Abstract
We prove local hypoellipticity of the complex Laplacian \(\Box \) and of the Kohn Laplacian \(\Box _b\) in a pseudoconvex boundary when, for a system of cut-off \(\eta \), the gradient \(\partial _b\eta \) and the Levi form \(\frac{1}{2}(\partial _b\bar{\partial }_b-\bar{\partial }_b\partial _b)\eta \) are subelliptic multipliers in the sense of Kohn (Acta Math 142:79–122, 1979).
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References
Baracco, L., Khanh, T.V., Zampieri, G.: Propagation of regularity for solutions of the Kohn Laplacian in a flat boundary. Adv. Math. 230, 1972–1978 (2012)
Baracco, L., Khanh, T.V., Zampieri, G.: Hypoellipticity of the \(\bar{\partial }\)-Neumann problem at a point of infinite type. Asian J. Math. 18(4), 623–632 (2014)
Catlin, D.: Subelliptic estimates for the \(\bar{\partial }\)-Neumann problem on pseudoconvex domains. Ann. Math. 126, 131–191 (1987)
Christ, M.: Hypoellipticity: geometrization and speculation. Prog. Math. Birkh”auser Basel 188, 91–109 (2000)
Christ, M.: Hypoellipticity of the Kohn Laplacian for three-dimensional tubular Cauchy–Riemann structures. J. Inst. Math. Jussieu 1, 279–291 (2002)
Derridj, M., Tartakoff, D.S.: Microlocal analiticity for \(\Box _b\) in block-decoupled pseudoconvex domains. Math. Z. 220, 477–493 (1995)
Folland, G.B., Kohn, J.J.: The Neumann problem for the Cauchy–Riemann complex. Ann. Math. Studies, Princeton Univ. Press, Princeton NJ 75 (1972)
Hanges, J., Treves, F.: Propagation of holomorphic extandibility of CR functions. Math. Ann. 263(2), 157–177 (1983)
Khanh, T.V., Zampieri, G.: Regularity of the \(\bar{\partial }\)-Neumann problem at a flat point. J. Funct. Anal. 259(11), 2760–2775 (2010)
Kohn, J.J.: Subellipticity of the \(\bar{\partial }\)-Neumann problem on pseudoconvex domains: sufficient conditions. Acta Math. 142, 79–122 (1979)
Kohn, J.J.: Hypoellipticity at points of infinite type. Contemp. Math. 251, 393–398 (2000)
Kohn, J.J.: Superlogarithmic estimates on pseudoconvex domains and CR manifolds. Ann. Math. 156, 213–248 (2002)
Kohn, J.J., Nirenberg, L.: Non-coercive boundary value problems. Commun. Pure Appl. Math. 18, 443–492 (1965)
Pinton, S.: Hypoellipticity of the \(\bar{\partial }\)-Neumann problem at a set of infinite type with positive CR dimension (to appear)
Straube, E.: Lectures on the \(L^2\)-Sobolev theory of the \(\bar{\partial }\)-Neumann problem. ESI Lect. in Math. and Physics (2010)
Acknowledgments
The paper was accomplished at Sao Paulo USP in November 2013. The authors are grateful to Paulo Domingo Cordaro for friendly hospitality and fruitful discussions.
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Baracco, L., Pinton, S. & Zampieri, G. Hypoellipticity of the Kohn-Laplacian \(\Box _b\) and of the \(\bar{\partial }\)-Neumann problem by means of subelliptic multipliers. Math. Ann. 362, 887–901 (2015). https://doi.org/10.1007/s00208-014-1144-1
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DOI: https://doi.org/10.1007/s00208-014-1144-1