Hartogs theorem for forms : solvability of Cauchy-Riemann operator at critical degree

Chang, Chin-Huei; Lee, Hsuan-Pei

Chang, Chin-Huei ; Lee, Hsuan-Pei

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 5 (2006) no. 1, pp. 21-37.

Résumé

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 $C^{k}$ $\bar{\partial}$ -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.

MR Zbl

Classification : 32A26, 32W10

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     author = {Chang, Chin-Huei and Lee, Hsuan-Pei},
     title = {Hartogs theorem for forms : solvability of {Cauchy-Riemann} operator at critical degree},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     pages = {21--37},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 5},
     number = {1},
     year = {2006},
     mrnumber = {2240164},
     zbl = {1170.32303},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2006_5_5_1_21_0/}
}

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Chang, Chin-Huei; Lee, Hsuan-Pei. Hartogs theorem for forms : solvability of Cauchy-Riemann operator at critical degree. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 5 (2006) no. 1, pp. 21-37. http://www.numdam.org/item/ASNSP_2006_5_5_1_21_0/

Bibliographie
Cité par

[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 $ℂ^{n}$ $(n \geq 3)$ , Publ. Mat. 43 (1999), 535-570. | MR | Zbl

[2] C. H. Chang and H. P. Lee, Hartogs theorem for CR functions, Bull. Inst. Math. Acad. Sinica 32 (2004), 221-227. | MR | Zbl

[3] J. E. Fornæss, Embedding strictly pseudoconvex domains in convex domains, Amer. J. Math. 98 (1976), 529-569. | MR | Zbl

[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. | JFM | MR

[5] R. Harvey and J. Polking, Fundamental solutions in complex analysis, I, II, Duke Math. J. 46 (1979), 253-340. | MR | Zbl

[6] G. M. Henkin, The Lewy equation and analysis on pseudoconvex manifolds, Russian Math. Surveys 32 (1977), 59-130. | MR | Zbl

[7] L. Hörmander, “Notions of Convexity”, Progress in Math., Vol. 127, Birkhäuser, 1994. | MR | Zbl

[8] S. G. Krantz, “Function Theory of Several Complex Variables”, 2ed. Wadsworth Books/Cole Mathematics Series, 1992. | MR | Zbl

[9] R. Narasimhan, “Several Complex Variables”, Chicago Lecture Notes in Math., The University of Chicago Press, 1971. | MR | Zbl

[10] M. R. Range, “Holomorphic Functions and Integral Representations in Several Complex Variables”, Springer-Verlag, New York, 1986. | MR | Zbl

[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. | Numdam | MR | Zbl

[12] Y. T. Siu, “Techniques of Extension of Analytic Objects”, Lecture Notes in Pure and Applied Math., Vol. 8, Marcel Dekker, 1974. | MR | Zbl

[13] Y. T. Siu and G. Trautmann, “Gap-sheaves and Extension of Coherent Analytic Subsheaves”, Lecture Notes in Mathematics, Vol. 172, Springer-Verlag, 1971. | MR | Zbl

[14] S. M. Webster, On the local solution of the tangential Cauchy-Riemann equations, Nonlinear Anal. 6 (1989), 167-182. | Numdam | MR | Zbl