On local solvability of certain differential complexes

Nicola, Fabio

doi:10.1090/S0002-9939-07-09062-4

On local solvability of certain differential complexes
HTML articles powered by AMS MathViewer

by Fabio Nicola PDF

Proc. Amer. Math. Soc. 136 (2008), 351-358 Request permission

Abstract:

In any locally integrable structure a differential complex induced by the de Rham differential is naturally defined. We give necessary conditions, in terms of the signature of the Levi form, for its local solvability with a prescribed rate of shrinking.

References

Aldo Andreotti, Gregory Fredricks, and Mauro Nacinovich, On the absence of Poincaré lemma in tangential Cauchy-Riemann complexes, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 8 (1981), no. 3, 365–404. MR 634855
A. Andreotti, C.D. Hill, E.E. Levi convexity and the Hans Lewy problem, I and II, Ann. Scuola Norm. Sup. Pisa, Sci. Fis. Mat., 26 (1972), 325–363, 747–806.
Sagun Chanillo and François Treves, Local exactness in a class of differential complexes, J. Amer. Math. Soc. 10 (1997), no. 2, 393–426. MR 1423030, DOI 10.1090/S0894-0347-97-00231-2
Paulo Cordaro and Jorge Hounie, On local solvability of underdetermined systems of vector fields, Amer. J. Math. 112 (1990), no. 2, 243–270. MR 1047299, DOI 10.2307/2374715
Paulo D. Cordaro and Jorge Hounie, Local solvability for top degree forms in a class of systems of vector fields, Amer. J. Math. 121 (1999), no. 3, 487–495. MR 1738411, DOI 10.1353/ajm.1999.0017
Paulo D. Cordaro and Jorge G. Hounie, Local solvability for a class of differential complexes, Acta Math. 187 (2001), no. 2, 191–212. MR 1879848, DOI 10.1007/BF02392616
Paulo Cordaro and François Trèves, Homology and cohomology in hypo-analytic structures of the hypersurface type, J. Geom. Anal. 1 (1991), no. 1, 39–70. MR 1097935, DOI 10.1007/BF02938114
Paulo D. Cordaro and François Trèves, Hyperfunctions on hypo-analytic manifolds, Annals of Mathematics Studies, vol. 136, Princeton University Press, Princeton, NJ, 1994. MR 1311923
So-Chin Chen and Mei-Chi Shaw, Partial differential equations in several complex variables, AMS/IP Studies in Advanced Mathematics, vol. 19, American Mathematical Society, Providence, RI; International Press, Boston, MA, 2001. MR 1800297, DOI 10.1090/amsip/019
C. Denson Hill and Mauro Nacinovich, On the failure of the Poincaré lemma for $\overline \partial _M$. II, Math. Ann. 335 (2006), no. 1, 193–219. MR 2217688, DOI 10.1007/s00208-005-0746-z
Lars Hörmander, Differential operators of principal type, Math. Ann. 140 (1960), 124–146. MR 130574, DOI 10.1007/BF01360085
Masaki Kashiwara and Pierre Schapira, A vanishing theorem for a class of systems with simple characteristics, Invent. Math. 82 (1985), no. 3, 579–592. MR 811552, DOI 10.1007/BF01388871
Hans Lewy, An example of a smooth linear partial differential equation without solution, Ann. of Math. (2) 66 (1957), 155–158. MR 88629, DOI 10.2307/1970121
Vincent Michel, Sur la régularité $C^\infty$ du $\overline \partial$ au bord d’un domaine de $\mathbf C^n$ dont la forme de Levi a exactement $s$ valeurs propres strictement négatives, Math. Ann. 295 (1993), no. 1, 135–161 (French). MR 1198845, DOI 10.1007/BF01444880
M. Nacinovich, Poincaré lemma for tangential Cauchy-Riemann complexes, Math. Ann. 268 (1984), no. 4, 449–471. MR 753407, DOI 10.1007/BF01451852
Mauro Nacinovich, On boundary Hilbert differential complexes, Ann. Polon. Math. 46 (1985), 213–235. MR 841829, DOI 10.4064/ap-46-1-213-235
Mauro Nacinovich, On strict Levi $q$-convexity and $q$-concavity on domains with piecewise smooth boundaries, Math. Ann. 281 (1988), no. 3, 459–482. MR 954153, DOI 10.1007/BF01457157
Fabio Nicola, On the absence of the one-sided Poincaré lemma in Cauchy-Riemann manifolds, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 4 (2005), no. 4, 587–600. MR 2207735
Marco M. Peloso and Fulvio Ricci, Tangential Cauchy-Riemann equations on quadratic CR manifolds, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 13 (2002), no. 3-4, 285–294. Harmonic analysis on complex homogeneous domains and Lie groups (Rome, 2001). MR 1984107
François Trèves, A remark on the Poincaré lemma in analytic complexes with nondegenerate Levi form, Comm. Partial Differential Equations 7 (1982), no. 12, 1467–1482. MR 679951, DOI 10.1080/03605308208820259
François Trèves, Homotopy formulas in the tangential Cauchy-Riemann complex, Mem. Amer. Math. Soc. 87 (1990), no. 434, viii+121. MR 1028234, DOI 10.1090/memo/0434
François Trèves, Hypo-analytic structures, Princeton Mathematical Series, vol. 40, Princeton University Press, Princeton, NJ, 1992. Local theory. MR 1200459
François Treves, A treasure trove of geometry and analysis: the hyperquadric, Notices Amer. Math. Soc. 47 (2000), no. 10, 1246–1256. MR 1784240