Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)

 
 

 

Local exactness
in a class of differential complexes


Authors: Sagun Chanillo and François Treves
Journal: J. Amer. Math. Soc. 10 (1997), 393-426
MSC (1991): Primary 35A07, 35F05
DOI: https://doi.org/10.1090/S0894-0347-97-00231-2
MathSciNet review: 1423030
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The article studies the local exactness at level $q$ $(1\le q\le n)$ in the differential complex defined by $n$ commuting, linearly independent real-analytic complex vector fields $L_1,\dotsc ,L_n$ in $n+1$ independent variables. Locally the system $\{L_1,\dotsc ,L_n\}$ admits a first integral $Z$, i.e., a $\mathcal {C}^\omega $ complex function $Z$ such that $L_1Z=\cdots =L_nZ=0$ and $dZ\ne 0$. The germs of the ``level sets'' of $Z$, the sets $Z=z_0\in \mathbb {C}$, are invariants of the structure. It is proved that the vanishing of the (reduced) singular homology, in dimension $q-1$, of these level sets is sufficient for local exactness at the level $q$. The condition was already known to be necessary.


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

  • 1. M. S. Baouendi and F. Treves, A property of the functions and distributions annihilated by a locally integrable system of complex vector fields, Ann. of Math. 113 (1981), 387-421. MR 82f:35057
  • 2. S. Chanillo, The first eigenvalue of analytic level surfaces on spheres, Math. Research Letters 1 (1994), 159-166. MR 95g:58241
  • 3. S. Chanillo and F. Treves, On the lowest eigenvalue of the Hodge Laplacian, J. Diff. Geom., to appear.
  • 4. P. D. Cordaro and J. Hounie, On local solvability of underdetermined systems of vector fields, Amer. J. Math. 112 (1990), 243-270. MR 91j:58150
  • 5. P. D. Cordaro and F. Treves, Homology and cohomology in hypo-analytic structures of the hypersurface type, J. Geometric Analysis I (1991), 39-70. MR 92h:32034
  • 6. -, Hyperfunctions on Hypo-Analytic Manifolds, Annals of Math. Studies # 136, Princeton University Press, Princeton, NJ, 1994. MR 96h:58163
  • 7. -, Necessary and sufficient conditions for the local solvability in hyperfunctions of a class of systems of complex vector fields, Invent. Math. 120 (1995), 339-360. MR 96j:58157
  • 8. R. Hardt, Slicing and intersection theory for chains associated with real analytic varieties, Acta Math. 129 (1972), 75-136. MR 47:4110
  • 9. -, Triangulations of subanalytic sets and proper light subanalytic maps, Invent. Math. 38 (1977), 207-217. MR 56:12302
  • 10. L. Hörmander, Propagation of singularities and semi-global existence theorems for (pseudo)-differential operators of principal type, Annals of Math. 108 (1978), 569-609. MR 81j:35110
  • 11. -, Pseudo-differential operators of principal type, Nato Advanced Study Inst. on Singularities in Boundary Value Problems, Reidel Publ. Co., Dordrecht, 1981, 69-96. MR 83m:35003
  • 12. S. Lojasiewicz, Sur les problème de la division, Studia Math. 18 (1959), 87-136. MR 21:5893
  • 13. -, Ensembles semi-analytiques, IHES Notes, Paris, 1965.
  • 14. G. Mendoza and F. Treves, Local solvability in a class of overdetermined systems of linear PDE, Duke Math. J. 63 (1991), 355-377. MR 92i:58177
  • 15. L. Nirenberg and F. Treves, Solvability of a first-order linear partial differential equation, Comm. Pure Appl. Math. 16 (1963), 331-351. MR 29:348
  • 16. Séminaire Schwartz, Unicité du problème de Cauchy. Division des distributions. Secrét. Math. Fac. Sci. Paris 4, 1959/60. MR 23:A2275
  • 17. H. Sussmann, Real-analytic desingularization and subanalytic sets: an elementary approach, Trans. Amer. Math. Soc. 317 (1990), 417-461. MR 90e:32007
  • 18. F. Treves, Study of a model in the theory of overdetermined pseudodifferential equations, Ann. of Math. 104 (1976), 269-324. MR 54:14014; MR 82i:35044
  • 19. -, On the local solvability and local integrability of systems of vector fields, Acta Math. 151 (1983), 1-38. MR 85j:35009
  • 20. -, Hypo-analytic Structures. Local Theory, Princeton University Press, Princeton, NJ, 1992. MR 94e:35014
  • 21. R. Weitzenböck, Invariantentheorie, Groningen, Noordhoff, 1923.

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (1991): 35A07, 35F05

Retrieve articles in all journals with MSC (1991): 35A07, 35F05


Additional Information

Sagun Chanillo
Affiliation: Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903
Email: chanillo@math.rutgers.edu

François Treves
Affiliation: Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903
Email: treves@math.rutgers.edu

DOI: https://doi.org/10.1090/S0894-0347-97-00231-2
Keywords: Differential complex, local solvability, singular homology, subanalytic sets
Received by editor(s): May 17, 1996
Received by editor(s) in revised form: November 11, 1996
Additional Notes: The first author was partially supported by NSF Grant DMS-9401782, and the second author by NSF Grant DMS-9201980
Article copyright: © Copyright 1997 American Mathematical Society

American Mathematical Society