Siberian Mathematical Journal

, Volume 48, Issue 6, pp 984–997 | Cite as

Local stability of mappings with bounded distortion on Heisenberg groups

Article

Abstract

This is the second of the author’s three papers on stability in the Liouville theorem on the Heisenberg group. The aim is to prove that each mapping with bounded distortion of a John domain on the Heisenberg group is close to a conformal mapping with order of closeness \(\sqrt {K - 1} \) in the uniform norm and order of closeness K − 1 in the Sobolev norm L p 1 for all \(p < \tfrac{C}{{K - 1}}\).

In this paper we prove a local variant of the desired result: each mapping on a ball with bounded distortion and distortion coefficient K near to 1 is close on a smaller ball to a conformal mapping with order of closeness \(\sqrt {K - 1} \) in the uniform norm and order of closeness K − 1 in the Sobolev norm L p 1 for all \(p < \tfrac{C}{{K - 1}}\). We construct an example that demonstrates the asymptotic sharpness of the order of closeness of a mapping with bounded distortion to a conformal mapping in the Sobolev norm.

Keywords

Heisenberg group mapping with bounded distortion coercive estimate stability 

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References

  1. 1.
    Lavrent’ev M. A., “Sur une classe de représentations continués,” Mat. Sb., 42, No. 4, 407–424 (1935).MathSciNetGoogle Scholar
  2. 2.
    Lavrent’ev M. A., “On stability in Liouville’s theorem,” Dokl. Akad. Nauk USSR, 95, No. 5, 925–926 (1954).MATHMathSciNetGoogle Scholar
  3. 3.
    Reshetnyak Yu. G., Stability Theorems in Geometry and Analysis, Kluwer Academic Publishers, Dordrecht (1994) (Mathematics and Its Applications; 304).MATHGoogle Scholar
  4. 4.
    Korányi A. and Reimann H. M., “Quasiconformal mappings on the Heisenberg group,” Invent. Math., 80, 309–338 (1985).MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    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).MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Capogna L., “Regularity of quasilinear equations in the Heisenberg group,” Comm. Pure Appl. Math., 50, 867–889 (1997).MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    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
  8. 8.
    Dairbekov N. S., “Mappings with bounded distortion on Heisenberg groups,” Siberian Math. J., 41, No. 3, 465–487 (2000).CrossRefMathSciNetGoogle Scholar
  9. 9.
    Isangulova D. V., “The class of mappings with bounded specific oscillation and integrability of mappings with bounded distortion on Carnot groups,” Siberian Math. J., 48, No. 2, 249–267 (2007).CrossRefMathSciNetGoogle Scholar
  10. 10.
    Heinonen J. and Holopainen I., “Quasiregular maps on Carnot groups,” J. Geom. Anal., 7, No. 1, 109–148 (1997).MATHMathSciNetGoogle Scholar
  11. 11.
    Astala K., “Area distortion of quasiconformal mappings,” Acta Math., 173, 37–60 (1994).MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Romanovskii N. N., “Integral representations and embedding theorems for functions on the Heisenberg groups ℍn,” Algebra and Analysis, 16, No. 2, 82–119 (2004).MathSciNetGoogle Scholar
  13. 13.
    Romanovskii N. N., “On Mikhlin’s problem on Carnot groups,” Siberian Math. J., 49, No. 1 (2008) (to appear).Google Scholar
  14. 14.
    Isangulova D. V., “Stability in the Liouville theorem on Heisenberg groups,” Dokl. Math., 72, No. 3, 912–916 (2005).MATHGoogle Scholar
  15. 15.
    Isangulova D. V., Stability in the Liouville Theorem on Heisenberg Groups [Preprint No. 158], Sobolev Institute of Mathematics, Novosibirsk (2005).Google Scholar
  16. 16.
    Vodop’yanov S. K., “P-differentiability on Carnot groups in different topologies and related topics,” in: Proceedings on Analysis and Geometry, Sobolev Institute Press, Novosibirsk, 2000, pp. 603–670.Google Scholar
  17. 17.
    Vodop’yanov S. K., “Closure of classes of mappings with bounded distortion on Carnot groups,” Siberian Adv. Math., 14, No. 1, 84–125 (2005).MathSciNetGoogle Scholar
  18. 18.
    Folland G. B., Harmonic Analysis in Phase Space, Princeton Univ. Press, Princeton, NJ (1989) (Ann. Math. Stud.; 122).MATHGoogle Scholar
  19. 19.
    Vodop’yanov S. K., “Foundations of the theory of mappings with bounded distortion on Carnot groups,” Dokl. Math., 72, No. 3, 829–833 (2005).MATHGoogle Scholar
  20. 20.
    Vodopyanov S. K., “Foundations of the theory of mappings with bounded distortion on Carnot groups,” in: The Interaction of Analysis and Geometry, Amer. Math. Soc., Providence, RI, 2007, pp. 303–344 (Contemp. Math.; 424).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  1. 1.Sobolev Institute of MathematicsNovosibirskRussia

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