Ck-regularity for the $$\bar \partial $$
-equation with a support condition |
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Authors: | Shaban Khidr Osama Abdelkader |
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Institution: | 1.Mathematics Department, Faculty of Science,University of Jeddah,Jeddah,Saudi Arabia;2.Mathematics Department, Faculty of Science,Beni-Suef University,Beni-Suef,Egypt;3.Mathematics Department, Faculty of Science,Minia University,Minia,Egypt |
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Abstract: | Let D be a C d q-convex intersection, d ≥ 2, 0 ≤ q ≤ n ? 1, in a complex manifold X of complex dimension n, n ≥ 2, and let E be a holomorphic vector bundle of rank N over X. In this paper, C k -estimates, k = 2, 3,...,∞, for solutions to the \(\bar \partial \)-equation with small loss of smoothness are obtained for E-valued (0, s)-forms on D when n ? q ≤ s ≤ n. In addition, we solve the \(\bar \partial \)-equation with a support condition in C k -spaces. More precisely, we prove that for a \(\bar \partial \)-closed form f in C 0,q k (X D,E), 1 ≤ q ≤ n ? 2, n ≥ 3, with compact support and for ε with 0 < ε < 1 there exists a form u in C 0,q?1 k?ε (X D,E) with compact support such that \(\bar \partial u = f\) in \(X\backslash \bar D\). Applications are given for a separation theorem of Andreotti-Vesentini type in C k -setting and for the solvability of the \(\bar \partial \)-equation for currents. |
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