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Subellipticity of the $\overline\partial$-Neumann Problem on Nonpseudoconvex Domains

Lop-Hing Ho
Transactions of the American Mathematical Society
Vol. 291, No. 1 (Sep., 1985), pp. 43-73
DOI: 10.2307/1999894
Stable URL: http://www.jstor.org/stable/1999894
Page Count: 31
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Subellipticity of the $\overline\partial$-Neumann Problem on Nonpseudoconvex Domains
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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 Leviform 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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