Czechoslovak Mathematical Journal

, Volume 67, Issue 2, pp 515–523

Ck-regularity for the \(\bar \partial \) -equation with a support condition

Article

Abstract

Let D be a Cdq-convex intersection, d ≥ 2, 0 ≤ qn − 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, Ck-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 nqsn. In addition, we solve the \(\bar \partial \) -equation with a support condition in Ck-spaces. More precisely, we prove that for a \(\bar \partial \) -closed form f in C0,qk(XD,E), 1 ≤ qn − 2, n ≥ 3, with compact support and for ε with 0 < ε < 1 there exists a form u in C0,q−1kε(XD,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 Ck-setting and for the solvability of the \(\bar \partial \) -equation for currents.

Keywords

\(\bar \partial \) -equation q-convexity Ck-estimate 

MSC 2010

32F10 32W05 

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Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2017

Authors and Affiliations

  1. 1.Mathematics Department, Faculty of ScienceUniversity of JeddahJeddahSaudi Arabia
  2. 2.Mathematics Department, Faculty of ScienceBeni-Suef UniversityBeni-SuefEgypt
  3. 3.Mathematics Department, Faculty of ScienceMinia UniversityMiniaEgypt

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