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

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...