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The \(\overline\partial\)-Neumann operator on Lipschitz pseudoconvex domains with plurisubharmonic defining functions. (English) Zbl 1020.32030

If \(D\) is a bounded pseudoconvex Lipschitz domain in \(\mathbb{C}^n\) with a plurisubharmonic Lipschitz defining function, the authors prove that the \(\overline\partial\)-Neumann operator and the canonical Kohn solution operator of the \(\overline\partial\)-equation are bounded from the Sobolev space \(H^{1/2}(D)\) to itself.
In a more general setting a slightly weaker result has been obtained by Bo Berndtsson and Ph. Charpentier [Math. Z. 235, 1-10 (2000; Zbl 0969.32015)] who proved that the Kohn operator and the Bergman projection are bounded on \(H^s\) for \(s< \eta/2\) provided that \(D\) has a defining function \(\rho\) such that \(-(-\rho)^\eta\) is plurisubharmonic.

MSC:

32W05 \(\overline\partial\) and \(\overline\partial\)-Neumann operators
32U10 Plurisubharmonic exhaustion functions
35N15 \(\overline\partial\)-Neumann problems and formal complexes in context of PDEs

Citations:

Zbl 0969.32015
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References:

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