For a principal type pseudodifferential operator, we prove that condition implies local solvability with a loss of 3/2 derivatives. We use many elements of Dencker’s paper on the proof of the Nirenberg-Treves conjecture and we provide some improvements of the key energy estimates which allows us to cut the loss of derivatives from for any (Dencker’s most recent result) to 3/2 (the present paper). It is already known that condition does not imply local solvability with a loss of 1 derivative, so we have to content ourselves with a loss .
Pour un opérateur de type principal, nous démontrons que la condition () implique la résolubilité locale avec perte de 3/2 dérivées. Nous utilisons beaucoup d’éléments de la démonstration par Dencker de la conjecture de Nirenberg-Treves et nous limitons la perte de dérivées à 3/2, améliorant le résultat le plus récent de Dencker (perte de dérivées pour tout ). La condition () n’impliquant pas la résolubilité locale avec perte d’une dérivée, nous devons nous contenter d’une perte .
Keywords: solvability, a priori estimates, pseudodifferential operators
Mot clés : résolubilité, estimations a priori, op´erateurs pseudodifférentiels
@article{BSMF_2006__134_4_559_0, author = {Lerner, Nicolas}, title = {Cutting the loss of derivatives for solvability under condition $(\Psi )$}, journal = {Bulletin de la Soci\'et\'e Math\'ematique de France}, pages = {559--631}, publisher = {Soci\'et\'e math\'ematique de France}, volume = {134}, number = {4}, year = {2006}, doi = {10.24033/bsmf.2522}, mrnumber = {2364944}, zbl = {1181.35355}, language = {en}, url = {http://archive.numdam.org/articles/10.24033/bsmf.2522/} }
TY - JOUR AU - Lerner, Nicolas TI - Cutting the loss of derivatives for solvability under condition $(\Psi )$ JO - Bulletin de la Société Mathématique de France PY - 2006 SP - 559 EP - 631 VL - 134 IS - 4 PB - Société mathématique de France UR - http://archive.numdam.org/articles/10.24033/bsmf.2522/ DO - 10.24033/bsmf.2522 LA - en ID - BSMF_2006__134_4_559_0 ER -
%0 Journal Article %A Lerner, Nicolas %T Cutting the loss of derivatives for solvability under condition $(\Psi )$ %J Bulletin de la Société Mathématique de France %D 2006 %P 559-631 %V 134 %N 4 %I Société mathématique de France %U http://archive.numdam.org/articles/10.24033/bsmf.2522/ %R 10.24033/bsmf.2522 %G en %F BSMF_2006__134_4_559_0
Lerner, Nicolas. Cutting the loss of derivatives for solvability under condition $(\Psi )$. Bulletin de la Société Mathématique de France, Volume 134 (2006) no. 4, pp. 559-631. doi : 10.24033/bsmf.2522. http://archive.numdam.org/articles/10.24033/bsmf.2522/
[1] « On local solvability of linear partial differential equations », Ann. of Math. 97 (1973), p. 482-498. | MR | Zbl
& -[2] « Espaces fonctionnels associés au calcul de Weyl-Hörmander », Bull Soc. Math. France 122 (1994), p. 77-118. | Numdam | MR | Zbl
& -[3] « Quantification asymptotique et microlocalisations d'ordre supérieur », Ann. Sci. Éc. Norm. Sup. 22 (1989), p. 377-433. | Numdam | MR | Zbl
& -[4] « Estimates and solvability », Ark. Mat. 37 (1999), p. 221-243. | MR | Zbl
-[5] -, « On the sufficiency of condition », preprint, May 22 2001.
[6] -, « The solvability of pseudo-differential operators », Phase space analysis of partial differential equations, Pubbl. Cent. Ric. Mat. Ennio Giorgi, vol. 1, Sc. Norm. Sup., Pisa, 2004, p. 175-200. | MR | Zbl
[7] -, « The resolution of the Nirenberg-Treves conjecture », Ann. of Math. 163 (2006), p. 405-444. | MR | Zbl
[8] « On positivity of pseudo-differential equations », Proc. Nat. Acad. Sci. 75 (1978), p. 4673-4674. | MR | Zbl
& -[9] « On the theory of general partial differential operators », Acta Math. 94 (1955), p. 161-248. | MR | Zbl
-[10] -, « Differential equations without solutions », Math. Ann. 140 (1960), p. 169-173. | EuDML | MR
[11] -, « Pseudo-differential operators and non-elliptic boundary value problems », Ann. of Math. 83 (1966), p. 129-209. | MR | Zbl
[12] -, « Propagation of singularities and semiglobal existence theorems for (pseudo-)differential operators of principal type », Ann. of Math. 108 (1978), p. 569-609. | MR | Zbl
[13] -, « Pseudo-differential operators of principal type », Singularities in boundary value problems, D. Reidel Publ. Co., Dortrecht, Boston, London, 1981.
[14] -, The analysis of linear partial differential operators I-IV, Springer Verlag, 1983-85. | Zbl
[15] -, Notions of convexity, Birkhäuser, 1994. | MR
[16] -, « On the solvability of pseudodifferential equations », Structure of solutions of differential equations (M. Morimoto & T. Kawai, éds.), World Sci. Publishing, River Edge, NJ, 1996, p. 183-213. | MR | Zbl
[17] -, « private communications », september 2002 - august 2004.
[18] « Sufficiency of condition for local solvability in two dimensions », Ann. of Math. 128 (1988), p. 243-258. | MR | Zbl
-[19] -, « An iff solvability condition for the oblique derivative problem », Séminaire EDP, École polytechnique, 1990-91, exposé 18.
[20] -, « Nonsolvability in for a first order operator satisfying condition », Ann. of Math. 139 (1994), p. 363-393. | MR | Zbl
[21] -, « Energy methods via coherent states and advanced pseudo-differential calculus », Multidimensional complex analysis and partial differential equations (P. Cordaro, H. Jacobowitz & S. Gindikin, éds.), Amer. Math. Soc., 1997, p. 177-201. | MR | Zbl
[22] -, « Perturbation and energy estimates », Ann. Sci. Éc. Norm. Sup. 31 (1998), p. 843-886. | EuDML | Numdam | MR | Zbl
[23] -, « When is a pseudo-differential equation solvable? », Ann. Inst. Fourier (Grenoble) 50 (2000), p. 443-460. | EuDML | Numdam | MR | Zbl
[24] -, « Solving pseudo-differential equations », Proceedings of the ICM 2002 in Beijing, vol. II, Higher Education Press, 2002, p. 711-720. | MR | Zbl
[25] « An example of a smooth linear partial differential equation without solution », Ann. of Math. 66 (1957), p. 155-158. | MR | Zbl
-[26] « Solutions nulles et solutions non analytiques », J. Math. Kyoto Univ. 1 (1962), p. 271-302. | MR | Zbl
-[27] « Local solvability in two dimensions: necessary conditions for the principal type case », mimeographed manuscript, University of Kansas, 1978.
-[28] « Solvability of a first order linear partial differential equation », Comm. Pure Appl. Math. 16 (1963), p. 331-351. | MR | Zbl
& -[29] -, « On local solvability of linear partial differential equations. I. Necessary conditions », Comm. Pure Appl. Math. 23 (1970), p. 1-38. | MR | Zbl
[30] -, « On local solvability of linear partial differential equations. II.Sufficient conditions », Comm. Pure Appl. Math. 23 (1970), p. 459-509. | MR | Zbl
[31] -, « On local solvability of linear partial differential equations. Correction », Comm. Pure Appl. Math. 24 (1971), p. 279-288. | MR | Zbl
[32] « Sur la résolubilité analytique microlocale des opérateurs pseudo-différentiels de type principal », Thèse, Université de Reims, 1984.
-Cited by Sources: