Compactness for the $${\overline{\partial}}$$ -Neumann problem: a functional analysis approach

Abstract

We characterize compactness of the ∂-Neumann operator for a smoothly bounded pseudoconvex domain and in the setting of weighted L 2 -spaces on C n . For this purpose we use a description of relatively compact subsets of L 2 -spaces. We also point out how to use this method to show that property (P) implies compactness for the ∂-Neumann operator on a smoothly bounded pseudoconvex domain.

Compactness of the $$\bar{\partial}$$-Neumann Operator on Weighted (0, q)-forms

“…In this paper we continue the investigations of [12] ans [11] concerning existence and compactness of the canonical solution operator to ∂ on weighted L 2 -spaces over C n . Let ϕ : C n −→ R + be a plurisubharmonic C 2 -weight function and define the space…”

Compactness of the ∂¯-Neumann operator on singular complex spaces

“…As a byproduct, we also obtain the following characterization of compactness of the ∂-Neumann operator on singular spaces with arbitrary singularities (in the spirit of some recent work of Gansberger [13] and Haslinger [15] about compactness of the ∂-Neumann operator on domains in C n ): Theorem 1.3. Let Z be a Hermitian complex space of pure dimension n, X ⊂ Z an open Hermitian submanifold and ∂ a closed L 2 -extension of the ∂ cpt -operator on smooth forms with compact support in X, for example ∂ = ∂ w the ∂-operator in the sense of distributions.…”

“…The criterion is inspired by the work of Gansberger [13] who treats domains in C n . Part of his criterion goes back to an earlier work of Haslinger (see [15]). …”

Sobolev Inequalities and the $$\overline{\partial }$$ ∂ ¯ -Neumann Operator

“…Here we apply a general characterization of compactness of the ∂-Neumann operator N using a description of precompact subsets in L 2 -spaces (see [10]). for each u ∈ dom (∂) ∩ dom (∂ * ).…”