Geodesic
The $L^2$ $\bar \partial $-Cauchy problem on weakly $q$-pseudoconvex domains in Stein manifolds
Czechoslovak Mathematical Journal, Tome 65 (2015) no. 3, pp. 739-745 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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Let $X$ be a Stein manifold of complex dimension $n\ge 2$ and $\Omega \Subset X$ be a relatively compact domain with $C^2$ smooth boundary in $X$. Assume that $\Omega $ is a weakly \mbox {$q$-pseudoconvex} domain in $X$. The purpose of this paper is to establish sufficient conditions for the closed range of $\bar \partial $ on $\Omega $. Moreover, we study the \mbox {$\bar \partial $-problem} on $\Omega $. Specifically, we use the modified weight function method to study the weighted \mbox {$\bar \partial $-problem} with exact support in $\Omega $. Our method relies on the \mbox {$L^2$-estimates} by Hörmander (1965) and by Kohn (1973).
Let $X$ be a Stein manifold of complex dimension $n\ge 2$ and $\Omega \Subset X$ be a relatively compact domain with $C^2$ smooth boundary in $X$. Assume that $\Omega $ is a weakly \mbox {$q$-pseudoconvex} domain in $X$. The purpose of this paper is to establish sufficient conditions for the closed range of $\bar \partial $ on $\Omega $. Moreover, we study the \mbox {$\bar \partial $-problem} on $\Omega $. Specifically, we use the modified weight function method to study the weighted \mbox {$\bar \partial $-problem} with exact support in $\Omega $. Our method relies on the \mbox {$L^2$-estimates} by Hörmander (1965) and by Kohn (1973).
DOI : 10.1007/s10587-015-0205-2
Classification : 32F10, 32W05
Keywords: $\bar \partial $ operator; $\bar \partial $-Neumann operator; $q$-convex domain; Stein manifold 
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     title = {The $L^2$ $\bar \partial ${-Cauchy} problem on weakly $q$-pseudoconvex domains in {Stein} manifolds},
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Saber, Sayed. The $L^2$ $\bar \partial $-Cauchy problem on weakly $q$-pseudoconvex domains in Stein manifolds. Czechoslovak Mathematical Journal, Tome 65 (2015) no. 3, pp. 739-745. doi: 10.1007/s10587-015-0205-2

[1] Abdelkader, O., Saber, S.: Solution to $\bar\partial$-equations with exact support on pseudo-convex manifolds. Int. J. Geom. Methods Mod. Phys. 4 (2007), 339-348. | DOI | MR

[2] Cao, J., Shaw, M.-C., Wang, L.: Estimates for the $\bar\partial$-Neumann problem and nonexistence of $C^2$ Levi-flat hypersurfaces in {${\Bbb C} P^n$}. Math. Z. 248 (2004), 183-221 errata dtto 248 223-225 (2004). | MR

[3] Chen, S.-C., Shaw, M.-C.: Partial Differential Equations in Several Complex Variables. AMS/IP Studies in Advanced Mathematics 19 American Mathematical Society, Providence; International Press, Somerville (2001). | MR | Zbl

[4] Demailly, J.-P.: Complex analytic and differential geometry. Preprint (2009) available at http://www-fourier.ujf-grenoble.fr/\char126 demailly/manuscripts/agbook.pdf.

[5] Demailly, J.-P.: Estimations $L^2$ pour l'opérateur $\bar\partial$ d'un fibré vectoriel holomorphe semi-positif au-dessus d'une variété kählérienne complète. Ann. Sci. Éc. Norm. Supér. (4) 15 French (1982), 457-511. | MR

[6] Derridj, M.: Inégalités de Carleman et extension locale des fonctions holomorphes. Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 9 French (1982), 645-669. | MR | Zbl

[7] Derridj, M.: Regularité pour $\bar\partial$ dans quelques domaines faiblement pseudo-convexes. J. Differ. Geom. 13 French (1978), 559-576. | DOI | MR

[8] Harrington, P. S., Raich, A.: Closed range for $\bar\partial$ and $\bar\partial_b$ on bounded hypersurfaces in Stein manifolds. arXiv:1106.0629 (2011). | MR

[9] Ho, L.-H.: $\bar\partial$-problem on weakly $q$-convex domains. Math. Ann. 290 (1991), 3-18. | DOI | MR | Zbl

[10] Hörmander, L.: $L^2$ estimates and existence theorems for the $\bar\partial$ operator. Acta Math. 113 (1965), 89-152. | DOI | MR

[11] Kohn, J. J.: Harmonic integrals on strongly pseudo-convex manifolds. II. Ann. Math. (2) 79 (1964), 450-472. | DOI | MR | Zbl

[12] Kohn, J. J.: Harmonic integrals on strongly pseudo-convex manifolds. I. Ann. Math. (2) 78 (1963), 112-148. | DOI | MR | Zbl

[13] Morrow, J., Kodaira, K.: Complex Manifolds. Athena Series. Selected Topics in Mathematics. Holt, Rinehart and Winston, New York (1971). | MR

[14] Saber, S.: The $\bar\partial$-problem on $q$-pseudoconvex domains with applications. Math. Slovaca 63 (2013), 521-530. | DOI | MR

[15] Saber, S.: Solution to $\bar\partial$ problem with exact support and regularity for the $\bar\partial$-Neumann operator on weakly $q$-convex domains. Int. J. Geom. Methods Mod. Phys. 7 (2010), 135-142. | DOI | MR

[16] Sambou, S.: Résolution du $\bar\partial$ pour les courants prolongeables définis dans un anneau. Ann. Fac. Sci. Toulouse, Math. (6) 11 French (2002), 105-129. | MR

[17] Shaw, M.-C.: Local existence theorems with estimates for $\bar\partial_b$ on weakly pseudo-convex CR manifolds. Math. Ann. 294 (1992), 677-700. | DOI | MR

[18] Zampieri, G.: Complex Analysis and CR Geometry. University Lecture Series 43 American Mathematical Society, Providence (2008). | DOI | MR | Zbl

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