On the absence of the one-sided Poincaré lemma in Cauchy-Riemann manifolds

Fabio Nicola[1]

  • [1] Dipartimento di Matematica Politecnico di Torino Corso Duca degli Abruzzi, 24 10129 Torino, Italy

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2005)

  • Volume: 4, Issue: 4, page 587-600
  • ISSN: 0391-173X

Abstract

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Given an embeddable C R manifold M and a non-characteristic hypersurface S ⊂ M we present a necessary condition for the tangential Cauchy-Riemann operator ∂ ¯ M on M to be locally solvable near a point x 0 ∈ S in one of the sidesdetermined by S .

How to cite

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Nicola, Fabio. "On the absence of the one-sided Poincaré lemma in Cauchy-Riemann manifolds." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 4.4 (2005): 587-600. <http://eudml.org/doc/84572>.

@article{Nicola2005,
abstract = {Given an embeddable $CR$ manifold $M$ and a non-characteristic hypersurface $S\subset M$ we present a necessary condition for the tangential Cauchy-Riemann operator $\overline\{\partial \}_M$ on $M$ to be locally solvable near a point $x_0\in S$in one of the sidesdetermined by $S$.},
affiliation = {Dipartimento di Matematica Politecnico di Torino Corso Duca degli Abruzzi, 24 10129 Torino, Italy},
author = {Nicola, Fabio},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
language = {eng},
number = {4},
pages = {587-600},
publisher = {Scuola Normale Superiore, Pisa},
title = {On the absence of the one-sided Poincaré lemma in Cauchy-Riemann manifolds},
url = {http://eudml.org/doc/84572},
volume = {4},
year = {2005},
}

TY - JOUR
AU - Nicola, Fabio
TI - On the absence of the one-sided Poincaré lemma in Cauchy-Riemann manifolds
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2005
PB - Scuola Normale Superiore, Pisa
VL - 4
IS - 4
SP - 587
EP - 600
AB - Given an embeddable $CR$ manifold $M$ and a non-characteristic hypersurface $S\subset M$ we present a necessary condition for the tangential Cauchy-Riemann operator $\overline{\partial }_M$ on $M$ to be locally solvable near a point $x_0\in S$in one of the sidesdetermined by $S$.
LA - eng
UR - http://eudml.org/doc/84572
ER -

References

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  1. [1] A. Andreotti, G. Fredricks and M. Nacinovich, On the absence of the Poincaré lemma in tangential Cauchy-Riemann complexes, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 8 (1981), 365–404. Zbl0482.35061MR634855
  2. [2] A. Andreotti and C. D. Hill, E.E. Levi convexity and the Hans Lewy problem, I and II, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 26 (1972), 325–363, 747–806. Zbl0283.32013
  3. [3] P. D. Cordaro and J. Hounie, Local solvability for a class of differential complexes, Acta Math. 187 (2001), 191–212. Zbl1004.58012MR1879848
  4. [4] P.D. Cordaro and F. Treves, “Hyperfunctions on Hypo-Analytic Manifolds”, Annals of Mathematics Studies, Vol. 136. Princeton University Press, Princeton, NJ, 1994. Zbl0817.32001MR1311923
  5. [5] S.-C. Chen and M.-C. Shaw, “Partial Differential Equations in Several Complex Variables”, Studies in Advanced Mathematics, Vol. 19, Amer. Math. Soc., International Press, Providence, RI, 2001. Zbl0963.32001MR1800297
  6. [6] G. B. Folland and E.M. Stein, Estimates for the ∂ ¯ b complex and analysis on the Heisenberg group, Comm. Pure Appl. Math. 27 (1974), 429–522. Zbl0293.35012MR367477
  7. [7] C. D. Hill and M. Nacinovich, On the failure of the Poincaré lemma for ∂ ¯ M II, preprint. Zbl1092.32503MR2217688
  8. [8] L. Hörmander, Differential operators of principal type, Math. Ann. 140 (1960), 124–146. Zbl0090.08101MR130574
  9. [9] M. Kashiwara and P. Schapira, A vanishing theorem for a class of systems with simple characteristics, Invent. Math. 82 (1985), 579–592. Zbl0626.58028MR811552
  10. [10] H. Lewy, An example of a smooth linear partial differential equation without solution, Ann. of Math. 66 (1957), 155–158. Zbl0078.08104MR88629
  11. [11] R. B. Melrose, “Differential Analysis on Manifolds with Corners”, book in preparation, available at www.math.mit.edu/ ∼ rbm/book.html. 
  12. [12] V. Michel, Sur la régularité C ∞ du ∂ ¯ au bord d’un domaine de ℂ n dont la forme de Levi a exactement s valeus propres négatives, Math. Ann. 295 (1993), 135–161. Zbl0788.32010MR1198845
  13. [13] M. Nacinovich, Poincaré lemma for tangential Cauchy-Riemann complexes, Math. Ann. 268 (1984), 449–471. Zbl0574.32045MR753407
  14. [14] M. Nacinovich, On strict Levi q -convexity and q -concavity on domains with piecewise smooth boundaries, Math. Ann. 281 (1988), 459–482. Zbl0628.32021MR954153
  15. [15] M. M. Peloso and F. Ricci, Tangential Cauchy-Riemann equations on quadratic CR manifolds, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei 13 (2002), 125–134. Zbl1225.32037MR1984107
  16. [16] F. Treves, “Homotopy Formulas in the Tangential Cauchy-Riemann Complex”, Memoirs, Vol. 87, No. 434, Amer. Math. Soc., Providence, RI, 1990. Zbl0707.35105MR1028234
  17. [17] F. Treves, “Hypo-analytic Strucures: Local Theory”, Princeton University Press, Princeton, 1992. Zbl0787.35003MR1200459
  18. [18] F. Treves, A treasure trove of geometry and analysis: The hyperquadric, Notices Amer. Math. Soc. 47 (2000), 1246–1256. Zbl0986.51038MR1784240

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