Analyticity for singular sums of squares of degenerate vector fields

Tartakoff, David

doi:10.1090/S0002-9939-06-08419-X

Analyticity for singular sums of squares of degenerate vector fields
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by David S. Tartakoff PDF

Proc. Amer. Math. Soc. 134 (2006), 3343-3352 Request permission

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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