Abstract
Let Ω be a weakly q-convex domain in ℂn. We establish the L2 existence theorem for the
$\overline{\partial}$
-Neumann operator N when the boundary of Ω is C1. Using this result, we study the
$\overline{\partial}$
problem with exact support on such domains. Furthermore, there exists a number ℓ0 > 0 such that the operators N,
$\overline{\partial}^{\star}N$
and the Bergman projection are regular in the Sobolev space Wℓ(Ω) for ℓ < ℓ0 when the boundary of Ω is C∞.