The verification of the Nirenberg-Treves conjecture

Nicolas Lerner

Séminaire Bourbaki (2005-2006)

  • Volume: 48, page 211-236
  • ISSN: 0303-1179

Abstract

top
In a series of recent papers, Nils Dencker proves that condition ( ψ ) implies the local solvability of principal type pseudodifferential operators (with loss of 3 2 + ϵ derivatives for all positive ϵ ), verifying the last part of the Nirenberg-Treves conjecture, formulated in 1971. The origin of this question goes back to the Hans Lewy counterexample, published in 1957. In this text, we follow the pattern of Dencker’s papers, and we provide a proof of local solvability with a loss of 3 2 derivatives.

How to cite

top

Lerner, Nicolas. "The verification of the Nirenberg-Treves conjecture." Séminaire Bourbaki 48 (2005-2006): 211-236. <http://eudml.org/doc/252146>.

@article{Lerner2005-2006,
abstract = {In a series of recent papers, Nils Dencker proves that condition $(\psi )$ implies the local solvability of principal type pseudodifferential operators (with loss of $\frac\{3\}\{2\}+\epsilon $ derivatives for all positive $\epsilon $), verifying the last part of the Nirenberg-Treves conjecture, formulated in 1971. The origin of this question goes back to the Hans Lewy counterexample, published in 1957. In this text, we follow the pattern of Dencker’s papers, and we provide a proof of local solvability with a loss of $\frac\{3\}\{2\}$ derivatives.},
author = {Lerner, Nicolas},
journal = {Séminaire Bourbaki},
keywords = {résolubilité; opérateurs pseudodifférentiels; estimations d’énergie; opérateurs non autoadjoints},
language = {eng},
pages = {211-236},
publisher = {Association des amis de Nicolas Bourbaki, Société mathématique de France},
title = {The verification of the Nirenberg-Treves conjecture},
url = {http://eudml.org/doc/252146},
volume = {48},
year = {2005-2006},
}

TY - JOUR
AU - Lerner, Nicolas
TI - The verification of the Nirenberg-Treves conjecture
JO - Séminaire Bourbaki
PY - 2005-2006
PB - Association des amis de Nicolas Bourbaki, Société mathématique de France
VL - 48
SP - 211
EP - 236
AB - In a series of recent papers, Nils Dencker proves that condition $(\psi )$ implies the local solvability of principal type pseudodifferential operators (with loss of $\frac{3}{2}+\epsilon $ derivatives for all positive $\epsilon $), verifying the last part of the Nirenberg-Treves conjecture, formulated in 1971. The origin of this question goes back to the Hans Lewy counterexample, published in 1957. In this text, we follow the pattern of Dencker’s papers, and we provide a proof of local solvability with a loss of $\frac{3}{2}$ derivatives.
LA - eng
KW - résolubilité; opérateurs pseudodifférentiels; estimations d’énergie; opérateurs non autoadjoints
UR - http://eudml.org/doc/252146
ER -

References

top
  1. [1] G. Bachelard – La formation de l’esprit scientifique, Vrin, Paris, 1938. 
  2. [2] R. Beals & C. Fefferman – “On local solvability of linear partial differential equations”, Ann. of Math. (2) 97 (1973), p. 482–498. Zbl0256.35002MR352746
  3. [3] J.-M. Bony – “Principe du maximum, inégalité de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés”, Ann. Inst. Fourier (Grenoble) 19 (1969), no. 1, p. 277–304. Zbl0176.09703MR262881
  4. [4] J.-M. Bony & J.-Y. Chemin – “Espaces fonctionnels associés au calcul de Weyl-Hörmander”, Bull. Soc. Math. France 122 (1994), no. 1, p. 77–118. Zbl0798.35172MR1259109
  5. [5] J.-M. Bony & N. Lerner – “Quantification asymptotique et microlocalisations d’ordre supérieur. I”, Ann. Sci. École Norm. Sup. (4) 22 (1989), no. 3, p. 377–433. Zbl0753.35005MR1011988
  6. [6] H. Brézis – “On a characterization of flow-invariant sets”, Comm. Pure Appl. Math.23 (1970), p. 261–263. Zbl0191.38703MR257511
  7. [7] N. Dencker – “On the sufficiency of condition ( ψ ) ”, preprint, May 22, 2001. 
  8. [8] —, The solvability of non- L 2 -solvable operators, 1996, Saint Jean de Monts meeting. Zbl0885.35151
  9. [9] —, “Estimates and solvability”, Ark. Mat. 37 (1999), no. 2, p. 221–243. Zbl1021.35137MR1714771
  10. [10] —, “The solvability of pseudodifferential operators”, in Phase space analysis of PDE, Centro de Giorgi, Scuola Normale Superiore, Pisa, 2004, p. 175–200. 
  11. [11] —, “The resolution of the Nirenberg-Treves conjecture”, Ann. of Math. (2) 163 (2006), p. 405–444. Zbl1104.35080MR2199222
  12. [12] C. Fefferman & D. H. Phong – “On positivity of pseudo-differential operators”, Proc. Nat. Acad. Sci. U.S.A. 75 (1978), no. 10, p. 4673–4674. Zbl0391.35062MR507931
  13. [13] L. Hörmander – “Private communications”, september 2002 – august 2004. 
  14. [14] —, “On the theory of general partial differential operators”, Acta Math.94 (1955), p. 161–248. Zbl0067.32201MR76151
  15. [15] —, “Differential equations without solutions”, Math. Ann.140 (1960), p. 169–173. Zbl0093.28903MR147765
  16. [16] —, “Pseudo-differential operators and non-elliptic boundary problems”, Ann. of Math. (2) 83 (1966), p. 129–209. Zbl0132.07402MR233064
  17. [17] —, “Propagation of singularities and semiglobal existence theorems for (pseudo)differential operators of principal type”, Ann. of Math. (2) 108 (1978), no. 3, p. 569–609. Zbl0396.35087MR512434
  18. [18] —, Pseudo-differential operators of principal type. Singularities in boundary value problems, D. Reidel Publ. Co., Dortrecht, Boston, London, 1981. 
  19. [19] —, The analysis of linear partial differential operators I–IV, Grundlehren der Mathematischen Wissenschaften, vols. 256-257, 274-275, Springer-Verlag, Berlin, 1983. Zbl0612.35001MR717035
  20. [20] —, Notions of convexity, Progress in Mathematics, vol. 127, Birkhäuser Boston Inc., Boston, MA, 1994. Zbl0835.32001MR1301332
  21. [21] —, “On the solvability of pseudodifferential equations. Structure of solutions of differential equations”, in Proceedings of the Taniguchi Symposium held in Katata, June 26–30, 1995, and the RIMS Symposium held at Kyoto University, Kyoto, July 3–7, 1995 (M. Morimoto & T. Kawai, éds.), World Scientific Publishing Co. Inc., River Edge, NJ, 1996. Zbl0882.00037MR1445329
  22. [22] N. Lerner – “Cutting the loss of derivatives for solvability under condition ( ψ )”, http://hal.ccsd.cnrs.fr/ccsd-00016103, december 2005, to appear in Bull. Soc. Math. France. Zbl1181.35355MR2364944
  23. [23] —, “Sufficiency of condition ( ψ ) for local solvability in two dimensions”, Ann. of Math. (2) 128 (1988), no. 2, p. 243–258. Zbl0682.35112MR960946
  24. [24] —, “An iff solvability condition for the oblique derivative problem”, Séminaire EDP, École polytechnique, exposé 18, 1990–91. 
  25. [25] —, “Nonsolvability in L 2 for a first order operator satisfying condition ( ψ ) ”, Ann. of Math. (2) 139 (1994), no. 2, p. 363–393. Zbl0818.35152MR1274095
  26. [26] —, “Energy methods via coherent states and advanced pseudo-differential calculus”, in Multidimensional complex analysis and partial differential equations (São Carlos, 1995), Contemp. Math., vol. 205, Amer. Math. Soc., Providence, 1997, p. 177–201. Zbl0885.35152MR1447224
  27. [27] —, “Perturbation and energy estimates”, Ann. Sci. École Norm. Sup. (4) 31 (1998), no. 6, p. 843–886. Zbl0927.35139MR1664214
  28. [28] —, “When is a pseudo-differential equation solvable?”, Ann. Inst. Fourier (Grenoble) 50 (2000), no. 2, p. 443–460. Zbl0952.35166MR1775357
  29. [29] —, “Solving pseudo-differential equations”, in Proceedings of the International Congress of Mathematicians II (Beijing 2002), Higher Ed. Press, 2002, p. 711–720. Zbl1106.35337MR1957078
  30. [30] H. Lewy – “An example of a smooth linear partial differential equation without solution”, Ann. of Math. (2) 66 (1957), p. 155–158. Zbl0078.08104MR88629
  31. [31] S. Mizohata – “Solutions nulles et solutions non analytiques”, J. Math. Kyoto Univ. 1 (1961/1962), p. 271–302. Zbl0106.29601MR142873
  32. [32] R. D. Moyer – “Local solvability in two dimensions: necessary conditions for the principal type case”, mimeographed manuscript, University of Kansas, 1978. 
  33. [33] L. Nirenberg & F. Treves – “Solvability of a first order linear partial differential equation”, Comm. Pure Appl. Math.16 (1963), p. 331–351. Zbl0117.06104MR163045
  34. [34] —, “On local solvability of linear partial differential equations I. Necessary conditions”, Comm. Pure Appl. Math.23 (1970), p. 1–38. Zbl0191.39103MR264470
  35. [35] —, “On local solvability of linear partial differential equations II. Sufficient conditions”, Comm. Pure Appl. Math.23 (1970), p. 459–509. Zbl0208.35902MR264471
  36. [36] —, “A correction to: “On local solvability of linear partial differential equations II. Sufficient conditions” (Comm. Pure Appl. Math. 23 (1970), p. 459–509)”, Comm. Pure Appl. Math. 24 (1971), no. 2, p. 279–288. Zbl0221.35019MR435641
  37. [37] J.-M. Trépreau – “Sur la résolubilité analytique microlocale des opérateurs pseudo-différentiels de type principal”, Thèse, Université de Reims, 1984. 

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.