Compactness of the Complex Green Operator
| dc.creator | Raich, Andrew S. | |
| dc.creator | Straube, Emil J. | |
| dc.date | 2007-06-18 | |
| dc.date | 2007-10-18 | |
| dc.date.accessioned | 2026-07-07T12:54:56Z | |
| dc.date.available | 2026-07-07T12:54:56Z | |
| dc.description | Let $Ω\subset\C^n$ be a bounded smooth pseudoconvex domain. We show that compactness of the complex Green operator $G_{q}$ on $(0,q)$-forms on $bΩ$ implies compactness of the $\bar{\partial}$-Neumann operator $N_{q}$ on $Ω$. We prove that if $1 \leq q \leq n-2$ and $bΩ$ 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. | |
| dc.description | 17 pages. We added an appendix, fixed the proof of a main theorem, and revised the statement of another theorem. Also, we fixed some other typos | |
| dc.identifier | https://arxiv.org/abs/0706.2645 | |
| dc.identifier | http://arxiv.org/abs/0706.2645 | |
| dc.identifier | Math. Res. Lett. 15 (2008), no. 4, 761--778 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/224107 | |
| dc.subject | Complex Variables | |
| dc.subject | Analysis of PDEs | |
| dc.subject | 32W10, 32W05, 35N15 | |
| dc.title | Compactness of the Complex Green Operator | |
| dc.type | text |