### ${\mathcal{C}}^{k}$-regularity for the $\overline{\partial}$-equation with a support condition

Shaban Khidr, Osama Abdelkader (2017)

Czechoslovak Mathematical Journal

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Let $D$ be a ${\mathcal{C}}^{d}$ $q$-convex intersection, $d\ge 2$, $0\le q\le n-1$, in a complex manifold $X$ of complex dimension $n$, $n\ge 2$, and let $E$ be a holomorphic vector bundle of rank $N$ over $X$. In this paper, ${\mathcal{C}}^{k}$-estimates, $k=2,3,\cdots ,\infty $, for solutions to the $\overline{\partial}$-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 $\overline{\partial}$-equation with a support condition in ${\mathcal{C}}^{k}$-spaces. More precisely, we prove that for a $\overline{\partial}$-closed form $f$ in ${\mathcal{C}}_{0,q}^{k}(X\setminus D,E)$, $1\le q\le n-2$, $n\ge 3$, with compact support and for $\epsilon $ with $0<\epsilon <1$ there...