Abstract and Applied Analysis

Distortion of Quasiregular Mappings and Equivalent Norms on Lipschitz-Type Spaces

Miodrag Mateljević

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Abstract

We prove a quasiconformal analogue of Koebe’s theorem related to the average Jacobian and use a normal family argument here to prove a quasiregular analogue of this result in certain domains in n -dimensional space. As an application, we establish that Lipschitz-type properties are inherited by a quasiregular function from its modulo. We also prove some results of Hardy-Littlewood type for Lipschitz-type spaces in several dimensions, give the characterization of Lipschitz-type spaces for quasiregular mappings by the average Jacobian, and give a short review of the subject.

Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2013), Article ID 895074, 20 pages.

Dates
First available in Project Euclid: 27 February 2015

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1425049702

Digital Object Identifier
doi:10.1155/2014/895074

Mathematical Reviews number (MathSciNet)
MR3273917

Zentralblatt MATH identifier
07023261

Citation

Mateljević, Miodrag. Distortion of Quasiregular Mappings and Equivalent Norms on Lipschitz-Type Spaces. Abstr. Appl. Anal. 2014, Special Issue (2013), Article ID 895074, 20 pages. doi:10.1155/2014/895074. https://projecteuclid.org/euclid.aaa/1425049702


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References

  • K. Astala and F. W. Gehring, “Injectivity, the BMO norm and the universal Teichmüller space,” Journal d'Analyse Mathématique, vol. 46, pp. 16–57, 1986.
  • F. W. Gehring and O. Martio, “Lipschitz classes and quasiconformal mappings,” Annales Academiæ Scientiarum Fennicæ. Series A I. Mathematica, vol. 10, pp. 203–219, 1985.
  • C. A. Nolder, “A quasiregular analogue of a theorem of Hardy and Littlewood,” Transactions of the American Mathematical Society, vol. 331, no. 1, pp. 215–226, 1992.
  • K. Astala and F. W. Gehring, “Quasiconformal analogues of theorems of Koebe and Hardy-Littlewood,” The Michigan Mathe-matical Journal, vol. 32, no. 1, pp. 99–107, 1985.
  • K. Kim, “Hardy-Littlewood property with the inner length metric,” Korean Mathematical Society, vol. 19, no. 1, pp. 53–62, 2004.
  • K. M. Dyakonov, “Equivalent norms on Lipschitz-type spaces of holomorphic functions,” Acta Mathematica, vol. 178, no. 2, pp. 143–167, 1997.
  • A. Abaob, M. Arsenović, M. Mateljević, and A. Shkheam, “Moduli of continuity of harmonic quasiregular mappings on bounded domains,” Annales Academiæ Scientiarum Fennicæ, vol. 38, no. 2, pp. 839–847, 2013.
  • M. Mateljević and M. Vuorinen, “On harmonic quasiconformal quasi-isometries,” Journal of Inequalities and Applications, vol. 2010, Article ID 178732, 2010.
  • J. Qiao and X. Wang, “Lipschitz-type spaces of pluriharmonic mappings,” Filomat, vol. 27, no. 4, pp. 693–702, 2013.
  • T. Iwaniec and G. Martin, Geometric Function Theory and Nonlinear Analysis, Syraccuse, Aucland, New Zealand, 2000.
  • M. Vuorinen, Conformal Geometry and Quasiregular Mappings, vol. 1319 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1988.
  • F. W. Gehring and B. G. Osgood, “Uniform domains and the quasihyperbolic metric,” Journal d'Analyse Mathématique, vol. 36, pp. 50–74, 1979.
  • M. Arsenović, V. Manojlović, and M. Mateljević, “Lipschitz-type spaces and harmonic mappings in the space,” Annales Academiæ Scientiarum Fennicæ. Mathematica, vol. 35, no. 2, pp. 379–387, 2010.
  • Sh. Chen, M. Mateljević, S. Ponnusamy, and X. Wang, “Lipschitz type spaces and Landau-BlochčommentComment on ref. [12?]: Please update the information of this reference, if possible. type theorems for harmonic functions and solutions to Poisson equations,” to appear.
  • K. M. Dyakonov, “HolomorphicčommentComment on ref. [17?]: We split these references [48a?]. Please check. functions and quasiconformal mappings with smooth moduli,” Advances in Mathematics, vol. 187, no. 1, pp. 146–172, 2004.
  • A. Hinkkanen, “Modulus of continuity of harmonic functions,” Journal d'Analyse Mathématique, vol. 51, pp. 1–29, 1988.
  • V. Lappalainen, “Lip$_{h}$-extension domains,” Annales Academiæ Scientiarum Fennicæ. Series A I. Mathematica Dissertationes, no. 56, p. 52, 1985.
  • O. Martio and R. Näkki, “Boundary Hölder continuity and quasiconformal mappings,” Journal of the London Mathematical Society. Second Series, vol. 44, no. 2, pp. 339–350, 1991.
  • S. Chen, S. Ponnusamy, and X. Wang, “On planar harmonic Lipschitz and planar harmonic Hardy classes,” Annales Academiæ Scientiarum Fennicæ. Mathematica, vol. 36, no. 2, pp. 567–576, 2011.
  • Sh. Chen, S. Ponnusamy, and X. Wang, “Equivalent moduli of continuity, Bloch's theorem for pluriharmonic mappings in ${B}^{n}$,” Proceedings of Indian Academy of Sciences–-Mathematical Sciences, vol. 122, no. 4, pp. 583–595, 2012.
  • Sh. Chen, S. Ponnusamy, M. Vuorinen, and X. Wang, “Lipschitz spaces and bounded mean oscillation of harmonic mappings,” Bulletin of the Australian Mathematical Society, vol. 88, no. 1, pp. 143–157, 2013.
  • S. Chen, S. Ponnusamy, and A. Rasila, “Coefficient estimates, Landau's theorem and Lipschitz-type spaces on planar harmonic mappings,” Journal of the Australian Mathematical Society, vol. 96, no. 2, pp. 198–215, 2014.
  • F. W. Gehring, “Rings and quasiconformal mappings in space,” Transactions of the American Mathematical Society, vol. 103, pp. 353–393, 1962.
  • J. Väisälä, Lectures on n-Dimensional Quasiconformal Mappings, Springer, New York, NY, USA, 1971.
  • M. Mateljević, “Quasiconformality of harmonic mappings between Jordan domains,” Filomat, vol. 26, no. 3, pp. 479–510, 2012.
  • D. Kalaj and M. S. Mateljević, “Harmonic quasiconformal self-mappings and Möbius transformations of the unit ball,” Pacific Journal of Mathematics, vol. 247, no. 2, pp. 389–406, 2010.
  • D. Kalaj, “On harmonic diffeomorphisms of the unit disc onto a convex domain,” Complex Variables, vol. 48, no. 2, pp. 175–187, 2003.
  • M. Mateljević, Topics in Conformal, Quasiconformal and Harmonic Maps, Zavod za Udzbenike, Belgrade, Serbia, 2012.
  • M. Mateljević, “Distortion of harmonic functions and harmonic quasiconformal quasi-isometry,” Revue Roumaine de Mathématiques Pures et Appliquées, vol. 51, no. 5-6, pp. 711–722, 2006.
  • M. Knežević and M. Mateljević, “On the quasi-isometries of harmonic quasiconformal mappings,” Journal of Mathematical Analysis and Applications, vol. 334, no. 1, pp. 404–413, 2007.
  • V. Manojlović, “Bi-Lipschicity of quasiconformal harmonic mappings in the plane,” Filomat, vol. 23, no. 1, pp. 85–89, 2009.
  • M. Mateljević, Communications at Analysis Seminar, University of Belgrade, 2012.
  • M. Mateljević, “Quasiconformal and Quasiregular harmonic mappings and Applications,” Annales Academiæ Scientiarum Fennicæ Mathematica, vol. 32, pp. 301–315, 2007.
  • I. Anić, V. Marković, and M. Mateljević, “Uniformly bounded maximal $\varphi $-disks, Bers space and harmonic maps,” Proceedings of the American Mathematical Society, vol. 128, no. 10, pp. 2947–2956, 2000.
  • S. Rickman, Quasiregular Mappings, vol. 26, Springer, Berlin, Germany, 1993.
  • O. Lehto and K. I. Virtanen, Quasiconformal Mappings in the Plane, Springer, New York, NY, USA, 2nd edition, 1973.
  • D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, vol. 224 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer, Berlin, Germany, 2nd edition, 1983.
  • B. Burgeth, “A Schwarz lemma for harmonic and hyperbolic-harmonic functions in higher dimensions,” Manuscripta Mathematica, vol. 77, no. 2-3, pp. 283–291, 1992.
  • S. Axler, P. Bourdon, and W. Ramey, Harmonic Function Theory, vol. 137 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 1992.
  • W. Rudin, Function Theory in the Unit Ball of C$^{n}$, Springer, Berlin, Germany, 1980.
  • R. Näkki and B. Palka, “Lipschitz conditions and quasiconformal mappings,” Indiana University Mathematics Journal, vol. 31, no. 3, pp. 377–401, 1982.
  • M. Mateljević, “Versions of Koebe 1/4 theorem for analytic and quasiregular harmonic functions and applications,” Institut Mathématique. Publications. Nouvelle Série, vol. 84, pp. 61–72, 2008.
  • P. Duren, Harmonic Mappings in the Plane, vol. 156 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, UK, 2004.
  • D. Kalaj, Harmonic functions and quasiconformal mappings [M.S. thesis], Harmonijske Funkcije i Kvazikonformna Preslikavanja, Belgrade, Serbia, 1998.
  • E. Heinz, “On one-to-one harmonic mappings,” Pacific Journal of Mathematics, vol. 9, no. 1, pp. 101–105, 1959.
  • G. Martin and K. Peltonen, “Stoïlow factorization for quasiregular mappings in all dimensions,” Proceedings of the American Mathematical Society, vol. 138, no. 1, pp. 147–151, 2010.
  • A. N. Fletcher and D. A. Nicks, “Iteration of quasiregular tangent functions in three dimensions,” Conformal Geometry and Dynamics, vol. 16, pp. 1–21, 2012.
  • V. A. Zorich, “The global homeomorphism theorem for space quasiconformal mappings, its development and related open problems,” in Quasiconformal Space Mappings, vol. 1508 of Lecture Notes in Mathematics, pp. 132–148, Springer, Berlin, Germany, 1992.
  • O. Martio, S. Rickman, and J. Väisälä, “Topological and metric properties of quasiregular mappings,” Annales Academiæ Scientiarum Fennicæ Mathematica, vol. 488, pp. 1–31, 1971.
  • Yu. G. Reshetnyak, “Space mappings with bounded distortion,” Siberian Mathematical Journal, vol. 8, no. 3, pp. 466–487, 1967.
  • F. W. Gehring and J. Väisälä, “The coefficients of quasiconformality of domains in space,” Acta Mathematica, vol. 114, pp. 1–70, 1965.
  • M. Pavlović, “On Dyakonov's paper `Equivalent norms on lipschitz-type spaces of holomorphic functions',” Acta Mathematica, vol. 183, no. 1, pp. 141–143, 1999.
  • O. Martio, “On harmonic quasiconformal mappings,” Annales Academiæ Scientiarum Fennicæ Mathematica, vol. 425, pp. 3–10, 1968.
  • M. Arsenović, V. Kojić, and M. Mateljević, “On Lipschitz conti-nuity of harmonic quasiregular maps on the unit ball in ${R}^{n}$,” Annales Academiæ Scientiarum Fennicæ. Mathematica, vol. 33, no. 1, pp. 315–318, 2008.
  • M. Mateljević, “A version of Bloch's theorem for quasiregular harmonic mappings,” Revue Roumaine de Mathématiques Pures et Appliquées, vol. 47, no. 5-6, pp. 705–707, 2002. \endinput