Compactness of the canonical solution operator on Lipschitz q-pseudoconvex boundaries

Abstract

Let Ω ⊂ ℂn be a bounded Lipschitz q-pseudoconvex domain that admit good weight functions. We shall prove that the canonical solution operator for the the ∂¯-equation is compact on the boundary of Ω and is bounded in the Sobolev space Wkr,s(Ω) for some values of k. Moreover, we show that the Bergman projection and the ∂¯-Neumann operator are bounded in the Sobolev space Wkr,s(Ω) for some values of k. If Ω is smooth, we shall give sufficient conditions for compactness of the ∂¯-Neumann operator.