Czechoslovak Mathematical Journal, online first, 9 pp.

$\mathcal C^k$-regularity for the $\bar\partial$-equation with a support condition

Shaban Khidr, Osama Abdelkader

Received January 27, 2016.   First published March 20, 2017.

Shaban Khidr, Mathematics Department, Faculty of Science, University of Jeddah, Asfan St., Jeddah 21589, Saudi Arabia, and Mathematics Department, Faculty of Science, Beni-Suef University, Salah Salem St., Beni-Suef 62511, Egypt, e-mail: skhidr@yahoo.com; Osama Abdelkader, Mathematics Department, Faculty of Science, Minia University, Main Road St., Minia 61915, Egypt, e-mail: usamakader882000@yahoo.com

Abstract: Let $D$ be a $\mathcal{C}^d$ $q$-convex intersection, $d\geq2$, $0\le q\le n-1$, in a complex manifold $X$ of complex dimension $n$, $n\ge2$, and let $E$ be a holomorphic vector bundle of rank $N$ over $X$. In this paper, $\mathcal C^k$-estimates, $k=2, 3, \dots, \infty$, 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\le s\le n$. In addition, we solve the $\bar\partial$-equation with a support condition in $\mathcal C^k$-spaces. More precisely, we prove that for a $\bar\partial$-closed form $f$ in $\mathcal C_{0,q}^k(X\setminus D, E)$, $1\le q\le n-2$, $n\ge3$, with compact support and for $\varepsilon$ with $0<\varepsilon<1$ there exists a form $u$ in $\mathcal C_{0,q-1}^{k-\varepsilon}(X\setminus D, E)$ with compact support such that $\bar{\partial}u=f$ in $X\setminus\ol D$. Applications are given for a separation theorem of Andreotti-Vesentini type in $\mathcal C^k$-setting and for the solvability of the $\bar\partial$-equation for currents.

Keywords: $\bar\partial$-equation; $q$-convexity; $\mathcal C^k$-estimate

Classification (MSC 2010): 32F10, 32W05

DOI: 10.21136/CMJ.2017.0039-16

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