manuscripta mathematica

, Volume 94, Issue 1, pp 151–167 | Cite as

Local a priori estimates in Lp for first order linear operators with nonsmooth coefficients

  • Jorge Hounie
  • Maria Eulália Moraes Melo


We prove local a priori estimates inL p , 1<p<∞, for first-order linear operators that satisfy the Nirenberg-Treves condition (p) and whose coefficients have Lipschitz continuous derivatives of order one. When the number of variables is two, only Lipschitz continuity of the coefficients is assumed. This extends toL p spaces estimates that were previously known forp=2. Examples show that the regularity required from the coefficients is essentially minimal.


Lipschitz Function Lipschitz Continuity Principal Symbol Principal Type Local Solvability 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [BF]
    Beals, R. and Fefferman, C.On local solvability of partial differential equations, Ann. Math. 97 (1973),552–571.MathSciNetCrossRefGoogle Scholar
  2. [Gu]
    Guan, P.Hölder regularity of subelliptic pseudo-differential operators, Dissertation for the degree of Doctor of Philosophy, (1989), Princeton University.Google Scholar
  3. [Hor]
    Hörmander, L.Pseudo-differential equations of principal type, Singularities in Boundary Value Problems, NATO Adv. Study Inst. Ser., Nijhoff, The Haghe(1981).Google Scholar
  4. [H]
    Hounie, J.Local solvability of first order linear operators with Lipschitz coefficients, Duke Math. J. 62 (1991), 467–477.zbMATHCrossRefMathSciNetGoogle Scholar
  5. [HM]
    Hounie, J. and Moraes Melo, M.E.,Local solvability of first order linear operators with Lipschitz coefficients in two variables, J. of Diff. Equations,121 (1995), 406–416.zbMATHCrossRefMathSciNetGoogle Scholar
  6. [HP]
    Hounie, J. and Perdigão, E.,On local solvability in L p of first-order equations, J. of Math. An. and Appl., to appear.Google Scholar
  7. [J]
    Jacobowitz, H.A nonsolvable complex vector field with Hölder coefficients, Proc. Amer. Math. Soc. 116 (1992), 787–795.zbMATHCrossRefMathSciNetGoogle Scholar
  8. [K]
    Kenig, C.Progress on two problems posed by Rivière, Contemp. Math. 107 (1990), 101–107.MathSciNetGoogle Scholar
  9. [L]
    Lerner, N.Sufficiency of condition (ω) for local solvability in two dimensions, Ann. of Math. 128 (1988), 243–258.CrossRefMathSciNetGoogle Scholar
  10. [Mo]
    Moyer, R.D.Local solvability in two dimensions: Necessary conditions for the principal type case, Mimeographed manuscript.University of Kansas, (1978), (9 pages).Google Scholar
  11. [NT1]
    Nirenberg, L. and Treves F.,Solvability of a first order linear partial differental equation, Comm. Pure Appl. Math. 16 (1963), 331–351.zbMATHCrossRefMathSciNetGoogle Scholar
  12. [NT2]
    Nirenberg, L. and Treves, F.On local solvability of linear partial differential equations, I:Necessary conditions, II:Sufficient conditions, Comm. Pure Applied Math. 23 (1970), 1–38; 459–510. Correction, ibid.24 (1971), 279–278.zbMATHCrossRefMathSciNetGoogle Scholar
  13. [P]
    Perdigão, E.,Resolubilidade local em L p de operadores diferenciais de primeira ordem, Dissertation for the degree of Doctor of Philosophy, (1993), UFPE.Google Scholar
  14. [S]
    Smith, H.An elementary proof of local solvability in two dimensions under condition (ω), Ann. of Math.136 (1992), 335–337.CrossRefMathSciNetGoogle Scholar
  15. [St]
    Stein, E. M.,Singular integrals and differentiability properties of functions, Princeton, N.J., Princeton Univ. Press, 1970.zbMATHGoogle Scholar
  16. [T]
    Treves, F.,Local solvability in L 2 of first order linear PDEs,Amer. J. Math. 92 (1970), 369–380.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 1997

Authors and Affiliations

  • Jorge Hounie
    • 1
  • Maria Eulália Moraes Melo
    • 1
    • 2
  1. 1.Departamento de MatemáticaUniv. Fed. de São CarlosSão Carlos, SPBrasil
  2. 2.Recife, PEBrasil

Personalised recommendations