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Coherence and other properties of sheaves in the Kohn algorithm. (English) Zbl 1314.32013
In this paper, techniques by J.-C. Tougeron [C. R. Acad. Sci., Paris 260, 2971–2973 (1965; Zbl 0131.20603)] are used to show the following result: let \(\Omega \subset \mathbb C^n\) be a domain with real-analytic boundary \(b\Omega,\) let \(\tilde U\) be any open subset of \(b\Omega\) such that \(\tilde U\) is contained in a compact semianalytic subset \(Y\) of \(b\Omega;\) then the ideal sheaf \(\tilde \mathcal I^q\) of real-analytic subelliptic multipliers for the \(\overline \partial\)-Neumann problem on \((p,q)\) forms defined on \(\tilde U\) is coherent; additionally, if \(\Omega\) is pseudoconvex, the multiplier ideal sheaf \(\tilde \mathcal I^q_k\) given by the modified Kohn algorithm on \(\tilde U\) at step \(k\) for each \(k\geq 1\) is also coherent; if \(\Omega\) is bounded, \(b\Omega\) itself may be taken as \(\tilde U.\) It is shown that this implies a much stronger result than Kohn’s original method in the real-analytic case for a domain of finite D’Angelo type [J. J. Kohn, Acta Math. 142, 79–122 (1979; Zbl 0395.35069)].

MSC:
32C05 Real-analytic manifolds, real-analytic spaces
32W05 \(\overline\partial\) and \(\overline\partial\)-Neumann operators
35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
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References:
[1] DOI: 10.1215/S0012-7094-75-04258-1 · Zbl 0357.46032
[2] DOI: 10.1007/978-1-4757-3849-0
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