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 | References | Similar Articles | Additional Information

Abstract: Recently J. J. Kohn (2005) proved hypoellipticity for

*loss*of 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

**1.**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.**2.**A. BOVE AND D.S. TARTAKOFF,*Analytic hypo-ellipticity at non-symplectic characteristics when the symplectic form changes its rank,*preprint.**3.**M CHRIST,*A remark on sums of squares of complex vector fields,*preprint, arXiv:math.CV/0503506.**4.**Makhlouf Derridj and David S. Tartakoff,*Local analyticity for □_{𝑏} and the \overline∂-Neumann problem at certain weakly pseudoconvex points*, Comm. Partial Differential Equations**13**(1988), no. 12, 1521–1600. MR**970155**, https://doi.org/10.1080/03605308808820586**5.**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.**6.**F. Hirzebruch,*Topological methods in algebraic geometry*, Third enlarged edition. New appendix and translation from the second German edition by R. L. E. Schwarzenberger, with an additional section by A. Borel. Die Grundlehren der Mathematischen Wissenschaften, Band 131, Springer-Verlag New York, Inc., New York, 1966. MR**0202713****7.**J.J. KOHN,*Hypoellipticity and loss of derivatives,*Annals of Mathematics**162(2)**(2005).**8.**C. PARENTI AND A. PARMEGGIANI,*A Note on Kohn's and Christ's Examples,*preprint.**9.**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**

Hajime Tsuji,*Deformation invariance of plurigenera*, Nagoya Math. J.**166**(2002), 117–134. MR**1908576****10.**David S. Tartakoff,*Local analytic hypoellipticity for □_{𝑏} on nondegenerate Cauchy-Riemann manifolds*, Proc. Nat. Acad. Sci. U.S.A.**75**(1978), no. 7, 3027–3028. MR**499657****11.**David S. Tartakoff,*The local real analyticity of solutions to □_{𝑏} and the ∂-Neumann problem*, Acta Math.**145**(1980), no. 3-4, 177–204. MR**590289**, https://doi.org/10.1007/BF02414189**12.**François Trèves,*Analytic hypo-ellipticity of a class of pseudodifferential operators with double characteristics and applications to the \overline∂-Neumann problem*, Comm. Partial Differential Equations**3**(1978), no. 6-7, 475–642. MR**0492802**, https://doi.org/10.1080/03605307808820074

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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

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

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.