The proof of the Nirenberg-Treves conjecture

Dencker, Nils

Journées équations aux dérivées partielles (2003), article no. 5, 25 p.

We prove the Nirenberg-Treves conjecture : that for principal type pseudo-differential operators local solvability is equivalent to condition ($\Psi$). This condition rules out certain sign changes of the imaginary part of the principal symbol along the bicharacteristics of the real part. We obtain local solvability by proving a localizable estimate for the adjoint operator with a loss of two derivatives (compared with the elliptic case). The proof involves a new metric in the Weyl (or Beals-Fefferman) calculus. This makes it possible to reduce to the case when the gradient of the imaginary part is non-vanishing, and then the zeroes form a smooth submanifold. The estimate uses a new type of weight, which measures the change of the distance to the zeroes of the imaginary part along the bicharacteristics of the real part between the minima of the curvature of this submanifold. By using condition ($\Psi$) and this weight, we can construct a multiplier which gives the estimate.

[1] Richard Beals and Charles Fefferman, On local solvability of linear partial differential equations, Ann. of Math. 97 (1973), 482-498. | MR | Zbl

[2] Jean-Michel Bony and Jean-Yves Chemin, Espace fonctionnels associés au calcul de Weyl-Hörmander, Bull. Soc. Math. France 122 (1994), 77-118. | Numdam | MR | Zbl

[3] Nils Dencker, On the propagation of singularities for pseudo-differential operators of principal type, Ark. Mat. 20 (1982), 23-60. | MR | Zbl

[4] Nils Dencker, The solvability of non $L^2$ solvable operators, Journees ''Équations aux Dérivées Partielles'', St. Jean de Monts, France, 1996. | Numdam | MR | Zbl

[5] Nils Dencker, A sufficient condition for solvability, International Mathematics Research Notices 1999:12 (1999), 627-659. | MR | Zbl

[6] Nils Dencker, On the sufficiency of condition (Ψ), Report 2001:11, Centre for Mathematical Sciences, Lund University.

[7] Nils Dencker, The resolution of the Nirenberg-Treves conjecture, Report 22, Institute Mittag-Leffler, 2002/2003 fall.

[8] Jean Dieudonné, Foundations of modern analysis, Academic Press, New York and London, 1960. | MR | Zbl

[9] Lars Hörmander, The Weyl calculus of pseudo-differential operators, Comm. Partial Differential Equations 32 (1979), 359-443. | MR | Zbl

[10] Lars Hörmander, The analysis of linear partial differential operators, vol. I-IV, Springer Verlag, Berlin, 1983-1985. | MR

[11] Lars Hörmander, Notions of convexity, Birkhäuser, Boston, 1994. | MR | Zbl

[12] Lars Hörmander, On the solvability of pseudodifferential equations, Structure of solutions of differential equations (M. Morimoto and T. Kawai, eds.), World Scientific, New Jersey, 1996, pp. 183-213. | MR | Zbl

[13] Nicolas Lerner, Sufficiency of condition (Ψ) for local solvability in two dimensions, Ann. of Math. 128 (1988), 243-258. | MR | Zbl

[14] Nicolas Lerner, Nonsolvability in $L^2$ for a first order operator satisfying condition (Ψ), Ann. of Math. 139 (1994), 363-393. | MR | Zbl

[15] Nicolas Lerner, Energy methods via coherent states and advanced pseudo-differential calculus, Multidimensional complex analysis and partial differential equations (P. D. Cordaro, H. Jacobowitz, and S. Gidikin, eds.), Amer. Math. Soc., Providence, R.I., USA, 1997, pp. 177-201. | MR | Zbl

[16] Nicolas Lerner, Perturbation and energy estimates, Ann. Sci. École Norm. Sup. 31 (1998), 843-886. | Numdam | MR | Zbl

[17] Nicolas Lerner, Factorization and solvability, Preprint.

[18] Nicolas Lerner, Private communication.

[19] Louis Nirenberg and François Treves, On local solvability of linear partial differential equations. Part I : Necessary conditions, Comm. Partial Differential Equations 23 (1970), 1-38, Part II: Sufficient conditions, Comm. Pure Appl. Math. 23 (1970), 459-509; Correction, Comm. Pure Appl. Math. 24 (1971), 279-288.

[20] Jean-Marie Trépreau, Sur la résolubilité analytique microlocale des opérateurs pseudodifférentiels de type principal, Ph.D. thesis, Université de Reims, 1984.