On compactness of the \overline{∂}-Neumann problem and Hankel operators

Abstract

Let Ω = Ω 1 ∖ Ω ¯ 2 \Omega =\Omega _1\setminus \overline {\Omega }_2 , where Ω 1 \Omega _1 and Ω 2 \Omega _2 are two smooth bounded pseudoconvex domains in C n , n ≥ 3 , \mathbb {C}^n, n\geq 3, such that Ω ¯ 2 ⊂ Ω 1 . \overline {\Omega }_2\subset \Omega _1. Assume that the ∂ ¯ \overline {\partial } -Neumann operator of Ω 1 \Omega _1 is compact and the interior of the Levi-flat points in the boundary of Ω 2 \Omega _2 is not empty (in the relative topology). Then we show that the Hankel operator on Ω \Omega with symbol ϕ , H ϕ Ω , \phi , H^{\Omega }_{\phi }, is compact for every ϕ ∈ C ( Ω ¯ ) \phi \in C(\overline {\Omega }) but the ∂ ¯ \overline {\partial } -Neumann operator on Ω \Omega is not compact.