Let D be a Cd q-convex intersection, d > 2, 0 6 q 6 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, . . . ,1, for solutions to the -equation with small loss of smoothness are obtained for E-valued (0, s)-forms on D when n − q 6 s 6 n. In addition, we solve the -equation with a support condition in Ck-spaces. More precisely, we prove that for a -closed form f in Ck 0,q(X \ D,E), 1 6 q 6 n − 2, n > 3, with compact support and for " with 0 < " < 1 there exists a form u in Ck−ε 0,q−1(X \ D,E) with compact support such that u = f in X \ D. Applications are given for a separation theorem of Andreotti-Vesentini type in Ck-setting and for the solvability of the -equation for currents., Shaban Khidr, Osama Abdelkader., and Seznam literatury