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On Ahlfors–Beurling Operator

Abstract

We investigate regularity properties of solutions of Beltrami equation expressed in terms of moduli of continuity. In particular, we prove that a class of Calderon–Zygmund operators, including Ahlfors–Beurling operator, preserves certain type of modulus of continuity of compactly supported functions. We also prove a purely topological result which easily gives injectivity of normal solutions of Beltrami equation.

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References

  1. 1.

    L. V. Ahlfors. Lectures on Quasiconformal mappings with additional chapters by C.J. Earle and I. Kra, M. Shishikura, J.H. Hubbard. Univ. Lectures Series, v. 38, Providence, R. I., 2006.

  2. 2.

    K. Astala, T. Iwaniec, and G. J. Martin Elliptic partial differential equations and quasiconformal mappings in the plane. Princeton University Press, MR 2472875 (2010j:30040), 2009.

  3. 3.

    A. P. Calderon and A. Zygmund. “Singular integrals and periodic functions,” Studia Mathematica, 14, 349–371 (1954).

    MathSciNet  Article  Google Scholar 

  4. 4.

    Chung-Wu Ho. “A note on proper mapsm,” Proc. AMS, 51(1), (1975).

  5. 5.

    V. Gutlyanskii, V. Ryazanov, U. Srebro, and E. Yakubov. The Beltrami equation. A geometric approach, Developments in Mathematics, 26. Springer, New York, 2012.

  6. 6.

    D. Kalaj. “On Kellogg’s theorem for quasiconformal mappings,” Glasgow Mathematical Journal, 54(03), 599–603 (2012).

    MathSciNet  Article  Google Scholar 

  7. 7.

    D. Kalaj. “Corrigendum to the paper David Kalaj: On Kellogg’s theorem for quasiconformal mappings. Glasgow Mathematical Journal, Vol. 54, No. 3, 599–603 (2012),” Glasgow Math. J., 1, (2020).

  8. 8.

    S. B. Klimentov. “Another version of Kellogg’s theorem,” Complex Variables and Elliptic Equations, 60(12), 1647–1657 (2015).

    MathSciNet  Article  Google Scholar 

  9. 9.

    L. O. Rode and I. B. Simonenko. “Multidimensional singular integrals in classes of common leading moduli of continuity,” Siberian Mathematical Journal, 9, 690–696 (1968).

    Article  Google Scholar 

  10. 10.

    L. O. Rode and I. B. Simonenko. “Corrections to the article Multidimensional Singular Integrals in Classes of General Leading Modules of Smoothness, Sib. Matem. Zh., 9, 4, 690–696 (1968),” Siberian Mathematical Journal, 11(3), 543 (1970).

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Correspondence to Miloš Arsenović.

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Dedicated to 80th anniversary of Professor Vladimir Gutlyanskii

Translated from Ukrains’kiĭ Matematychnyĭ Visnyk, Vol. 18, No. 3, pp. 292–302, July–September, 2021.

This investigation was supported Supported by Ministry of Science, Serbia, project OI 174017 and project OI 174032.

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Arsenović, M., Mateljević, M. On Ahlfors–Beurling Operator. J Math Sci 259, 1–9 (2021). https://doi.org/10.1007/s10958-021-05596-9

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Keywords

  • Beltrami equation
  • Ahlfors–Beurlling operator
  • weakly closed forms