Czechoslovak Mathematical Journal, Vol. 67, No. 2, pp. 515-523, 2017

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

Shaban Khidr, Osama Abdelkader

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

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 $X\setminus\overline D$. 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:  32F10, 32W05
DOI:  10.21136/CMJ.2017.0039-16

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Affiliations:   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:; Osama Abdelkader, Mathematics Department, Faculty of Science, Minia University, Main Road St., Minia 61915, Egypt, e-mail:

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