Siberian Mathematical Journal

, Volume 53, Issue 6, pp 984–995 | Cite as

Quasispaces induced by vector fields measurable in ℝ3

Article

Abstract

We study some metric functions that are induced by a class of basis vector fields in ℝ3 with measurable coordinates. These functions are proved to be quasimetrics in the domain of definition of the vector fields. Under some natural constraints, the Rashevsky-Chow Theorem and the Ball-Box Theorem are established for the classes of vector fields we consider.

Keywords

vector field quasimetric generalized triangle inequality horizontal curve 

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References

  1. 1.
    Citti G. and Montanari A., “Regularity properties of solutions of a class of elliptic-parabolic nonlinear Levi type equations,” Trans. Amer. Math. Soc., 354,No. 7, 2819–2848 (2002).MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Montanari A. and Morbidelli D., “Nonsmooth Hörmander vector fields and their controlled balls,” arxiv.org.2008.0812.2369v1.Google Scholar
  3. 3.
    Slodkowski Z. and Tomassini G., “Weak solutions for the Levi equation and envelope of holomorphy,” J. Funct. Anal., 101,No. 2, 392–407 (1991).MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Slodkowski Z. and Tomassini G., “Evolution of subsets in ℂ2 and a parabolic problem for the Levi equation,” Ann. Sc. Norm. Super. Pisa Cl. Sci., 25, No. 4, 757–784 (1997).MathSciNetMATHGoogle Scholar
  5. 5.
    Hörmander L., “Hypoelliptic second order differential equations,” Acta Math., 119, No. 3–4, 147–171 (1967).MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Gromov M., “Carnot-Carathéodory spaces seen from within,” in: Sub-Riemannian Geometry, Birkhäuser, Basel, 1996, pp. 79–323.CrossRefGoogle Scholar
  7. 7.
    Rashevsky P. K., “About the possibility to connect any two points of a completely nonholonomic space by an admissible curve,” Uchen. Zap. Mosk. Ped. Inst. Ser. Fiz.-Mat. Nauk, 3, No. 2, 83–94 (1938).Google Scholar
  8. 8.
    Chow W. L., “Über Systeme von linearen partiellen Differentialgleichungen erster Ordnung,” Math. Ann., 117, 98–105 (1939).MathSciNetCrossRefGoogle Scholar
  9. 9.
    Nagel A., Stein E. M., and Wainger S., “Balls and metrics defined by vector fields. I: Basic properties,” Acta Math., 155, No. 1–2, 103–147 (1985).MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Stein E. M., “Some geometrical concepts arising in harmonic analysis,” in: GAFA 2000. Visions in Mathematics-Towards 2000. Proceedings of a Meeting, Tel Aviv, Israel, August 25–September 3, 1999. Part I, Birkhäuser, Basel, 2000, pp. 434–453.Google Scholar
  11. 11.
    Stein E. M., Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Univ. Press, Princeton (1993).MATHGoogle Scholar
  12. 12.
    Nemytskiĭ V. V. and Stepanov V. V., Qualitative Theory of Differential Equations [in Russian], Gostekhizdat, Moscow and Leningrad (1949).Google Scholar
  13. 13.
    Helgason S., Differential Geometry and Symmetric Spaces, Academic Press, New York and London (1962).MATHGoogle Scholar
  14. 14.
    Bellaïche A., “The tangent space in sub-Riemannian geometry,” in: Sub-Riemannian Geometry, Birkhäuser, Basel, 1996, pp. 1–78.CrossRefGoogle Scholar
  15. 15.
    Citti G., “C -regularity of solutions of a quasilinear equation related to the Levi operator,” Ann. Sc. Norm. Super. Pisa Cl. Sci., 23, No. 4, 483–529 (1996).MathSciNetMATHGoogle Scholar
  16. 16.
    Franchi B. and Lanconelli E., “Holder regularity theorem for a class of linear nonuniformly elliptic operators with measurable coefficients,” Ann. Sc. Norm. Super. Pisa Cl. Sci., 10, No. 4, 523–541 (1983).MathSciNetMATHGoogle Scholar
  17. 17.
    Vodopyanov S. K. and Karmanova M. B., “Subriemannian geometry under minimal smoothness of vector fields,” Dokl. Akad. Nauk, 422, No. 5, 583–588 (2008).MathSciNetGoogle Scholar
  18. 18.
    Montanari A. and Morbidelli D., “Step-s involutive families of vector fields, their orbits and the Poincaré inequality,” available at arxiv.org. 2012. 1106.2410v2.Google Scholar
  19. 19.
    Bonfiglioli A., Lanconelli E., and Uguzzoni F., Stratified Lie Groups and Potential Theory for Their Sub-Laplacian, Springer-Verlag, Berlin and Heidelberg (2007).Google Scholar
  20. 20.
    Korányi A. and Reimann H. M., “Foundations for the theory of quasiconformal mappings on the Heisenberg group,” Adv. Math., 111, No. 1, 1–87 (1995).MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Greshnov A. V., “On one class of Lipschitz vector fields in ℝ3,” Siberian Math. J., 51, No. 3, 410–418 (2010).MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Sansone G., Ordinary Differential Equations. Vol. 2 [Russian translation], Izdat. Inostr. Lit., Moscow (1954).Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2012

Authors and Affiliations

  1. 1.Sobolev Institute of MathematicsNovosibirsk State UniversityNovosibirskRussia

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