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Transactions of the American Mathematical Society

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Subellipticity of the $ \bar \partial$-Neumann problem on nonpseudoconvex domains

Author: Lop-Hing Ho
Journal: Trans. Amer. Math. Soc. 291 (1985), 43-73
MSC: Primary 32F20; Secondary 35N15
MathSciNet review: 797045
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Abstract: Following the work of Kohn, we give a sufficient condition for subellipticity of the $ \overline \partial $-Neumann problem for not necessarily pseudoconvex domains. We define a sequence of ideals of germs and show that if $ 1$ is in any of them, then there is a subelliptic estimate. In particular, we prove subellipticity under some specific conditions for $ n - 1$ forms and for the case when the Levi-form is diagonalizable. For the necessary conditions, we use another method to prove that there is no subelliptic estimate for $ q$ forms if the Levi-form has $ n - q - 1$ positive eigenvalues and $ q$ negative eigenvalues. This was proved by Derridj. Using similar techniques, we prove a necessary condition for subellipticity for some special domains. Finally, we give a remark to Catlin's theorem concerning the hypoellipticity of the $ \overline \partial $-Neumann problem in the case of nonpseudoconvex domains.

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Article copyright: © Copyright 1985 American Mathematical Society