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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Local integrability of Mizohata structures


Authors: Jorge Hounie and Pedro Malagutti
Journal: Trans. Amer. Math. Soc. 338 (1993), 337-362
MSC: Primary 35N15; Secondary 32F20, 32F40, 35F05
MathSciNet review: 1106189
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Abstract: In this work we study the local integrability of strongly pseudoconvex Mizohata structures of rank $ n > 2$ (and co-rank $ 1$). These structures are locally generated in an appropriate coordinate system $ ({t_1}, \ldots ,{t_n},x)$ by flat perturbations of Mizohata vector fields $ {M_j} = \frac{\partial } {{\partial {t_j}}} - i{t_j}\frac{\partial } {{\partial x}}$, $ j = 1, \ldots ,n$. For this, we first prove the global integrability of small perturbations of the structure generated by $ \frac{\partial } {{\partial \bar z}} + {\sigma _1}\frac{\partial } {{\partial z}}$, $ \frac{\partial } {{\partial {\theta _{n - 1}}}} + {\sigma _j}\frac{\partial } {{\partial z}}$, $ j = 2, \ldots ,n$, defined over a manifold $ {\mathbf{C}} \times S$, where $ S$ is simply connected.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1993-1106189-4
PII: S 0002-9947(1993)1106189-4
Keywords: overdetermined systems, localintegrability, Mizohata structures, differential complex
Article copyright: © Copyright 1993 American Mathematical Society