Cutting the loss of derivatives for solvability under condition ( Ψ )

Nicolas Lerner

Bulletin de la Société Mathématique de France (2006)

  • Volume: 134, Issue: 4, page 559-631
  • ISSN: 0037-9484

Abstract

top
For a principal type pseudodifferential operator, we prove that condition  ( ψ ) 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 ϵ + 3 / 2 for any ϵ > 0 (Dencker’s most recent result) to 3/2 (the present paper). It is already known that condition  ( ψ ) doesnotimply local solvability with a loss of 1 derivative, so we have to content ourselves with a loss > 1 .

How to cite

top

Lerner, Nicolas. "Cutting the loss of derivatives for solvability under condition $(\Psi )$." Bulletin de la Société Mathématique de France 134.4 (2006): 559-631. <http://eudml.org/doc/272406>.

@article{Lerner2006,
abstract = {For a principal type pseudodifferential operator, we prove that condition $(\psi )$ 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 $\epsilon +3/2$ for any $\epsilon &gt;0$ (Dencker’s most recent result) to 3/2 (the present paper). It is already known that condition $(\psi )$ doesnotimply local solvability with a loss of 1 derivative, so we have to content ourselves with a loss $&gt;1$.},
author = {Lerner, Nicolas},
journal = {Bulletin de la Société Mathématique de France},
keywords = {solvability; a priori estimates; pseudodifferential operators},
language = {eng},
number = {4},
pages = {559-631},
publisher = {Société mathématique de France},
title = {Cutting the loss of derivatives for solvability under condition $(\Psi )$},
url = {http://eudml.org/doc/272406},
volume = {134},
year = {2006},
}

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
PB - Société mathématique de France
VL - 134
IS - 4
SP - 559
EP - 631
AB - For a principal type pseudodifferential operator, we prove that condition $(\psi )$ 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 $\epsilon +3/2$ for any $\epsilon &gt;0$ (Dencker’s most recent result) to 3/2 (the present paper). It is already known that condition $(\psi )$ doesnotimply local solvability with a loss of 1 derivative, so we have to content ourselves with a loss $&gt;1$.
LA - eng
KW - solvability; a priori estimates; pseudodifferential operators
UR - http://eudml.org/doc/272406
ER -

References

top
  1. [1] R. Beals & C. Fefferman – « On local solvability of linear partial differential equations », Ann. of Math.97 (1973), p. 482–498. Zbl0256.35002MR352746
  2. [2] J. Bony & J. Chemin – « Espaces fonctionnels associés au calcul de Weyl-Hörmander », Bull Soc. Math. France122 (1994), p. 77–118. Zbl0798.35172MR1259109
  3. [3] J. Bony & N. Lerner – « Quantification asymptotique et microlocalisations d’ordre supérieur », Ann. Sci. Éc. Norm. Sup. 22 (1989), p. 377–433. Zbl0753.35005MR1011988
  4. [4] N. Dencker – « Estimates and solvability », Ark. Mat.37 (1999), p. 221–243. Zbl1021.35137MR1714771
  5. [5] —, « On the sufficiency of condition ( ψ ) », preprint, May 22 2001. 
  6. [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. Zbl1079.35105MR2144409
  7. [7] —, « The resolution of the Nirenberg-Treves conjecture », Ann. of Math.163 (2006), p. 405–444. Zbl1104.35080MR2199222
  8. [8] C. Fefferman & D. Phong – « On positivity of pseudo-differential equations », Proc. Nat. Acad. Sci.75 (1978), p. 4673–4674. Zbl0391.35062MR507931
  9. [9] L. Hörmander – « On the theory of general partial differential operators », Acta Math.94 (1955), p. 161–248. Zbl0067.32201MR76151
  10. [10] —, « Differential equations without solutions », Math. Ann.140 (1960), p. 169–173. Zbl0093.28903MR147765
  11. [11] —, « Pseudo-differential operators and non-elliptic boundary value problems », Ann. of Math.83 (1966), p. 129–209. Zbl0132.07402MR233064
  12. [12] —, « Propagation of singularities and semiglobal existence theorems for (pseudo-)differential operators of principal type », Ann. of Math.108 (1978), p. 569–609. Zbl0396.35087MR512434
  13. [13] —, « Pseudo-differential operators of principal type », Singularities in boundary value problems, D. Reidel Publ. Co., Dortrecht, Boston, London, 1981. 
  14. [14] —, The analysis of linear partial differential operators I-IV, Springer Verlag, 1983-85. Zbl0601.35001
  15. [15] —, Notions of convexity, Birkhäuser, 1994. MR1301332
  16. [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. Zbl0897.35082MR1445329
  17. [17] —, « private communications », september 2002 – august 2004. 
  18. [18] N. Lerner – « Sufficiency of condition ( ψ ) for local solvability in two dimensions », Ann. of Math.128 (1988), p. 243–258. Zbl0682.35112MR960946
  19. [19] —, « An iff solvability condition for the oblique derivative problem », Séminaire EDP, École polytechnique, 1990-91, exposé 18. 
  20. [20] —, « Nonsolvability in L 2 for a first order operator satisfying condition ( ψ ) », Ann. of Math.139 (1994), p. 363–393. Zbl0818.35152MR1274095
  21. [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. Zbl0885.35152MR1447224
  22. [22] —, « Perturbation and energy estimates », Ann. Sci. Éc. Norm. Sup. 31 (1998), p. 843–886. Zbl0927.35139MR1664214
  23. [23] —, « When is a pseudo-differential equation solvable? », Ann. Inst. Fourier (Grenoble) 50 (2000), p. 443–460. Zbl0952.35166MR1775357
  24. [24] —, « Solving pseudo-differential equations », Proceedings of the ICM 2002 in Beijing, vol. II, Higher Education Press, 2002, p. 711–720. Zbl1156.35476MR1957078
  25. [25] H. Lewy – « An example of a smooth linear partial differential equation without solution », Ann. of Math.66 (1957), p. 155–158. Zbl0078.08104MR88629
  26. [26] S. Mizohata – « Solutions nulles et solutions non analytiques », J. Math. Kyoto Univ.1 (1962), p. 271–302. Zbl0106.29601MR142873
  27. [27] R. Moyer – « Local solvability in two dimensions: necessary conditions for the principal type case », mimeographed manuscript, University of Kansas, 1978. 
  28. [28] L. Nirenberg & F. Treves – « Solvability of a first order linear partial differential equation », Comm. Pure Appl. Math.16 (1963), p. 331–351. Zbl0117.06104MR163045
  29. [29] —, « On local solvability of linear partial differential equations. I. Necessary conditions », Comm. Pure Appl. Math.23 (1970), p. 1–38. Zbl0191.39103MR264470
  30. [30] —, « On local solvability of linear partial differential equations. II.Sufficient conditions », Comm. Pure Appl. Math.23 (1970), p. 459–509. Zbl0208.35902MR264471
  31. [31] —, « On local solvability of linear partial differential equations. Correction », Comm. Pure Appl. Math.24 (1971), p. 279–288. Zbl0221.35019MR435641
  32. [32] 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.