Siberian Mathematical Journal

, Volume 49, Issue 2, pp 339–352 | Cite as

Integral representations and the generalized Poincaré inequality on Carnot groups

  • E. A. PlotnikovaEmail author


We give some integral representations of the form f(x) = P(f)+K(∇f) on two-step Carnot groups, where P(f) is a polynomial and K is an integral operator with a specific singularity. We then obtain the weak Poincaré inequality and coercive estimates as well as the generalized Poincaré inequality on the general Carnot groups.


Carnot group integral representation Poincaré inequality 


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  1. 1.
    Sobolev S. L., Some Applications of Functional Analysis in Mathematical Physics [in Russian], Nauka, Moscow (1988).Google Scholar
  2. 2.
    Sobolev S. L., Selected Works. Vol. 2 [in Russian], Izdat. Inst. Mat. Sibirsk. Otdel. Akad. Nauk, Novosibirsk (2006).zbMATHGoogle Scholar
  3. 3.
    Nikol’skii S. M., Approximation of Functions in Several Variables and Embedding Theorems [in Russian], Nauka, Moscow (1977).Google Scholar
  4. 4.
    Stein E. M., Singular Integrals and Differentiability Properties of Functions [Russian translation], Mir, Moscow (1973).zbMATHGoogle Scholar
  5. 5.
    Besov O. V., Il’in V. P., and Nikol’skii S. M., Integral Representations of Functions and Embedding Theorems [in Russian], Nauka, Moscow (1975).zbMATHGoogle Scholar
  6. 6.
    Maz’ya V. G., Sobolev Spaces [in Russian], Leningrad Univ., Leningrad (1985).Google Scholar
  7. 7.
    Hörmander L., “Hypoelliptic second order differential equations,” Acta Math., 119, 147–171 (1967).zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Jerison D., “The Poincaré inequality for vector fields satisfying Hörmander’s condition,” Duke Math. J., 53, No. 2, 503–523 (1986).zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Stein E. M., Harmonic Analysis: Real-Variables Methods, Orthogonality, and Oscillatory Integrals, Princeton Univ. Press, Princeton (1993).Google Scholar
  10. 10.
    Hajłasz P. and Koskela P., “Sobolev met Poincaré,” Mem. Amer. Math. Soc., 145, No. 688, 1–101 (2000).Google Scholar
  11. 11.
    Hajłasz P., “Geometric approach to Sobolev spaces and badly degenerated elliptic equations,” Math. Sci. Appl., 7, 141–168 (1995).Google Scholar
  12. 12.
    Vodop’yanov S. K., “Mappings with bounded distortion and with finite distortion on Carnot groups,” Siberian Math. J., 40, No. 4, 644–677 (1999).CrossRefMathSciNetGoogle Scholar
  13. 13.
    Vodopyanov S. K., “Foundations of the theory of mappings with bounded distortion on Carnot groups,” The Interaction of Analysis and Geometry. Contemporary Mathematics, 424, 303–344 (2007).MathSciNetGoogle Scholar
  14. 14.
    Heinonen J. and Koskela P., “Quasiconformal maps on metric spaces with controlled geometry,” Acta Math., 181, 1–61 (1998).zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Pansu P., “Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un,” Ann. of Math. (2), 128, No. 2, 1–60 (1989).CrossRefMathSciNetGoogle Scholar
  16. 16.
    Romanovskii N. N., “Integral representations and embedding theorems for the functions given on the Heisenberg groups ℍn,” Dokl. Akad. Nauk, 382, No. 4, 456–459 (2002).MathSciNetGoogle Scholar
  17. 17.
    Gol’dshtein V. M. and Reshetnyak Yu. G., An Introduction to the Theory of Functions with Generalized Derivatives and the Quasiconformal Mappings [in Russian], Nauka, Moscow (1983).Google Scholar
  18. 18.
    Reshetnyak Yu. G., Stability Theorems in Geometry and Analysis, Kluwer Academic Publishers, Dordrecht (1994) (Mathematics and Its Applications; 304).zbMATHGoogle Scholar
  19. 19.
    Rotschild G. B., Stein I., “Hypoelliptic differential operators and nilpotent groups,” Acta Math., 137, 247–320 (1976).CrossRefMathSciNetGoogle Scholar
  20. 20.
    Capogna L., Danielli D., and Garofalo N., “An embedding theorem and the Harnack inequality for nonlinear subelliptic equations,” Comm. Partial Differential Equations, 18, No. 9–10, 1765–1794 (1993).zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Capogna L., Danielli D., and Garofalo N., “Capacitary estimates and the local behavior of solutions to nonlinear subelliptic equations,” Amer. J. Math., 118, No. 6, 1153–1196 (1996).zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Franchi B. and Lanconelli E., “Hölder regularity theorem for a class of linear nonuniformly elliptic operators with measurable coefficients,” Ann. Scuola Norm. Sup. Pisa Cl. Sci., 10, No. 4, 523–541 (1983).zbMATHMathSciNetGoogle Scholar
  23. 23.
    Lu G., “Weighted Poincaré and Sobolev inequalities for vector fields satisfying Hörmander’s condition and applications,” Rev. Mat. Iberoamericana, 8, No. 3, 367–439 (1992).zbMATHMathSciNetGoogle Scholar
  24. 24.
    Garofalo N. and Nhieu D. M., “Isoperimetric and Sobolev inequalities for Carnot-Carathéodory spaces and the existence of minimal surfaces,” Comm. Pure Appl. Math., 49, No. 10, 1081–1144 (1996).zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Folland G. B. and Stein E. M., Hardy Spaces on Homogeneous Groups, Princeton Univ. Press, Princeton NJ (1982).zbMATHGoogle Scholar
  26. 26.
    Vodop’yanov S. K. and Pupyshev I. M., “Whitney-type theorems on extension of functions on Carnot groups,” Siberian Math. J., 47, No. 4, 601–620 (2006).CrossRefMathSciNetGoogle Scholar
  27. 27.
    Plotnikova E. A., “Integral representations of the Sobolev type for the functions on Carnot groups,” Mat. Trudy, 11, No. 1 (2008).Google Scholar
  28. 28.
    Romanovskii N. N., “Mikhlin’s problem on Carnot groups,” Siberian Math. J., 49, No. 1, 193–206 (2008).zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Mikhlin S. G., Multidimensional Singular Integrals and Integral Equations [in Russian], Fizmatgiz, Moscow (1962).Google Scholar
  30. 30.
    Romanovskii N. N., “Coercive estimates for the linear differential operators with constant coefficients,” Math. Notes, 70, No. 2, 283–287 (2001).zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2008

Authors and Affiliations

  1. 1.Novosibirsk State UniversityNovosibirskRussia

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