Journal d’Analyse Mathématique

, Volume 83, Issue 1, pp 1–20 | Cite as

BMO-quasiconformal mappings



Plane BMO-quasiconformal and BMO-quasiregular mappings are introduced, and their basic properties are studied. This includes distortion, existence, uniqueness, representation, integrability, convergence and removability theorems, the reflection principle, boundary behavior and mapping properties.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [A] L. Ahlfors,Lectures on Quasiconformal Mappings, Van Nostrand Math. Studies #10, 1966.Google Scholar
  2. [AG] K. Astala and F. W. Gehring,Injectivity, the BMO norm and the universal Teichmüller space, J. Analyse Math.46 (1986), 16–57.MATHMathSciNetGoogle Scholar
  3. [AIKM] K. Astala, T. Iwaniec, P. Koskela and G. Martin,Mappings of BMO-bounded distortion, Math. Ann.317 (2000), 703–726.MATHCrossRefMathSciNetGoogle Scholar
  4. [As] K. Astala,A remark on quasiconformal mappings and BMO-functions, Michigan Math. J.80 (1983), 209–212.MathSciNetGoogle Scholar
  5. [B] B. Bojarski,Generalized solutions of systems of differential equations of first order and elliptic type with discontinuous coefficients, Mat. Sb.43 (85) (1957), 451–503 (Russian).MathSciNetGoogle Scholar
  6. [BA] A. Beurling and L. Ahlfors,The boundary correspondence under quasiconformal mappings, Acta Math.96 (1956), 125–142.MATHCrossRefMathSciNetGoogle Scholar
  7. [BJ] M. A. Brakalova and J. A. Jenkins,On solutions of the Beltrami equation, J. Analyse Math.76 (1998), 67–92.MATHMathSciNetCrossRefGoogle Scholar
  8. [C] C. Carathéodory,Theory of Functions of a Complex Variable, Vol. I, Chelsea Publ. Co., New York, 1964.Google Scholar
  9. [CL] E. F. Collingwood and A. J. Lohwater,The Theory of Cluster Sets, Cambridge University Press, 1966.Google Scholar
  10. [CR] R. R. Coifman and R. Rochberg,Another characterization of BMO, Proc. Amer. Math. Soc.79 (1980), 249–254.MATHCrossRefMathSciNetGoogle Scholar
  11. [D] G. David,Solutions de l'équation de Beltrami avec ‖μ‖ =1, Ann. Acad. Sci. Fenn. Ser. A I Math.13 (1988), 25–70.MATHMathSciNetGoogle Scholar
  12. [Du] J. Dugundji,Topology, Allen and Bacon Inc, Boston, 1966.MATHGoogle Scholar
  13. [G] F. W. Gehring,Characteristic Properties of Quasidisks, Les presses de l'Universite de Montreal, 1982.Google Scholar
  14. [GI] F. W. Gehring and T. Iwaniec,The limit of mappings with finite distorsion, Ann. Acad. Sci. Fenn. Ser. A I Math.24 (1999), 253–264.MATHMathSciNetGoogle Scholar
  15. [GV] V. Goldshtein and S. Vodop'yanov,Quasiconformal mappings and spaces of functions with generalized first derivatives, Sibirsk. Mat. Zh.17 (1976), 515–531.Google Scholar
  16. [HK] J. Heinonen and P. Koskela,Sobolev mappings with integrable dilatations, Arch. Rational Mech. Anal.125 (1993), 81–97.MATHCrossRefMathSciNetGoogle Scholar
  17. [IKM] T. Iwaniec, P. Koskela and G. Martin,Mappings of BMO-distortion and Beltrami type operators, Preprint, Univ. of Jyväkylä, 1998.Google Scholar
  18. [IM] T. Iwaniec and G. Martin,The Beltrami equation, Mem. Amer. Math. Soc., to appear.Google Scholar
  19. [IS] T. Iwaniec and V. Šverák,On mappings with integrable dilatation, Proc. Amer. Math. Soc.118 (1993), 181–188.MATHCrossRefMathSciNetGoogle Scholar
  20. [J] P. W. Jones,Extension theorems for BMO, Indiana Univ. Math. J.29 (1980), 41–66.MATHCrossRefMathSciNetGoogle Scholar
  21. [JN] F. John and L. Nirenberg,On functions of bounded mean oscillation, Comm. Pure Appl. Math.14 (1961), 415–426.MATHCrossRefMathSciNetGoogle Scholar
  22. [KK] S. L. Krushkal and R. Kühnau,Quasiconformal Mappings, New Methods and Applications, Nauka, Novosibirk, 1984.Google Scholar
  23. [LV] O. Lehto and K. Virtanen,Quasiconformal Mappings in the Plane, 2nd ed. Springer-Verlag, Berlin, 1973.MATHGoogle Scholar
  24. [MV] J. Manfredi and E. Villamor,An extension of Reshetnyak's theorem, Indiana Univ. Math. J.47 (1998), 1131–1145.MATHMathSciNetGoogle Scholar
  25. [R] H. M. Reimann,Functions of bounded mean oscillation and quasiconformal mappings, Comment. Math. Helv.49 (1974), 260–276.MATHCrossRefMathSciNetGoogle Scholar
  26. [RR] H. M. Reimann and T. Rychener,Funktionen Beschränkter Mittlerer Oscillation, Lecture Notes in Math.487, Springer-Verlag, Berlin, 1975.Google Scholar
  27. [Re] Yu. G. Reshetnyak,Space Mappings with Bounded Distortion, Transl. Math. Monographs, Vol. 73, Amer. Math. Soc., Providence, RI, 1989.MATHGoogle Scholar
  28. [R1] V. Ryazanov,On convergence and compactness theorems for ACL homeomorphisms, Rev. Roumaine Math. Pures Appl.41 (1996), 133–139.MATHMathSciNetGoogle Scholar
  29. [R2] V. Ryazanov,On convergence theorems for homeomorphisms of the Sobolev classes, Ukrain. Mat. Zh.47 (1995), 249–259.MATHMathSciNetGoogle Scholar
  30. [RSY1] V. Ryazanov, U. Srebro and E. Yakubov,BMO-quasiconformal mappings, Reports of the Department of Mathematics, University of Helsinki, Preprint 155, 1997, 22 pp. (fttp:// 155. ps).Google Scholar
  31. [RSY2] V. Ryazanov, U. Srebro and E. Yakubov,To the theory of BMO-quasiregular mappings, Dokl. Akad. Nauk Rossii369, No. 1 (1999), 13–15; Transl.: Dokl. Math.60, No. 3 (1999), 319–321.MathSciNetGoogle Scholar
  32. [RW] E. Reich and H. Walczak,On the behavior of quasiconformal mappings at a point, Trans. Amer. Math. Soc.117 (1965), 338–351.MATHCrossRefMathSciNetGoogle Scholar
  33. [Sa] S. Sastry,Boundary behavior of BMO-QC automorphisms, Israel J. Math., to appear.Google Scholar
  34. [T] P. Tukia,Compactness properties of μ-homeomorphisms, Ann. Acad. Sci. Fenn. Ser. A I Math.16 (1991), 47–69.MathSciNetGoogle Scholar
  35. [V] J. Väisälä,Lectures in n-Dimensional Quasiconformal Mappings, Lecture Notes in Math.229, Springer-Verlag, Berlin, 1971.Google Scholar

Copyright information

© The Hebrew University Magnes Press 2001

Authors and Affiliations

  1. 1.Institute of Applied Mathematics & MechanicsNasuDonetskUkraine
  2. 2.Technion-Israel Institute of TechnologyHaifaIsrael
  3. 3.Holon Academic Institute of TechnologyHolonIsrael

Personalised recommendations