Non completely solvable systems of complex first order PDEs

C. Denson Hill; Mauro Nacinovich

Rendiconti del Seminario Matematico della Università di Padova (2013)

  • Volume: 129, page 129-170
  • ISSN: 0041-8994

How to cite

top

Denson Hill, C., and Nacinovich, Mauro. "Non completely solvable systems of complex first order PDEs." Rendiconti del Seminario Matematico della Università di Padova 129 (2013): 129-170. <http://eudml.org/doc/275117>.

@article{DensonHill2013,
author = {Denson Hill, C., Nacinovich, Mauro},
journal = {Rendiconti del Seminario Matematico della Università di Padova},
keywords = {complex vector fields; manifolds; lack of local solvability; failure of local embeddability},
language = {eng},
pages = {129-170},
publisher = {Seminario Matematico of the University of Padua},
title = {Non completely solvable systems of complex first order PDEs},
url = {http://eudml.org/doc/275117},
volume = {129},
year = {2013},
}

TY - JOUR
AU - Denson Hill, C.
AU - Nacinovich, Mauro
TI - Non completely solvable systems of complex first order PDEs
JO - Rendiconti del Seminario Matematico della Università di Padova
PY - 2013
PB - Seminario Matematico of the University of Padua
VL - 129
SP - 129
EP - 170
LA - eng
KW - complex vector fields; manifolds; lack of local solvability; failure of local embeddability
UR - http://eudml.org/doc/275117
ER -

References

top
  1. [1] Takao Akahori, A new approach to the local embedding theorem of CR-structures for n 4 (the local solvability for the operator ¯ b in the abstract sense), Mem. Amer. Math. Soc. 67, no. 366 (1987), pp. xvi+257. Zbl0628.32025MR888499
  2. [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. MR2680822
  3. [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. Zbl0482.35061MR634855
  4. [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. Zbl0283.32013MR477150
  5. [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. Zbl0619.58001MR916277
  6. [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. MR1055709
  7. [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. Zbl0491.35036MR607899
  8. [8] Shiferaw Berhanu - Paulo D. Cordaro - Jorge Hounie, An introduction to involutive structures, New Mathematical Monographs, vol. 6, Cambridge University Press, Cambridge, 2008. Zbl1151.35011MR2397326
  9. [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. MR409893
  10. [10] J. Brinkschulte - C. Denson Hill - M. Nacinovich, Obstructions to generic embeddings, Ann. Inst. Fourier (Grenoble), 52, no. 6 (2002), pp. 1785–1792. Zbl1029.32018MR1952531
  11. [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. Zbl1150.32010MR1988212
  12. [12] David Catlin, Sufficient conditions for the extension of CR structures, J. Geom. Anal. 4, no. 4 (1994), pp. 467–538. Zbl0841.32012MR1305993
  13. [13] Ja. M. Èliašberg, Singularities of folding type, Izv. Akad. Nauk SSSR Ser. Mat. 34 (1970), pp. 1110–1126. MR278321
  14. [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. MR992454
  15. [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. MR1128593
  16. [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. Zbl0809.53063MR1249115
  17. [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. Zbl0929.32020MR1794546
  18. [18] C. Denson Hill - Mauro Nacinovich, On the failure of the Poincaré lemma for ¯ M . II, Math. Ann. 335, no. 1 (2006), pp. 193–219. MR2217688
  19. [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. MR124601
  20. [20] Howard Jacobowitz, Simple examples of nonrealizable CR hypersurfaces, Proc. Amer. Math. Soc. 98, no. 3 (1986), pp. 467–468. Zbl0605.32009MR857942
  21. [21] Howard Jacobowitz, Homogeneous solvability and CR structures, Notas de Curso [Course Notes], vol. 25, Universidade Federal de Pernambuco Departamento de Matemática, Recife, 1988. Zbl0675.35002MR934571
  22. [22] Howard Jacobowitz - François Trèves, Nonrealizable CR structures, Invent. Math. 66, no. 2 (1982), pp. 231–249. MR656622
  23. [23] Howard Jacobowitz - François Trèves, Nowhere solvable homogeneous partial differential equations, Bull. Amer. Math. Soc. (N.S.), 8, no. 3 (1983), 467–469. Zbl0527.35015MR693964
  24. [24] Howard Jacobowitz - François Trèves, Aberrant CR structures, Hokkaido Math. J.12 (1983), pp. 276–292. MR719968
  25. [25] Walter Koppelman, The Cauchy integral for functions of several complex variables, Bull. Amer. Math. Soc.73 (1967), pp. 373–377. Zbl0177.11103MR209519
  26. [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. Zbl0505.32018MR657236
  27. [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. Zbl0505.32019MR662117
  28. [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. Zbl0576.32033MR672837
  29. [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. Zbl0074.06204MR81952
  30. [30] Hans Lewy, An example of a smooth linear partial differential equation without solution, Ann. of Math. (2), 66 (1957), pp. 155–158. MR88629
  31. [31] Lan Ma - Joachim Michel, Regularity of local embeddings of strictly pseudoconvex CR structures, J. Reine Angew. Math.447 (1994), pp. 147–164. Zbl0791.32007MR1263172
  32. [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. Zbl0795.32006MR1237994
  33. [33] M. Nacinovich, Poincaré lemma for tangential Cauchy-Riemann complexes, Math. Ann. 268, no. 4 (1984), pp. 449–471. Zbl0574.32045MR753407
  34. [34] M. Nacinovich - E. Porten, 𝒞 -hypoellipticity and extension of cr functions, arXiv:1107.3374 (2011). 
  35. [35] A. Newlander - L. Nirenberg, Complex analytic coordinates in almost complex manifolds, Ann. of Math. (2), 65 (1957), pp. 391–404. Zbl0079.16102MR88770
  36. [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. MR492752
  37. [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. MR264470
  38. [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. MR264471
  39. [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. Zbl0639.32005MR933702
  40. [40] H. H. Schaefer - M. P. Wolff, Topological vector spaces, second ed., Graduate Texts in Mathematics, vol. 3, Springer-Verlag, New York, 1999. Zbl0983.46002MR1741419
  41. [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. Zbl0515.58030MR716137
  42. [42] François Trèves, Hypo-analytic structures, Princeton Mathematical Series, vol. 40, Princeton University Press, Princeton, NJ, 1992, Local theory. 
  43. [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. MR945904
  44. [44] A. E. Tumanov, Extension of CR-functions into a wedge, Mat. Sb. 181, no. 7 (1990), pp. 951–964. Zbl0714.32005MR1070489

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.