Premium
Compactness of the Canonical Solution Operator of $ \bar \partial $ on Bounded Pseudoconvex Domains
Author(s) -
Knirsch Wolfgang
Publication year - 2002
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/1522-2616(200211)245:1<94::aid-mana94>3.0.co;2-h
Subject(s) - mathematics , compact space , bounded function , holomorphic function , operator (biology) , compact operator , bounded operator , pure mathematics , finite rank operator , mathematical analysis , banach space , computer science , chemistry , repressor , gene , programming language , biochemistry , transcription factor , extension (predicate logic)
On bounded pseudoconvex domains Ω the orthogonal projection P q : L 2 ( p , q ) (Ω) → ker $ \bar \partial $ q is given by P q = Id – S q +1 $ \bar \partial $ q = Id – $ \bar \partial $ * q +1 N q +1 $ \bar \partial $ q , where S q is the canonical solution operator of the $ \bar \partial $ ‐equation and N q is the $ \bar \partial $ ‐Neumann operator. We prove a formula for the solution operator S q restricted on (0, q )‐forms with holomorphic coefficients. And as an application we get a characterization of compactness of the solution operator restricted on (0, q )‐forms with holomorphic coefficients. On general (0, q )‐forms we show that this condition is necessary for compactness of the solution operator.