Browse > Article
http://dx.doi.org/10.4134/CKMS.c170022

SOLUTION TO ${\bar{\partial}}$-PROBLEM WITH SUPPORT CONDITIONS IN WEAKLY q-CONVEX DOMAINS  

Saber, Sayed (Mathematics Department Faculty of Science Beni-Suef University)
Publication Information
Communications of the Korean Mathematical Society / v.33, no.2, 2018 , pp. 409-421 More about this Journal
Abstract
Let X be a complex manifold of dimension n $n{\geqslant}2$ and let ${\Omega}{\Subset}X$ be a weakly q-convex domain with smooth boundary. Assume that E is a holomorphic line bundle over X and $E^{{\otimes}m}$ is the m-times tensor product of E for positive integer m. If there exists a strongly plurisubharmonic function on a neighborhood of $b{\Omega}$, then we solve the ${\bar{\partial}}$-problem with support condition in ${\Omega}$ for forms of type (r, s), $s{\geqslant}q$ with values in $E^{{\otimes}m}$. Moreover, the solvability of the ${\bar{\partial}}_b$-problem on boundaries of weakly q-convex domains with smooth boundary in $K{\ddot{a}}hler$ manifolds are given. Furthermore, we shall establish an extension theorem for the ${\bar{\partial}}_b$-closed forms.
Keywords
$\bar{\partial}$ and $\bar{\partial}$-Neumann operators; pseudoconvex domains; line bundle;
Citations & Related Records
연도 인용수 순위
  • Reference
1 A. Andreotti and E. Vesentini, Sopra un teorema di Kodaira, Ann. Scuola Norm. Sup. Pisa (3) 15 (1961), 283-309.
2 J. Cao, M.-C. Shaw, and L. Wang, Estimates for the $\partial$-Neumann problem and nonexistence of $C^2$ Levi-flat hypersurfaces in $P^n$, Math. Z. 248 (2004), no. 1, 183-221.   DOI
3 M. Derridj, Regularitepour $\partial$ dans quelques domaines faiblement pseudo-convexes, J. Differential Geom. 13 (1978), no. 4, 559-576.   DOI
4 G. B. Folland and J. J. Kohn, The Neumann Problem for the Cauchy-Riemann Complex, Princeton University Press, Princeton, NJ, 1972.
5 P. A. Griffths, The extension problem in complex analysis. II. Embeddings with positive normal bundle, Amer. J. Math. 88 (1966), 366-446.   DOI
6 L.-H. Ho, $\partial$-problem on weakly q-convex domains, Math. Ann. 290 (1991), no. 1, 3-18.   DOI
7 L. Hormander, $L^2$ estimates and existence theorems for the $\partial$ operator, Acta Math. 113 (1965), 89-152.   DOI
8 K. Kodaira, On Kahler varieties of restricted type (an intrinsic characterization of algebraic varieties), Ann. of Math. (2) 60 (1954), 28-48.   DOI
9 T. Ohsawa, Pseudoconvex domains in $P^n$: a question on the 1-convex boundary points, in Analysis and geometry in several complex variables (Katata, 1997), 239-252, Trends Math, Birkhauser Boston, Boston, MA, 1997.
10 S. Saber, Solution to $\partial$ problem with exact support and regularity for the $\partial$-Neumann operator on weakly q-pseudoconvex domains, Inter. J. of Geometric Methods in Modern Physics 7 (2010), no. 1, 135-142.   DOI
11 S. Saber, The $L^2$ $\partial$-Cauchy problem on weakly q-pseudoconvex domains in Stein mani- folds, Czechoslovak Math. J. 65(140) (2015), no. 3, 739-745.   DOI
12 S. Saber, The $L^2$ $\partial$-cauchy problem on pseudoconvex domains and applications, Asian- European J. Math. 11 (2018), no. 1, 1850025, 8 pages.   DOI
13 S. Sambou, Resolution du $\partial$ pour les courants prolongeables definis dans un anneau, Ann. Fac. Sci. Toulouse Math. (6) 11 (2002), no. 1, 105-129.   DOI
14 E. Vesentini, Lectures on Levi convexity of complex manifolds and cohomology vanishing theorems, Notes by M. S. Raghunathan. Tata Institute of Fundamental Research Lectures on Mathematics, No. 39, Tata Institute of Fundamental Research, Bombay, 1967.
15 M.-C. Shaw, Local existence theorems with estimates for ${\partial}_b$ on weakly pseudo-convex CR manifolds, Math. Ann. 294 (1992), no. 4, 677-700.   DOI
16 K. Takegoshi, Representation theorems of cohomology on weakly 1-complete manifolds, Publ. Res. Inst. Math. Sci. 18 (1982), no. 2, 551-606.   DOI
17 K. Takegoshi, Global regularity and spectra of Laplace-Beltrami operators on pseudoconvex domains, Publ. Res. Inst. Math. Sci. 19 (1983), no. 1, 275-304.   DOI