Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Beals, R. and Fefferman, C.On local solvability of partial differential equations, Ann. Math. 97 (1973),552–571.
Guan, P.Hölder regularity of subelliptic pseudo-differential operators, Dissertation for the degree of Doctor of Philosophy, (1989), Princeton University.
Hörmander, L.Pseudo-differential equations of principal type, Singularities in Boundary Value Problems, NATO Adv. Study Inst. Ser., Nijhoff, The Haghe(1981).
Hounie, J.Local solvability of first order linear operators with Lipschitz coefficients, Duke Math. J. 62 (1991), 467–477.
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.
Hounie, J. and Perdigão, E.,On local solvability in L p of first-order equations, J. of Math. An. and Appl., to appear.
Jacobowitz, H.A nonsolvable complex vector field with Hölder coefficients, Proc. Amer. Math. Soc. 116 (1992), 787–795.
Kenig, C.Progress on two problems posed by Rivière, Contemp. Math. 107 (1990), 101–107.
Lerner, N.Sufficiency of condition (ω) for local solvability in two dimensions, Ann. of Math. 128 (1988), 243–258.
Moyer, R.D.Local solvability in two dimensions: Necessary conditions for the principal type case, Mimeographed manuscript.University of Kansas, (1978), (9 pages).
Nirenberg, L. and Treves F.,Solvability of a first order linear partial differental equation, Comm. Pure Appl. Math. 16 (1963), 331–351.
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.
Perdigão, E.,Resolubilidade local em L p de operadores diferenciais de primeira ordem, Dissertation for the degree of Doctor of Philosophy, (1993), UFPE.
Smith, H.An elementary proof of local solvability in two dimensions under condition (ω), Ann. of Math.136 (1992), 335–337.
Stein, E. M.,Singular integrals and differentiability properties of functions, Princeton, N.J., Princeton Univ. Press, 1970.
Treves, F.,Local solvability in L 2 of first order linear PDEs,Amer. J. Math. 92 (1970), 369–380.
Author information
Authors and Affiliations
Additional information
Research partially supported by CNPq.
Rights and permissions
About this article
Cite this article
Hounie, J., Moraes Melo, M.E. Local a priori estimates in Lp for first order linear operators with nonsmooth coefficients. Manuscripta Math 94, 151–167 (1997). https://doi.org/10.1007/BF02677844
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02677844