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Analyticity for singular sums of squares of degenerate vector fields

Author: David S. Tartakoff
Journal: Proc. Amer. Math. Soc. 134 (2006), 3343-3352
MSC (2000): Primary 35H10; Secondary 35N15
Published electronically: May 12, 2006
MathSciNet review: 2231919
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Abstract: Recently J. J. Kohn (2005) proved $ C^\infty$ hypoellipticity for

$\displaystyle P_k=L\overline{L} + \overline{L}\vert z\vert^{2k}L =-\overline{L}... ...quad} L={\partial \over \partial z} + i\overline{z}{\partial \over \partial t},$

(the negative of) a singular sum of squares of complex vector fields on the complex Heisenberg group, an operator which exhibits a loss of $ {k-1}$ derivatives. Subsequently, M. Derridj and D. S. Tartakoff proved analytic hypoellipticity for this operator using rather different methods going back to earlier methods of Tartakoff. Those methods also provide an alternate proof of the hypoellipticity given by Kohn.

In this paper, we consider the equation

$\displaystyle P_{m,k}=L_m\overline{L_m} + \overline{L_m}\,\vert z\vert^{2k}L_m ... ... \over \partial z} + i\overline{z}\vert z\vert^{2m}{\partial \over \partial t},$

for which the underlying manifold is only of finite type, and prove analytic hypoellipticity using methods of Derridj and Tartakoff. This operator is also subelliptic with large loss of derivatives, but the exact loss plays no role for analytic hypoellipticity. Nonetheless, these methods give a proof of $ C^\infty$ hypoellipticity with precise loss as well, which is to appear in a forthcoming paper by A. Bove, M. Derridj, J. J. Kohn and the author.

References [Enhancements On Off] (What's this?)

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Additional Information

David S. Tartakoff
Affiliation: Department of Mathematics, University of Illinois at Chicago, m/c 249, 851 S. Morgan Street, Chicago, Illinois 60607

Received by editor(s): June 1, 2005
Published electronically: May 12, 2006
Communicated by: Eric Bedford
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.