The verification of the Nirenberg-Treves conjecture
Séminaire Bourbaki : volume 2005/2006, exposés 952-966, Astérisque no. 311  (2007), Talk no. 960, p. 211-235

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.

Dans une série de preprints récents, Nils Dencker démontre que la condition (Ψ) implique la résolubilité locale des opérateurs pseudodifférentiels de type principal (complexe) avec une perte de deux dérivées, établissant la dernière partie de la conjecture de Nirenberg-Treves, formulée en 1971. L’origine de cette question remonte au contre-exemple de Hans Lewy, publié en 1957. Nous suivrons dans notre exposé une partie du développement de l’analyse microlocale afférente et mettrons en évidence les nouvelles idées géométriques apportées par Dencker.

Classification:  35S05,  47G30
Keywords: résolubilité, opérateurs pseudodifférentiels, estimations d'énergie, opérateurs non autoadjoints
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     author = {Lerner, Nicolas},
     title = {The verification of the Nirenberg-Treves conjecture},
     booktitle = {S\'eminaire Bourbaki : volume 2005/2006, expos\'es 952-966},
     author = {Collectif},
     series = {Ast\'erisque},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {311},
     year = {2007},
     note = {talk:960},
     pages = {211-235},
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Lerner, Nicolas. The verification of the Nirenberg-Treves conjecture, in Séminaire Bourbaki : volume 2005/2006, exposés 952-966, Astérisque, no. 311 (2007), Talk no. 960, pp. 211-235. http://www.numdam.org/item/SB_2005-2006__48__211_0/

[1] G. Bachelard - La formation de l'esprit scientifique, Vrin, Paris, 1938.

[2] R. Beals & C. Fefferman - “On local solvability of linear partial differential equations”, Ann. of Math. (2) 97 (1973), p. 482-498. | MR 352746 | Zbl 0256.35002

[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. | Numdam | MR 262881 | Zbl 0176.09703

[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. | Numdam | MR 1259109 | Zbl 0798.35172

[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. | Numdam | MR 1011988 | Zbl 0753.35005

[6] H. Brézis - “On a characterization of flow-invariant sets”, Comm. Pure Appl. Math. 23 (1970), p. 261-263. | Article | MR 257511 | Zbl 0191.38703

[7] N. Dencker - “On the sufficiency of condition (ψ), preprint, May 22, 2001.

[8] -, The solvability of non-L 2 -solvable operators, 1996, Saint Jean de Monts meeting. | Numdam | Zbl 0885.35151

[9] -, “Estimates and solvability”, Ark. Mat. 37 (1999), no. 2, p. 221-243. | MR 1714771 | Zbl 1021.35137

[10] -, “The solvability of pseudodifferential operators”, in Phase space analysis of PDE, Centro de Giorgi, Scuola Normale Superiore, Pisa, 2004, p. 175-200.

[11] -, “The resolution of the Nirenberg-Treves conjecture”, Ann. of Math. (2) 163 (2006), p. 405-444. | MR 2199222 | Zbl 1104.35080

[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. | MR 507931 | Zbl 0391.35062

[13] L. Hörmander - “Private communications”, september 2002 - august 2004.

[14] -, “On the theory of general partial differential operators”, Acta Math. 94 (1955), p. 161-248. | Article | MR 76151 | Zbl 0067.32201

[15] -, “Differential equations without solutions”, Math. Ann. 140 (1960), p. 169-173. | Article | MR 147765

[16] -, “Pseudo-differential operators and non-elliptic boundary problems”, Ann. of Math. (2) 83 (1966), p. 129-209. | MR 233064 | Zbl 0132.07402

[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. | MR 512434 | Zbl 0396.35087

[18] -, Pseudo-differential operators of principal type. Singularities in boundary value problems, D. Reidel Publ. Co., Dortrecht, Boston, London, 1981.

[19] -, The analysis of linear partial differential operators I-IV, Grundlehren der Mathematischen Wissenschaften, vols. 256-257, 274-275, Springer-Verlag, Berlin, 1983. | MR 717035 | Zbl 0612.35001

[20] -, Notions of convexity, Progress in Mathematics, vol. 127, Birkhäuser Boston Inc., Boston, MA, 1994. | MR 1301332

[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. | MR 1445329 | Zbl 0882.00037

[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. | Numdam | MR 2364944 | Zbl 1181.35355

[23] -, “Sufficiency of condition (ψ) for local solvability in two dimensions”, Ann. of Math. (2) 128 (1988), no. 2, p. 243-258. | MR 960946 | Zbl 0682.35112

[24] -, “An iff solvability condition for the oblique derivative problem”, Séminaire EDP, École polytechnique, exposé 18, 1990-91.

[25] -, “Nonsolvability in L 2 for a first order operator satisfying condition (ψ), Ann. of Math. (2) 139 (1994), no. 2, p. 363-393. | MR 1274095 | Zbl 0818.35152

[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. | MR 1447224 | Zbl 0885.35152

[27] -, “Perturbation and energy estimates”, Ann. Sci. École Norm. Sup. (4) 31 (1998), no. 6, p. 843-886. | Numdam | MR 1664214 | Zbl 0927.35139

[28] -, “When is a pseudo-differential equation solvable?”, Ann. Inst. Fourier (Grenoble) 50 (2000), no. 2, p. 443-460. | Numdam | MR 1775357 | Zbl 0952.35166

[29] -, “Solving pseudo-differential equations”, in Proceedings of the International Congress of Mathematicians II (Beijing 2002), Higher Ed. Press, 2002, p. 711-720. | MR 1957078 | Zbl 1106.35337

[30] H. Lewy - “An example of a smooth linear partial differential equation without solution”, Ann. of Math. (2) 66 (1957), p. 155-158. | MR 88629 | Zbl 0078.08104

[31] S. Mizohata - “Solutions nulles et solutions non analytiques”, J. Math. Kyoto Univ. 1 (1961/1962), p. 271-302. | MR 142873 | Zbl 0106.29601

[32] R. D. Moyer - “Local solvability in two dimensions: necessary conditions for the principal type case”, mimeographed manuscript, University of Kansas, 1978.

[33] L. Nirenberg & F. Treves - “Solvability of a first order linear partial differential equation”, Comm. Pure Appl. Math. 16 (1963), p. 331-351. | Article | MR 163045 | Zbl 0117.06104

[34] -, “On local solvability of linear partial differential equations I. Necessary conditions”, Comm. Pure Appl. Math. 23 (1970), p. 1-38. | Article | MR 264470 | Zbl 0191.39103

[35] -, “On local solvability of linear partial differential equations II. Sufficient conditions”, Comm. Pure Appl. Math. 23 (1970), p. 459-509. | Article | MR 264471 | Zbl 0208.35902

[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. | MR 435641 | Zbl 0221.35019

[37] J.-M. Trépreau - “Sur la résolubilité analytique microlocale des opérateurs pseudo-différentiels de type principal”, Thèse, Université de Reims, 1984.