On compactness of the dbar-Neumann problem and Hankel operators

Abstract

Let $\D=\D_1\setminus \Dc_2$, where $\D_1$ and $\D_2$ are two smooth bounded pseudoconvex domains in $\C^n, n\geq 3,$ such that $\Dc_2\subset \D_1.$ Assume that the $\dbar$-Neumann operator of $\D_1$ is compact and the interior of the Levi-flat points in the boundary of $\D_2$ is not empty (in the relative topology). Then we show that the Hankel operator on $\D$ with symbol $ϕ, H^{\D}_ϕ,$ is compact for every $ϕ\in C(\Dc)$ but the $\dbar$-Neumann operator on $\D$ is not compact.

8 pages, to appear in Proc. Amer. Math. Soc

References in corpus (2)

• Compactness of Hankel operators and analytic discs in the boundary of pseudoconvex domains
• Analytic discs, plurisubharmonic hulls, and non-compactness of the d-bar-Neumann operator

Cited by in corpus (3)

• Compactness of the dbar-Neumann operator and commutators of the Bergman projection with continuous functions
• On compactness of Hankel and the $\bar{\partial }$-Neumann operators on Hartogs domains in ${\mathbb{C}}^2$
• Localization of compactness of Hankel operators on pseudoconvex domains