Electron. J. Diff. Eqns., Vol. 2004(2004), No. 73, pp. 1-9.

Solutions to $\bar{\partial}$-equations on strongly pseudo-convex domains with $L^p$-estimates

Osama Abdelkader & Shaban Khidr

We construct a solution to the $\bar{\partial}$-equation on a strongly pseudo-convex domain of a complex manifold. This is done for forms of type $(0,s)$, $s\geq 1 $, with values in a holomorphic vector bundle which is Nakano positive and for complex valued forms of type $(r,s)$, $1\leq r\leq n$, when the complex manifold is a Stein manifold. Using Kerzman's techniques, we find the $L^p$-estimates, $1\leq p\leq \infty$, for the solution.

Submitted March 01, 2004. Published May 20, 2004.
Math Subject Classifications: 32F27, 32C35, 35N15.
Key Words: L^p-estimates, $\bar{\partial}$-equation, strongly pseudo-convex, smooth boundary, complex manifolds.

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Osama Abdelkader
Mathematics Department, Faculty of Science
Minia University, El-Minia, Egypt
email: usamakader882000@yahoo.com
Shaban Khidr
Mathematics Department, Faculty of Science
Cairo University, Beni- Suef, Egypt
email: skhidr@yahoo.com

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