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The \(\bar {\partial }\)-Neumann operator on Lipschitz \(q\)-pseudoconvex domains. (English) Zbl 1249.32016
The partial-Neumann operator \(N\) is the inverse of Laplace-Bertrami operator acting on \((p,q)\)-forms in a region \(\Omega \) of complex space \(\mathbb {C}^n\). The author proves that if the boundary of the region satisfies some properties, then the operator \(N\) can be extended to a bounded operator from the Sobolev space of \(r,s\)-forms \(W_{rs}^{-1/2}\) to \(W_{rs}^{1/2}\). O. Abdelkader and the author proved a similar result in an older paper [JIPAM, J. Inequal. Pure Appl. Math. 5, No. 3, Paper No. 70, 10 p., electronic only (2004; Zbl 1062.35048)], where they assumed that the region has a “bounded strictly pseudoconvex domain with Lipschitz boundary”. In the current paper he generalizes this to regions with “bounded \(q\)-pseudoconvex domain with Lipschitz boundary”.
MSC:
32F10 \(q\)-convexity, \(q\)-concavity
32W05 \(\overline\partial\) and \(\overline\partial\)-Neumann operators
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