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

DOI:
https://doi.org/10.1090/S0002-9939-06-08419-X

Published electronically:
May 12, 2006

MathSciNet review:
2231919

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Abstract: Recently J. J. Kohn (2005) proved $C^\infty$ hypoellipticity for \[ P_k=L\overline {L} + \overline {L}|z|^{2k}L =-\overline {L}^*\overline {L} - (\overline {z}^kL)^*\overline {z}^kL\mathrm { \quad with \quad } L={\partial \over \partial z} + i\overline {z}{\partial \over \partial t},\] (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 \[ P_{m,k}=L_m\overline {L_m} + \overline {L_m} |z|^{2k}L_m \mathrm { \; with \;} L_m={\partial \over \partial z} + i\overline {z}|z|^{2m}{\partial \over \partial t},\] 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.

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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

Email:
dst@uic.edu

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.