On the solvability of pseudodifferential operators
Séminaire Équations aux dérivées partielles (Polytechnique) (2005-2006), Talk no. 1, 27 p.
@article{SEDP_2005-2006____A1_0,
author = {Dencker, Nils},
title = {On the solvability of pseudodifferential operators},
journal = {S\'eminaire \'Equations aux d\'eriv\'ees partielles (Polytechnique)},
publisher = {Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
year = {2005-2006},
note = {talk:1},
mrnumber = {2276067},
language = {en},
url = {http://www.numdam.org/item/SEDP_2005-2006____A1_0}
}

Dencker, Nils. On the solvability of pseudodifferential operators. Séminaire Équations aux dérivées partielles (Polytechnique) (2005-2006), Talk no. 1, 27 p. http://www.numdam.org/item/SEDP_2005-2006____A1_0/

[1] Beals, R. and C. Fefferman, On local solvability of linear partial differential equations, Ann. of Math. 97 (1973), 482–498. | MR 352746 | Zbl 0256.35002

[2] Bony, J.-M. and J.-Y. Chemin, Espaces fonctionnels associés au calcul de Weyl-Hörmander, Bull. Soc. Math. France 122 (1994), 77–118. | Numdam | MR 1259109 | Zbl 0798.35172

[3] Dencker, N., On the propagation of singularities for pseudo-differential operators of principal type, Ark. Mat. 20 (1982), 23–60. | MR 660124 | Zbl 0503.58031

[4] —, The solvability of non ${L}^{2}$ solvable operators, Journées “Equations aux Dérivées Partielles” (Saint-Jean-de-Monts, 1996), Exp. No. X, 11 pp., Ecole Polytech., Palaiseau, 1996. | Numdam | Zbl 0885.35151

[5] —, A sufficient condition for solvability, International Mathematics Research Notices 1999:12 (1999), 627–659. | MR 1699215 | Zbl 0947.58019

[6] —, On the sufficiency of condition $\left(\psi \right)$, Report 2001:11, Centre for Mathematical Sciences, Lund University.

[7] —, The solvability of pseudo-differential operators, Phase space analysis of partial differential equations, Vol. I, Pubbl. Cent. Ric. Mat. Ennio Giorgi, Scuola Norm. Sup., Pisa, 2004, 175–200. | MR 2144409 | Zbl 1079.35105

[8] —, The resolution of the Nirenberg-Treves conjecture, Ann. of Math. 163 (2006), 405–444. | MR 2199222 | Zbl 1104.35080

[9] Hörmander, L., The Weyl calculus of pseudo-differential operators, Comm. Pure Appl. Math. 32 (1979), 359–443. | MR 517939 | Zbl 0388.47032

[10] —, Pseudo-differential operators of principal type, Singularities in boundary value problems (Proc. NATO Adv. Study Inst., Maratea, 1980) NATO Adv. Study Inst. Ser. C: Math. Phys. Sci., 65, Reidel, Dordrecht-Boston, Mass., 1981, 69–96. | Zbl 0459.35096

[11] —, The analysis of linear partial differential operators, vol. I–IV, Springer Verlag, Berlin, 1983–1985.

[12] —, Notions of convexity, Birkhäuser, Boston, 1994. | MR 1301332

[13] —, On the solvability of pseudodifferential equations, Structure of solutions of differential equations (M. Morimoto and T. Kawai, eds.), World Scientific, New Jersey, 1996, 183–213. | MR 1445329 | Zbl 0897.35082

[14] —, The proof of the Nirenberg–Treves conjecture according to N. Dencker och N. Lerner, Preprint.

[15] Lerner, N., Sufficiency of condition $\left(\psi \right)$ for local solvability in two dimensions, Ann. of Math. 128 (1988), 243–258. | MR 960946 | Zbl 0682.35112

[16] —, Nonsolvability in ${L}^{2}$ for a first order operator satisfying condition $\left(\psi \right)$, Ann. of Math. 139 (1994), 363–393. | MR 1274095 | Zbl 0818.35152

[17] —, 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, 177–201. | MR 1447224 | Zbl 0885.35152

[18] —, Perturbation and energy estimates, Ann. Sci. École Norm. Sup. 31 (1998), 843–886. | Numdam | MR 1664214 | Zbl 0927.35139

[19] —, The Wick calculus of pseudo-differential operators and some of its applications. Cubo Mat. Educ. 5 (2003), 213–236. | MR 1957713 | Zbl pre05508173

[20] —, Cutting the loss of derivatives for solvability under condition ($\Psi$), Preprint.

[21] Lewy, H. An example of a smooth linear partial differential equation without solution, Ann. of Math. 66 (1957), 155–158. | MR 88629 | Zbl 0078.08104

[22] Moyer, R.D., Local solvability in two dimensions: Necessary conditions for the principal-type case, Mimeographed manuscript, University of Kansas, 1978.

[23] Nirenberg, L. and F. Treves, On local solvability of linear partial differential equations. Part I: Necessary conditions, Comm. Pure Appl. Math. 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. | Zbl 0191.39103

[24] Trépreau, J.M., Sur la résolubilité analytique microlocale des opérateurs pseudodifférentiels de type principal, Ph.D. thesis, Université de Reims, 1984.