: solvability of Cauchy-Riemann operator at critical degree. (English) Zbl 1170.32303
Summary: The Hartogs theorem for holomorphic functions is generalized in two settings: a CR version (Theorem 1.2) and a corresponding theorem based on it for \(C^k\) \(\overline\partial\)-closed forms at the critical degree, \(0 \leq k\leq\infty\) (Theorem 1.1). Part of Frenkel’s lemma in \(C^k\) category is also proved.
32A26 Integral representations, constructed kernels (e.g., Cauchy, Fantappiè-type kernels)
32W10 \(\overline\partial_b\) and \(\overline\partial_b\)-Neumann operators
Full Text: EuDML
[1] C. H. Chang and H. P. Lee, Semi-global solution of \({\bar{\partial }_b}\) with \(L^p\) \((1\leq p \leq \infty )\) bounds on strongly pseudoconvex real hypersurfaces in \({{\mathbb{C}}^{n}}\) \((n\geq 3)\), Publ. Mat. 43 (1999), 535-570. Zbl0954.32027 MR1744621 · Zbl 0954.32027 · doi:10.5565/PUBLMAT_43299_05 · eudml:41602
[2] C. H. Chang and H. P. Lee, Hartogs theorem for CR functions, Bull. Inst. Math. Acad. Sinica 32 (2004), 221-227. Zbl1072.32025 MR2103960 · Zbl 1072.32025
[3] J. E. Fornæss, Embedding strictly pseudoconvex domains in convex domains, Amer. J. Math. 98 (1976), 529-569. Zbl0334.32020 MR422683 · Zbl 0334.32020 · doi:10.2307/2373900
[4] F. Hartogs, Zur Theorie der analytischen Funktionen mehrerer unabhängiger Veränderlichen, insbesondere über die Darstellung derselben durch Reihen, welche nach Potenzen einer Veränderlichen fortschreiten, Math. Ann. 62 (1906), 1-88. MR1511365 JFM37.0444.01 · JFM 36.0483.02
[5] R. Harvey and J. Polking, Fundamental solutions in complex analysis, I, II, Duke Math. J. 46 (1979), 253-340. Zbl0441.35043 MR534054 · Zbl 0441.35043 · doi:10.1215/S0012-7094-79-04613-1
[6] G. M. Henkin, The Lewy equation and analysis on pseudoconvex manifolds, Russian Math. Surveys 32 (1977), 59-130. Zbl0382.35038 MR454067 · Zbl 0382.35038 · doi:10.1070/RM1977v032n03ABEH001628
[7] L. Hörmander, “Notions of Convexity”, Progress in Math., Vol. 127, Birkhäuser, 1994. Zbl0835.32001 MR1301332 · Zbl 0835.32001
[8] S. G. Krantz, “Function Theory of Several Complex Variables”, 2ed. Wadsworth Books/Cole Mathematics Series, 1992. Zbl0471.32008 MR1162310 · Zbl 0471.32008
[9] R. Narasimhan, “Several Complex Variables”, Chicago Lecture Notes in Math., The University of Chicago Press, 1971. Zbl0223.32001 MR342725 · Zbl 0223.32001
[10] M. R. Range, “Holomorphic Functions and Integral Representations in Several Complex Variables”, Springer-Verlag, New York, 1986. Zbl0591.32002 MR847923 · Zbl 0591.32002
[11] J. P. Rosay, Some application of Cauchy-Fantappié forms to (local) problems in \({\bar{\partial }_b}\), Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 13 (1986), 225-243. Zbl0633.32007 MR876123 · Zbl 0633.32007 · numdam:ASNSP_1986_4_13_2_225_0 · eudml:83978
[12] Y. T. Siu, “Techniques of Extension of Analytic Objects”, Lecture Notes in Pure and Applied Math., Vol. 8, Marcel Dekker, 1974. Zbl0294.32007 MR361154 · Zbl 0294.32007
[13] Y. T. Siu and G. Trautmann, “Gap-sheaves and Extension of Coherent Analytic Subsheaves”, Lecture Notes in Mathematics, Vol. 172, Springer-Verlag, 1971. Zbl0208.10403 MR287033 · Zbl 0208.10403
[14] S. M. Webster, On the local solution of the tangential Cauchy-Riemann equations, Nonlinear Anal. 6 (1989), 167-182. Zbl0679.32019 MR995503 · Zbl 0679.32019 · numdam:AIHPC_1989__6_3_167_0 · eudml:78173
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