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The degree of regularity of a quasiconformal mapping

Author: Pekka Koskela
Journal: Proc. Amer. Math. Soc. 122 (1994), 769-772
MSC: Primary 30C65; Secondary 30C62
MathSciNet review: 1204381
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Abstract: T. Iwaniec has conjectured that the derivative of a locally $ \alpha $-Hölder continuous quasiconformal mapping of $ {\mathbb{R}^n}$ is locally integrable to any power $ p < \frac{n}{{1 - \alpha }}$. We disprove this conjecture by producing examples of quasiconformal mappings of the plane that are uniformly Hölder continuous with exponent $ \frac{1}{2} < \alpha < 1$ but whose derivatives are not locally integrable to the power $ \frac{1}{{1 - \alpha }}$.

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  • [A] K. Astala, Selfsimilar zippers, Holomorphic Functions and Moduli I (D. Drasin, F. W. Gehring, C. M. Earle, I. Kra, A. Marden, eds.), Springer-Verlag, New York, 1988, pp. 61-74. MR 955804 (89c:30001)
  • [AK] K. Astala and P. Koskela, Quasiconformal mappings and global integrability of the derivative, J. Analyse Math. 57 (1991), 203-220. MR 1191747 (94c:30026)
  • [B] B. Bojarski, Generalized solutions of a system of differential equations of first order and elliptic type with discontinuous coefficients, Mat. Sb. 85 (1957), 451-503. MR 0106324 (21:5058)
  • [FM] K. J. Falconer and D. T. Marsh, Classification of quasi-circles by Hausdorff dimension, Nonlinearity 2 (1989), 489-493. MR 1005062 (91a:58108)
  • [G1] F. W. Gehring, Rings and quasiconformal mappings in space, Trans. Amer. Math. Soc. 103 (1962), 353-393. MR 0139735 (25:3166)
  • [G2] -, Open problems, Proc. Romanian-Finnish Seminar on Teichmuller Spaces and Quasiconformal Mappings, Romania, 1969, p. 306.
  • [G3] -, The $ {L^P}$-integrability of the partial derivatives of a quasiconformal mapping, Acta Math. 130 (1973), 265-277. MR 0402038 (53:5861)
  • [GR] F. W. Gehring and E. Reich, Area distortion under quasiconformal mappings, Ann. Acad. Sci. Fenn. Ser. A I Math. 388 (1966), 1-14. MR 0201635 (34:1517)
  • [GT] D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Springer-Verlag, Berlin and Heidelberg, 1983. MR 737190 (86c:35035)
  • [H] J. E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), 713-747. MR 625600 (82h:49026)
  • [I] T. Iwaniec, $ {L^P}$-theory of quasiregular mappings, Quasiconformal Space Mappings. A Collection of Surveys 1960-90 (M. Vuorinen, ed.), Lecture Notes in Math., vol. 1508, Springer-Verlag, Berlin, 1992, pp. 39-64. MR 1187088
  • [KKM] T. Kilpeläinen, P. Koskela, and O. Martio, On the fusion problem for second order elliptic partial differential equations, preprint, 1993. MR 1263558 (95c:35082)
  • [M] J. McKemie, Quasiconformal groups with small dilatation, Ann. Acad. Sci. Fenn. Ser. A I Math. 12 (1987), 95-118. MR 877584 (88j:30045)
  • [S] S. Semmes, Bilipschitz mappings and strong $ {A_\infty }$-weights, Ann. Acad. Sci. Fenn. A I Math. 18 (1993), 211-248. MR 1234732 (95g:30032)
  • [T1] P. Tukia, Extension of Lipschitz embeddings of the real line into the plane, Ann. Acad. Sci. Fenn. Ser. A I Math. 6 (1981), 89-94. MR 639966 (83d:30022)
  • [T2] -, A quasiconformal group not isomorphic to a Möbius group, Ann. Acad. Sci. Fenn. Ser. A I Math. 6 (1981), 149-160. MR 639972 (83b:30019)

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Keywords: Quasiconformal mapping, Sobolev embedding, global integrability of the derivative
Article copyright: © Copyright 1994 American Mathematical Society

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