Abstract
Let D be a C (d) q-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, C (k) -estimates, k = 2, 3,...,a, for solutions to the -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 -equation with a support condition in C (k) -spaces. More precisely, we prove that for a -closed form f in C (0,q) (k) (X D,E), 1 <= q <= n - 2, n >= 3, with compact support and for epsilon with 0 < epsilon < 1 there exists a form u in C (0,q-1) (k-epsilon) (X D,E) with compact support such that in . Applications are given for a separation theorem of Andreotti-Vesentini type in C (k) -setting and for the solvability of the -equation for currents.