Shaw, Mei-Chi \(L^2\) estimates and existence theorems for \(\bar\partial_b\) on Lipschitz boundaries. (English) Zbl 1039.32049 Math. Z. 244, No. 1, 91-123 (2003). The purpose of this paper is to study the \(\overline \partial_b\) complex when the boundary of the domain is only Lipschitz. In the first part square-integrable \((p,q)\)- forms and the \(\overline \partial_b\) complex on the boundary of a Lipschitz domain are defined. The most important tools are the Bochner-Martinelli-Koppelman transform on the boundary of a Lipschitz domain and the \(\overline \partial \) Cauchy problems on Lipschitz domains.The main result is the following: if \(D\) is a domain having a Lipschitz plurisubharmonic defining function, the equation \(\overline \partial_b u=\alpha \) has a \(L^2\) solution on the boundary \(bD\) if and only if for any \(\overline \partial \)-closed smooth \((n-p,n-q-1)\)-form \(\phi\) in a neighborhood of \(bD\), \(\int_{bD}\alpha \wedge \phi =0, \;1\leq q \leq n-1.\) This result is used to develop a Hodge theory for \(\overline \partial_b\) and to show that the \(\overline \partial_b\) operator has closed range in \(L^2_{(p,q)}(bD).\) Reviewer: Fritz Haslinger (Wien) Cited in 7 Documents MSC: 32W10 \(\overline\partial_b\) and \(\overline\partial_b\)-Neumann operators 32W05 \(\overline\partial\) and \(\overline\partial\)-Neumann operators Keywords:\(\overline \partial_b\) operator; Lipschitz domain PDFBibTeX XMLCite \textit{M.-C. Shaw}, Math. Z. 244, No. 1, 91--123 (2003; Zbl 1039.32049) Full Text: DOI