Estimates for the
-Neumann problem and nonexistence of C
2 Levi-flat hypersurfaces in |
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Authors: | Jianguo Cao Mei-Chi Shaw Lihe Wang |
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Institution: | (1) Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, USA;(2) Department of Mathematics, University of Iowa, Iowa City, IA 52242, USA |
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Abstract: | Let be a pseudoconvex domain with C2 boundary in , n 2. We prove that the -Neumann operator N exists for square-integrable forms on . Furthermore, there exists a number 0>0 such that the operators and the Bergman projection are regular in the Sobolev space W ( ) for < 0. The -Neumann operator is used to construct -closed extension on for forms on the boundary b . This gives solvability for the tangential Cauchy-Riemann operators on the boundary. Using these results, we show that there exist no non-zero L2-holomorphic (p, 0)-forms on any domain with C2 pseudoconcave boundary in with p > 0 and n 2. As a consequence, we prove the nonexistence of C2 Levi-flat hypersurfaces in .This paper is a revision of our preprint (May 2003) formerly titled Estimates for the -Neumann problem and nonexistence of Levi-flat hypersurfaces in ![MediaObjects/s00209-004-0661-0flb3.gif](/content/b3y2cypne2qh5gvw/MediaObjects/s00209-004-0661-0flb3.gif) where the nonexistence of C2, Levi-flat hypersurfaces is proved for >0.All three authors are partially supported by NSF grants.An erratum to this article can be found at |
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