Compactness of the Complex Green Operator

TL;DR

This paper establishes that the compactness of the complex Green operator on boundary forms implies the compactness of the $ar{ ext{d}}$-Neumann operator inside the domain, providing new conditions for compactness in complex analysis.

Contribution

It proves that boundary compactness of the Green operator implies interior compactness and introduces new boundary conditions ensuring the Green operator's compactness.

Findings

Compactness of $G_q$ on boundary implies compactness of $N_q$ inside the domain.

Under $(P_q)$ and $(P_{n-q-1})$, $G_q$ is compact for certain $q$.

A new boundary representation formula and an auxiliary compactness result are developed.

Abstract

Let $\Omega\subset\C^n$ be a bounded smooth pseudoconvex domain. We show that compactness of the complex Green operator $G_{q}$ on $(0,q)$-forms on $b\Omega$ implies compactness of the $\bar{\partial}$-Neumann operator $N_{q}$ on $\Omega$. We prove that if $1 \leq q \leq n-2$ and $b\Omega$ satisfies $(P_q)$ and $(P_{n-q-1})$, then $G_{q}$ is a compact operator (and so is $G_{n-1-q}$). Our method relies on a jump type formula to represent forms on the boundary, and we prove an auxiliary compactness result for an `annulus' between two pseudoconvex domains. Our results, combined with the known characterization of compactness in the $\bar{\partial}$-Neumann problem on locally convexifiable domains, yield the corresponding characterization of compactness of the complex Green operator(s) on these domains.