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Keywords:
$\bar \partial $-equation; $q$-convexity; $\mathcal C^k$-estimate
Summary:
Let $D$ be a $\mathcal {C}^d$ $q$-convex intersection, $d\geq 2$, $0\le q\le n-1$, 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, \dots , \infty $, for solutions to the \hbox {$\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 $\mathcal C_{0,q}^{k}(X\setminus D, E)$, $1\le q\le n-2$, $n\ge 3$, 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.
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