Journal d’Analyse Mathématique

, Volume 74, Issue 1, pp 67–97 | Cite as

Lipschitz continuity, global smooth approximations and extension theorems for Sobolev functions in Carnot-Carathéodory spaces

  • Nicola Garofalo
  • Duy-Minh Nhieu


Vector Field Sobolev Space Heisenberg Group LIPSCHITZ Continuity Harnack Inequality 
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. [1]
    A. Bellaïche,Sub-Riemannian Geometry, Birkhäuser, Boston, 1996.zbMATHGoogle Scholar
  2. [2]
    L. Capogna, D. Danielli and N. Garofalo,An embedding theorem and the Harnack inequality for nonlinear subelliptic equations, Comm. Partial Differential Equations18 (1993), 1765–1794.zbMATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    L. Capogna, D. Danielli and N. Garofalo,Capacitary estimates and the local behavior of solutions of nonlinear subelliptic equations, Amer. J. Math.118 (1996), 1153–1196.zbMATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    L. Capogna and N. Garofalo,NTA domains for Carnot-Carathéodory metrics and a Fatou type theorem, preprint, 1995.Google Scholar
  5. [5]
    W. L. Chow,Uber System von linearen partiellen Differentialgleichungen erster Ordnug, Math. Ann.117 (1939), 98–105.zbMATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    M. Christ,The extension problem for certain function spaces involving fractional orders of differentiability, Ark. Mat.22 (1984), 63–81.zbMATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    S. K. Chua,Extension theorems on weighted sobolev spaces, Indiana Univ. Math. J.4 (1992), 1027–1076.CrossRefMathSciNetGoogle Scholar
  8. [8]
    R. Coifman and G. Weiss,Analyse harmonique non-commutative sur certains espaces homogenes, Springer-Verlag, Berlin, 1971.zbMATHGoogle Scholar
  9. [9]
    D. Danielli,Regularity at the boundary for solutions of nonlinear subelliptic equations, Indiana Univ. Math. J.44 (1995), 269–286.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    D. Danielli,A Fefferman-Phong type inequality and applications to quasilinear subelliptic equations, preprint.Google Scholar
  11. [11]
    D. Danielli, N. Garofalo and D. M. Nhieu,Trace inequalities for Carnot-Carathéodory spaces and applications to quasilinear subelliptic equations, preprint.Google Scholar
  12. [12]
    J. Deny and J. L. Lions,Les espaces du type de Beppo Levi, Ann. Inst. Fourier (Grenoble)5 (1953), 305–370.MathSciNetGoogle Scholar
  13. [13]
    L. C. Evans and R. F. Gariepy,Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, 1992.zbMATHGoogle Scholar
  14. [4]
    C. Fefferman and D. H. Phong,Subelliptic eigenvalue problems, inProceedings of the Conference in Harmonic Analysis in Honor of A. Zygmund, Wadsworth Math. Ser., Belmont, CA, 1981, pp. 530–606.Google Scholar
  15. [15]
    C. Fefferman and A. Sanchez-Calle,Fundamental solutions for second order subelliptic operators, Ann. of Math. (2)124 (1986), 247–272.CrossRefMathSciNetGoogle Scholar
  16. [16]
    G. B. Folland and E. M. Stein,Hardy Spaces on Homogeneous Groups, Princeton Univ. Press, 1982.Google Scholar
  17. [17]
    B. Franchi and E. Lanconelli,Une metrique associée à une classe d’operateurs elliptiques degénérés, inProceedings of the Meeting “Linear Partial and Pseudo Differential Operators”, Rend. Sem. Mat. Univ. Politec. Torino, 1982.Google Scholar
  18. [18]
    B. Franchi and E. Lanconelli,Hölder regularity theorem for a class of linear non uniform elliptic operators with measurable coefficients, Ann. Scuola Norm. Sup. Pisa10 (1983), 523–541.zbMATHMathSciNetGoogle Scholar
  19. [19]
    B. Franchi and E. Lanconelli,Une condition géométrique pour l'inégalité de Harnack, J. Math. Pures Appl.64 (1985), 237–256.zbMATHMathSciNetGoogle Scholar
  20. [20]
    B. Franchi, R. Serapioni and F. Serra Cassano,Meyers-Serrin type theorems and relaxation of variational integrals depending on vector fields, Houston J. Math.22 (1996), 859–890.zbMATHMathSciNetGoogle Scholar
  21. [21]
    B. Franchi, R. Serapioni and F. Serra Cassano,Approximation and imbedding theorems for weighted Sobolev spaces associated with Lipschitz continuous vector fields, Boll. Un. Mat. Ital. B (7)11 (1997), 83–117.zbMATHMathSciNetGoogle Scholar
  22. [22]
    K. O. Friedrichs,The identity of weak and strong extensions of differential operators, Trans. Amer. Math. Soc.55 (1944), 132–151.zbMATHCrossRefMathSciNetGoogle Scholar
  23. [23]
    N. Garofalo,Recent Developments in the Theory of Subelliptic Equations and Its Geometric Aspects, Birkhäuser, Boston, to appear.Google Scholar
  24. [24]
    N. Garofalo and D. M. Nhieu,Isoperimetric and Sobolev inequalities for Carnot-Carathéodory spaces and the existence of minimal surfaces, Comm. Pure Appl. Math.49 (1996), 1081–1144.zbMATHCrossRefMathSciNetGoogle Scholar
  25. [25]
    W. Hansen and H. Huber,The Dirichlet problem for sublaplacians on nilpotent groups—Geometric criteria for regularity, Math. Ann.246 (1984), 537–547.Google Scholar
  26. [26]
    P. Hartman,Ordinary Differential Equations, Birkhäuser, Boston, 1982.zbMATHGoogle Scholar
  27. [27]
    E. Hille,Lectures on Ordinary Differential Equations, Addison-Wesley, Reading, Mass., 1968.Google Scholar
  28. [28]
    L. Hörmander,Hypoelliptic second-order differential equations, Acta Math.119 (1967), 147–171.zbMATHCrossRefMathSciNetGoogle Scholar
  29. [29]
    D. Jerison,The Poincaré inequality for vector fields satisfying Hörmander's condition, Duke Math. J.53 (1986), 503–523.zbMATHCrossRefMathSciNetGoogle Scholar
  30. [30]
    D. Jerison and C. E. Kenig,Boundary behavior of harmonic functions in non-tangentially accessible domains, Adv. Math.46 (1982), 80–147.zbMATHCrossRefMathSciNetGoogle Scholar
  31. [31]
    P. W. Jones,Quasiconformal mappings and extendability of functions in Sobolev spaces, Acta Math.147 (1981), 71–88.zbMATHCrossRefMathSciNetGoogle Scholar
  32. [32]
    G. Lu,Embedding theorems into Lipschitz and BMO spaces and applications to quasilinear subelliptic differential equations, Publ. Mat.40 (1996), 301–329.zbMATHMathSciNetGoogle Scholar
  33. [33]
    O. Martio and J. Sarvas,Injectivity theorems in plane and space, Ann. Acad. Sci. Fenn. Ser. A I4 (1979), 384–401.MathSciNetGoogle Scholar
  34. [34]
    N. Meyers and J. Serrin,H=W, Proc. Nat. Acad. Sci. U.S.A.51 (1964), 1055–1056.zbMATHCrossRefMathSciNetGoogle Scholar
  35. [35]
    C. B. Morrey Jr.,Multiple Integrals in the Calculus of Variations, Springer-Verlag, New York, 1966.zbMATHGoogle Scholar
  36. [36]
    A. Nagel, E. M. Stein and S. Wainger,Balls and metrics defined by vector fields I: basic properties, Acta Math.155 (1985), 103–147.zbMATHCrossRefMathSciNetGoogle Scholar
  37. [37]
    D. M. Nhieu,The extension problem for Sobolev spaces on the Heisenberg group, Ph.D Thesis, Purdue University, 1996.Google Scholar
  38. [38]
    O. A. Oleinik and E. V. Radkevich,Second order equations with non-negative characteristic form, inMathematical Analysis 1969, Itogi Nauki, Moscow, 1971 [Russian]; English translation: Amer. Math. Soc., Providence, R.I., 1973.Google Scholar
  39. [39]
    R. S. Phillips and L. Sarason,Elliptic-parabolic equations of the second order, J. Math. Mech.17 (1967/8), 891–917.MathSciNetGoogle Scholar
  40. [40]
    L. Saloff-Coste,A note on Poincaré, Sobolev, and Harnack inequalities, Duke Math. J., I.M.R.N.2 (1992), 27–38.MathSciNetGoogle Scholar
  41. [4]
    N. Th. Varopoulos, L. Saloff-Coste and T. Coulhon,Analysis and Geometry on Groups, Cambridge Tracts in Mathematics 100, Cambridge University Press, 1992.Google Scholar
  42. [42]
    S. K. Vodop'yanov and A. V. Greshnov,On extension of functions of bounded mean oscillation from domains in a space of homogeneous type with intrinsic metric, Siberian Math. J. 36,5 (1995), 873–901.CrossRefMathSciNetGoogle Scholar

Copyright information

© Hebrew University of Jerusalem 1998

Authors and Affiliations

  • Nicola Garofalo
    • 1
    • 2
  • Duy-Minh Nhieu
    • 1
  1. 1.Department of MathematicsPurdue UniversityWest LafayetteUSA
  2. 2.Dipartimento di Metodi e Modelli MatematiciPadovaItaly

Personalised recommendations