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Sobolev estimates for the Cauchy-Riemann complex on \(C^{1}\) pseudoconvex domains. (English) Zbl 1165.32020

Summary: Let \({\Omega\subset\mathbb{C}^n}\) be a bounded pseudoconvex domain with \(C^{k}\) boundary, \(k \geq 1\). In this paper, we prove that the Cauchy-Riemann operator \(\overline \partial\) has a bounded solution operator in the Sobolev space \({W^s_{(p,q)}(\Omega)}\) for all \({0 \leq s < k-\frac{1}{2}}\).
It is worth noting (as the author points out) that although the results are stated only in \(\mathbb{C}^{n}\) for clarity, they apply on any Stein manifold.

MSC:

32W05 \(\overline\partial\) and \(\overline\partial\)-Neumann operators
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