Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

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

  • A. Bove, M. Derridj, J.J. Kohn and D.S. Tartakoff, Hypoellipticity for a sum of squares of complex vector fields with large loss of derivatives, preprint.
  • A. Bove and D.S. Tartakoff, Analytic hypo-ellipticity at non-symplectic characteristics when the symplectic form changes its rank, preprint.
  • M Christ, A remark on sums of squares of complex vector fields, preprint, arXiv:math.CV/0503506.
  • Makhlouf Derridj and David S. Tartakoff, Local analyticity for $\square _b$ and the $\overline \partial $-Neumann problem at certain weakly pseudoconvex points, Comm. Partial Differential Equations 13 (1988), no. 12, 1521–1600. MR 970155, DOI
  • M. Derridj and D.S. Tartakoff, Analyticity and loss of derivatives, Annals of Mathematics 162(2) (2005), as Appendix to Hypoellipticity and loss of derivatives, by J. J. Kohn in the same issue.
  • F. Hirzebruch, Topological methods in algebraic geometry, Third enlarged edition, Die Grundlehren der Mathematischen Wissenschaften, Band 131, Springer-Verlag New York, Inc., New York, 1966. New appendix and translation from the second German edition by R. L. E. Schwarzenberger, with an additional section by A. Borel. MR 0202713
  • J.J. Kohn, Hypoellipticity and loss of derivatives, Annals of Mathematics 162(2) (2005).
  • C. Parenti and A. Parmeggiani, A Note on Kohn’s and Christ’s Examples, preprint.
  • Yum-Tong Siu, Extension of twisted pluricanonical sections with plurisubharmonic weight and invariance of semipositively twisted plurigenera for manifolds not necessarily of general type, Complex geometry (Göttingen, 2000) Springer, Berlin, 2002, pp. 223–277. MR 1922108
  • David S. Tartakoff, Local analytic hypoellipticity for $\square _{b}$ on nondegenerate Cauchy-Riemann manifolds, Proc. Nat. Acad. Sci. U.S.A. 75 (1978), no. 7, 3027–3028. MR 499657, DOI
  • David S. Tartakoff, The local real analyticity of solutions to $\square _{b}$ and the $\bar \partial $-Neumann problem, Acta Math. 145 (1980), no. 3-4, 177–204. MR 590289, DOI
  • François Trèves, Analytic hypo-ellipticity of a class of pseudodifferential operators with double characteristics and applications to the $\overline \partial $-Neumann problem, Comm. Partial Differential Equations 3 (1978), no. 6-7, 475–642. MR 492802, DOI

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 35H10, 35N15

Retrieve articles in all journals with MSC (2000): 35H10, 35N15

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.