Non completely solvable systems of complex first order PDEs

Denson Hill, C. ; Nacinovich, Mauro

Rendiconti del Seminario Matematico della Università di Padova, Tome 130 (2013), p. 129-170 / Harvested from Numdam

Détail

Publié le : 2013-01-01

MR 3090635 | Zbl 1275.35074

Bibliographie

[1] Takao Akahori, A new approach to the local embedding theorem of CR-structures for $n\ge 4$ (the local solvability for the operator $\overline \partial _b$ in the abstract sense), Mem. Amer. Math. Soc. 67, no. 366 (1987), pp. xvi+257. | MR 888499

[2] Andrea Altomani - C. Denson Hill - Mauro Nacinovich - Egmont Porten, Complex vector fields and hypoelliptic partial differential operators, Ann. Inst. Fourier (Grenoble), 60, no. 3 (2010), pp. 987-1034. | Numdam | MR 2680822

[3] Aldo Andreotti - Gregory Fredricks - Mauro Nacinovich, On the absence of Poincaré lemma in tangential Cauchy-Riemann complexes, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 8, no. 3 (1981), pp. 365-404. | Numdam | MR 634855

[4] Aldo Andreotti - C. Denson Hill, E. E. Levi convexity and the Hans Lewy problem. II. Vanishing theorems, Ann. Scuola Norm. Sup. Pisa (3) 26 (1972), pp. 747-806. | Numdam | MR 477150

[5] M. S. Baouendi - L. P. Rothschild, Embeddability of abstract CR structures and integrability of related systems, Ann. Inst. Fourier (Grenoble), 37, no. 3 (1987), pp. 131-141. | MR 916277

[6] M. S. Baouendi - Linda Preiss Rothschild, Cauchy-Riemann functions on manifolds of higher codimension in complex space, Invent. Math. 101, no. 1 (1990), pp. 45-56. | MR 1055709

[7] M. S. Baouendi - F. Trèves, A property of the functions and distributions annihilated by a locally integrable system of complex vector fields, Ann. of Math. (2), 113, no. 2 (1981), pp. 387-421. | MR 607899

[8] Shiferaw Berhanu - Paulo D. Cordaro - Jorge Hounie, An introduction to involutive structures, New Mathematical Monographs, vol. 6, Cambridge University Press, Cambridge, 2008. | MR 2397326

[9] L. Boutet De Monvel, Intégration des équations de Cauchy-Riemann induites formelles, Séminaire Goulaouic-Lions-Schwartz 1974-1975; Équations aux derivées partielles linéaires et non linéaires, Centre Math., École Polytech., Paris, 1975, pp. Exp. No. 9, 14. | MR 409893

[10] J. Brinkschulte - C. Denson Hill - M. Nacinovich, Obstructions to generic embeddings, Ann. Inst. Fourier (Grenoble), 52, no. 6 (2002), pp. 1785-1792. | Numdam | MR 1952531

[11] Judith Brinkschulte - C. Denson Hill - Mauro Nacinovich, The Poincaré lemma and local embeddability, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8), 6, no. 2 (2003), pp. 393-398. | MR 1988212

[12] David Catlin, Sufficient conditions for the extension of CR structures, J. Geom. Anal. 4, no. 4 (1994), pp. 467-538. | MR 1305993

[13] Ja. M. Èliašberg, Singularities of folding type, Izv. Akad. Nauk SSSR Ser. Mat. 34 (1970), pp. 1110-1126. | MR 278321

[14] C. Denson Hill, What is the notion of a complex manifold with a smooth boundary?, Algebraic analysis, Vol. I, Academic Press (Boston, MA, 1988), pp.185-201. | MR 992454

[15] C. Denson Hill, Counterexamples to Newlander-Nirenberg up to the boundary, Several complex variables and complex geometry, Part 3 (Santa Cruz, CA, 1989), Proc. Sympos. Pure Math., vol. 52, Amer. Math. Soc., Providence (RI, 1991), pp. 191-197. | MR 1128593

[16] C. Denson Hill - Mauro Nacinovich, Embeddable CR manifolds with nonembeddable smooth boundary, Boll. Un. Mat. Ital. A (7), 7, no. 3 (1993), pp. 387-395. | MR 1249115

[17] C. Denson Hill - Mauro Nacinovich, Solvable Lie algebras and the embedding of CR manifolds, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8), 2, no. 1 (1999), pp. 121-126. | MR 1794546

[18] C. Denson Hill - Mauro Nacinovich, On the failure of the Poincaré lemma for $\overline{\partial }_M$ . II, Math. Ann. 335, no. 1 (2006), pp. 193-219. | MR 2217688

[19] Lars Hörmander, On existence of solutions of partial differential equations, Partial differential equations and continuum mechanics, Univ. of Wisconsin Press, Madison, Wis. (1961), pp. 233-240. | MR 124601

[20] Howard Jacobowitz, Simple examples of nonrealizable CR hypersurfaces, Proc. Amer. Math. Soc. 98, no. 3 (1986), pp. 467-468. | MR 857942

[21] Howard Jacobowitz, Homogeneous solvability and CR structures, Notas de Curso [Course Notes], vol. 25, Universidade Federal de Pernambuco Departamento de Matemática, Recife, 1988. | MR 934571

[22] Howard Jacobowitz - François Trèves, Nonrealizable CR structures, Invent. Math. 66, no. 2 (1982), pp. 231-249. | MR 656622

[23] Howard Jacobowitz - François Trèves, Nowhere solvable homogeneous partial differential equations, Bull. Amer. Math. Soc. (N.S.), 8, no. 3 (1983), 467-469. | MR 693964

[24] Howard Jacobowitz - François Trèves, Aberrant CR structures, Hokkaido Math. J. 12 (1983), pp. 276-292. | MR 719968

[25] Walter Koppelman, The Cauchy integral for functions of several complex variables, Bull. Amer. Math. Soc. 73 (1967), pp. 373-377. | MR 209519

[26] Masatake Kuranishi, Strongly pseudoconvex CR structures over small balls. I. An a priori estimate, Ann. of Math. (2), 115, no. 3 (1982), pp. 451-500. | MR 657236

[27] Masatake Kuranishi, Strongly pseudoconvex CR structures over small balls. II. A regularity theorem, Ann. of Math. (2), 116, no. 1 (1982), pp. 1-64. | MR 662117

[28] Masatake Kuranishi, Strongly pseudoconvex CR structures over small balls. III. An embedding theorem, Ann. of Math. (2), 116, no. 2 (1982), pp. 249-330. | MR 672837

[29] Hans Lewy, On the local character of the solutions of an atypical linear differential equation in three variables and a related theorem for regular functions of two complex variables, Ann. of Math. (2), 64 (1956), pp. 514-522. | MR 81952

[30] Hans Lewy, An example of a smooth linear partial differential equation without solution, Ann. of Math. (2), 66 (1957), pp. 155-158. | MR 88629

[31] Lan Ma - Joachim Michel, Regularity of local embeddings of strictly pseudoconvex CR structures, J. Reine Angew. Math. 447 (1994), pp. 147-164. | MR 1263172

[32] Abdelhamid Meziani, Perturbation of a class of CR structures of codimension larger than one, J. Funct. Anal. 116, no. 1 (1993), pp. 225-244. | MR 1237994

[33] M. Nacinovich, Poincaré lemma for tangential Cauchy-Riemann complexes, Math. Ann. 268, no. 4 (1984), pp. 449-471. | MR 753407

[34] M. Nacinovich - E. Porten, ${\mathcal {C}}^{\infty }$ -hypoellipticity and extension of cr functions, arXiv:1107.3374 (2011).

[35] A. Newlander - L. Nirenberg, Complex analytic coordinates in almost complex manifolds, Ann. of Math. (2), 65 (1957), pp. 391-404. | MR 88770

[36] Louis Nirenberg, A certain problem of Hans Lewy, Uspehi Mat. Nauk, 29, no. 2(176) (1974), pp. 241-251, Translated from the English by Ju. V. Egorov, Collection of articles dedicated to the memory of Ivan Georgievič PetrovskÉ (1901-1973), I. | MR 492752

[37] Louis Nirenberg - François Trèves, On local solvability of linear partial differential equations. I. Necessary conditions, Comm. Pure Appl. Math. 23 (1970), pp. 1-38. | MR 264470

[38] Louis Nirenberg - François Trèves, On local solvability of linear partial differential equations. II. Sufficient conditions, Comm. Pure Appl. Math. 23 (1970), pp. 459-509. | MR 264471

[39] S. I. Pinchuk - S. V. Khasanov, Asymptotically holomorphic functions and their applications, Mat. Sb. (N.S.), 134(176), no. 4 (1987), pp. 546-555, 576. | MR 933702

[40] H. H. Schaefer - M. P. Wolff, Topological vector spaces, second ed., Graduate Texts in Mathematics, vol. 3, Springer-Verlag, New York, 1999. | MR 1741419

[41] F. Trèves, Approximation and representation of functions and distributions annihilated by a system of complex vector fields, École Polytechnique Centre de Mathématiques, Palaiseau, 1981. | MR 716137

[42] François Trèves, Hypo-analytic structures, Princeton Mathematical Series, vol. 40, Princeton University Press, Princeton, NJ, 1992, Local theory.

[43] A. E. Tumanov, Extension of CR-functions into a wedge from a manifold of finite type, Mat. Sb. (N.S.), 136(178), no. 1 (1988), pp. 128-139. | MR 945904

[44] A. E. Tumanov, Extension of CR-functions into a wedge, Mat. Sb. 181, no. 7 (1990), pp. 951-964. | MR 1070489