\(L^2\) estimates and existence theorems for \(\bar\partial_b\) on Lipschitz boundaries
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 \(L^2_{(p,q)}(bD).\)
scientific article; zbMATH DE number 1944234