English

Abstract

Let D be a Cdq-convex intersection, d ≥ 2, 0≤ q ≤ n − 1, in a complex manifold X of complex dimension n, n ≥ 2, and let E be a holomorphic vector bundle of rank N over X. In this paper, Ck-estimates, k = 2, 3,...,∞, 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 ≤ s ≤ n. In addition, we solve the \(\bar \partial \)-equation with a support condition in Ck-spaces. More precisely, we prove that for a \(\bar \partial \)-closed form f in \(C_{0{,_q}}^k\left( {X\backslash D,E} \right),{\kern 1pt} 1 \leqslant q \leqslant n - 2,{\kern 1pt} n \geqslant 3\), with compact support and for e with 0 < e < 1 there exists a form u in \(C_{0{,_{q - 1}}}^k\left( {X\backslash D,E} \right)\) with compact support such that \(\bar \partial u = f{\kern 1pt} in{\kern 1pt} X\backslash \bar D\). Applications are given for a separation theorem of Andreotti-Vesentini type in Ck-setting and for the solvability of the \(\bar \partial \)-equation for currents.