Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Subelliptic estimates for some systems of complex vector fields: Quasihomogeneous case
HTML articles powered by AMS MathViewer

by M. Derridj and B. Helffer PDF
Trans. Amer. Math. Soc. 361 (2009), 2607-2630 Request permission

Abstract:

For about twenty five years it was a kind of folk theorem that complex vector-fields defined on $\Omega \times \mathbb R_t$ (with $\Omega$ open set in $\mathbb R^n$) by \[ L_j = \frac {\partial }{\partial t_j} + i \frac {\partial \varphi }{\partial t_j}(\mathbf {t}) \frac {\partial }{\partial x}\;,\; j=1,\dots , n\;,\; \mathbf {t}\in \Omega , x\in \mathbb R,\] with $\varphi$ analytic, were subelliptic as soon as they were hypoelliptic. This was the case when $n=1$, but in the case $n>1$, an inaccurate reading of the proof given by Maire (see also Trèves) of the hypoellipticity of such systems, under the condition that $\varphi$ does not admit any local maximum or minimum (through a nonstandard subelliptic estimate), was supporting the belief for this folk theorem. Quite recently, J.L. Journé and J.M. Trépreau show by examples that there are very simple systems (with polynomial $\varphi$’s) which are hypoelliptic but not subelliptic in the standard $L^2$-sense. So it is natural to analyze this problem of subellipticity which is in some sense intermediate (at least when $\varphi$ is $C^\infty$) between the maximal hypoellipticity (which was analyzed by Helffer-Nourrigat and Nourrigat) and the simple local hypoellipticity (or local microhypoellipticity) and to start first with the easiest nontrivial examples. The analysis presented here is a continuation of a previous work by the first author and is devoted to the case of quasihomogeneous functions.
References
  • Makhlouf Derridj, Subelliptic estimates for some systems of complex vector fields, Hyperbolic problems and regularity questions, Trends Math., Birkhäuser, Basel, 2007, pp. 101–108. MR 2298786, DOI 10.1007/978-3-7643-7451-8_{1}1
  • Bernard Helffer and Francis Nier, Hypoelliptic estimates and spectral theory for Fokker-Planck operators and Witten Laplacians, Lecture Notes in Mathematics, vol. 1862, Springer-Verlag, Berlin, 2005. MR 2130405, DOI 10.1007/b104762
  • Bernard Helffer and Jean Nourrigat, Hypoellipticité maximale pour des opérateurs polynômes de champs de vecteurs, Progress in Mathematics, vol. 58, Birkhäuser Boston, Inc., Boston, MA, 1985 (French). MR 897103
  • Lars Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147–171. MR 222474, DOI 10.1007/BF02392081
  • Lars Hörmander, Subelliptic operators, Seminar on Singularities of Solutions of Linear Partial Differential Equations (Inst. Adv. Study, Princeton, N.J., 1977/78) Ann. of Math. Stud., vol. 91, Princeton Univ. Press, Princeton, N.J., 1979, pp. 127–208. MR 547019
  • Jean-Lin Journé and Jean-Marie Trépreau, Hypoellipticité sans sous-ellipticité: le cas des systèmes de $n$ champs de vecteurs complexes en $(n+1)$ variables, Séminaire: Équations aux Dérivées Partielles. 2005–2006, Sémin. Équ. Dériv. Partielles, École Polytech., Palaiseau, 2006, pp. Exp. No. XIV, 19 (French). MR 2276079
  • J. J. Kohn, Lectures on degenerate elliptic problems, Pseudodifferential operator with applications (Bressanone, 1977) Liguori, Naples, 1978, pp. 89–151. MR 660652
  • J. J. Kohn, Hypoellipticity and loss of derivatives, Ann. of Math. (2) 162 (2005), no. 2, 943–986. With an appendix by Makhlouf Derridj and David S. Tartakoff. MR 2183286, DOI 10.4007/annals.2005.162.943
  • H.-M. Maire, Hypoelliptic overdetermined systems of partial differential equations, Comm. Partial Differential Equations 5 (1980), no. 4, 331–380. MR 567778, DOI 10.1080/0360530800882142
  • H.-M. Maire, Résolubilité et hypoellipticité de systèmes surdéterminés, Séminaire Goulaouic-Schwartz, 1979–1980 (French), École Polytech., Palaiseau, 1980, pp. Exp. No. 5, 12 (French). MR 600689
  • H.M. Maire. Necessary and sufficient conditions for maximal hypoellipticity of $\bar \partial _b$. Unpublished (1979).
  • H.-M. Maire, Régularité optimale des solutions de systèmes différentiels et du Laplacien associé; application au $cm_{b}$, Math. Ann. 258 (1981/82), no. 1, 55–63 (French). MR 641668, DOI 10.1007/BF01450346
  • F. Nier. Hypoellipticity for Fokker-Planck operators and Witten Laplacians. Course in China. Preprint September 2006.
  • J. Nourrigat. Subelliptic estimates for systems of pseudo-differential operators. Course in Recife (1982). University of Recife.
  • J. Nourrigat, Systèmes sous-elliptiques, Séminaire sur les équations aux dérivées partielles 1986–1987, École Polytech., Palaiseau, 1987, pp. Exp. No. V, 14 (French). MR 920023
  • J. Nourrigat, Systèmes sous-elliptiques. II, Invent. Math. 104 (1991), no. 2, 377–400 (French). MR 1098615, DOI 10.1007/BF01245081
  • François Trèves, A new method of proof of the subelliptic estimates, Comm. Pure Appl. Math. 24 (1971), 71–115. MR 290201, DOI 10.1002/cpa.3160240107
  • François Treves, Study of a model in the theory of complexes of pseudodifferential operators, Ann. of Math. (2) 104 (1976), no. 2, 269–324. MR 426068, DOI 10.2307/1971048
Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 35B65, 32N15
  • Retrieve articles in all journals with MSC (2000): 35B65, 32N15
Additional Information
  • M. Derridj
  • Affiliation: 5 rue de la Juvinière, 78 350 Les loges en Josas, France
  • MR Author ID: 56970
  • B. Helffer
  • Affiliation: Laboratoire de Mathématiques, Univ Paris-Sud and CNRS, F 91 405 Orsay Cedex, France
  • MR Author ID: 83860
  • Received by editor(s): January 8, 2007
  • Received by editor(s) in revised form: July 19, 2007
  • Published electronically: November 24, 2008
  • © Copyright 2008 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 361 (2009), 2607-2630
  • MSC (2000): Primary 35B65; Secondary 32N15
  • DOI: https://doi.org/10.1090/S0002-9947-08-04601-1
  • MathSciNet review: 2471931