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Composition operators on Hardy-Sobolev spaces and BMO-quasiconformal mappings

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Abstract

In this paper, we consider composition operators on Hardy-Sobolev spaces in connections with BMO-quasiconformal mappings. Using the duality of Hardy spaces and BMO-spaces, we prove that BMO-quasiconformal mappings generate bounded composition operators from Hardy–Sobolev spaces to Sobolev spaces.

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References

  1. P. Auscher, E. Russ, and P. Tchamitchian, “Hardy Sobolev spaces on strongly Lipschitz domains of ℝn,” Journal Funct. Anal., 218, 54–109 (2005).

  2. A. P. Calderón, On an inverse boundary value problem, In: Seminar on Numerical Analysis and its Applications to Continuum Physics, Rio de Janeiro, 1980, 65–73 (1980).

  3. H. Federer, Geometric measure theory, Springer Verlag, Berlin, 1969.

    MATH  Google Scholar 

  4. C. Fefferman and E. M. Stein, Hp spaces of several variables, Acta Math., 129, 137–193 (1972).

  5. V. Gutlyanskii, V. Ryazanov, U. Srebro, and E. Yakubov, The Beltrami Equation. A Geometric Approach, Springer-Verlag, New York, 2012.

    Book  MATH  Google Scholar 

  6. D. C. Chang, “The dual of Hardy spaces on a bounded domain in ℝn,” Forum Math., 6, 65–81 (1994).

  7. D. C. Chang, S. G. Krantz, and E. M. Stein, “Hp theory on a smooth domain in ℝN and elliptic boundary value problems,” J. Funct. Anal., 114, 286–347 (1993).

  8. D.-C. Chang, G. Dafni, and C. Sadosky, A Div-Curl Lemma in BMO on a Domain, Harmonic Analysis, Signal Processing, and Complexity, Progress in Mathematics, vol 238, Birkh¨auser, Boston, 55–65 (2005).

  9. P. W. Jones, “Extension theorems for BMO,” Indiana Univ. Math. J., 29, 41–66 (1980).

    Article  MathSciNet  MATH  Google Scholar 

  10. C. Cowen, B. MaCluer, Composition operators on spaces of analytic functions, CRCPress, New York, 1995.

    Google Scholar 

  11. M. Csörnyei, S. Hencl, and J. Malý, Homeomorphisms in the Sobolev space W1,n–1, J. Reine Angew. Math., 644, 221–235 (2010).

  12. A. Miyachi, “Hp spaces over open subsets of ℝn,” Studia Math., 95(3), 205–228 (1990).

  13. X. Chen, R. Jiang, and D. Yang, “Hardy and Hardy-Sobolev Spaces on Strongly Lipschitz Domains and Some Applications,” Anal. Geom. Metr. Spaces, 4(1), 336–362 (2016).

    MathSciNet  MATH  Google Scholar 

  14. V. Gol’dshtein and L. Gurov, “Applications of change of variables operators for exact embedding theorems,” Integral Equations Operator Theory, 19, 1–24 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  15. V. Gol’dshtein, L. Gurov, and A. Romanov, “Homeomorphisms that induce monomorphisms of Sobolev spaces,” Israel J. Math., 91, 31–60 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  16. V. Gol’dshtein, V. Pchelintsev, and A. Ukhlov, “On the First Eigenvalue of the Degenerate p-Laplace Operator in Non-convex Domains,” Integral Equations Operator Theory, 90, 43 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  17. V. Gol’dshtein and A. Ukhlov, “Weighted Sobolev spaces and embedding theorems,” Trans. Amer. Math. Soc., 361, 3829–3850 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  18. V. Gol’dshtein and A. Ukhlov, “About homeomorphisms that induce composition operators on Sobolev spaces,” Complex Var. Elliptic Equ., 55, 833–845 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  19. V. Gol’dshtein and A. Ukhlov, “On the first Eigenvalues of Free Vibrating Membranes in Conformal Regular Domains,” Arch. Rational Mech. Anal., 221(2), 893–915 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  20. V. Gol’dshtein and A. Ukhlov, “The spectral estimates for the Neumann-Laplace operator in space domains,” Adv. in Math., 315, 166–193 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  21. P. Hajlasz, “Change of variables formula under minimal assumptions,” Colloq. Math., 64, 93–101 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  22. S. Hencl and P. Koskela, “Composition of quasiconformal mappings and functions in Triebel–Lizorkin spaces,” Math. Nachr., 286, 669–678 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  23. S. Hencl and P. Koskela, Lectures on Mappings of Finite Distortion, Springer, Berlin/Heidelberg, 2014.

    Book  MATH  Google Scholar 

  24. H. Koch, P. Koskela, E. Saksman, and T. Soto, “Bounded compositions on scaling invariant Besov spaces,” J. Funct. Anal., 266, 2765–2788 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  25. P. Koskela, D. Yang, and Y. Zhou, “Pointwise characterizations of Besov and Triebel-Lizorkin spaces and quasiconformal mappings,” Adv. Math., 226, 3579–3621 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  26. S. C. Arora, G. Datt, and S. Verma, “Weighted composition operators on Orlicz–Sobolev spaces,” J. of the Australian Math. Soc., 83(3), 327–334 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  27. S. Hencl and L. Kleprlik, “Composition of q-quasiconformal mappings and function in Orlicz–Sobolev spaces,” Illinois J. Math., 56(3), 931–955 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  28. S. Hencl, L. Kleprlik, and J. Maly, “Composition operator and Sobolev–Lorentz spaces WLn,q,” Studia Mathematica, 221, 197–208 (2014).

  29. L. Kleprlik, “Composition operators on W1X are necessarily induced by quasiconformal mappings,” Cent. Eur. J. Math., 12(8), 1229–1238 (2014).

    MathSciNet  MATH  Google Scholar 

  30. A. S. Romanov, “The composition operators in Sobolev spaces with variable exponent of summability,” Sib. Elektron. Mat. Izv., 14, 794–806 (2017).

    MathSciNet  MATH  Google Scholar 

  31. P. Koskela and E. Saksman, “Pointwise characterizations of Hardy-Sobolev functions,” Math. Res. Lett., 15(4), 727–744 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  32. P. Koskela, J. Xiao, Yi. Ru-Ya Zhang, and Y. Zhou, “A quasiconformal composition problem for the Qspaces,” J. Eur. Math. Soc. (JEMS), 19, 1159–1187 (2017).

  33. L. G. Lewis, “Quasiconformal mappings and Royden algebras in space,” Trans. Amer. Math. Soc., 158, 481–492 (1971).

    Article  MathSciNet  MATH  Google Scholar 

  34. O. Martio, V. Ryazanov, U. Srebro, and E. Yakubov, “To the theory of Q-homeomorphisms,” Dokl. Akad. Nauk Rossii, 381, 20–22 (2001).

    MathSciNet  MATH  Google Scholar 

  35. O. Martio, V. Ryazanov, U. Srebro, and E. Yakubov, “Mappings with finite length distortion,” J. d’Anal. Math., 93, 215–236 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  36. O. Martio, V. Ryazanov, U. Srebro, and E. Yakubov, “On Q-homeomorphisms,” Ann. Acad. Sci. Fenn. Math., 30(1), 49–69 (2005).

    MathSciNet  MATH  Google Scholar 

  37. O. Martio, V. Ryazanov, U. Srebro, and E. Yakubov, Moduli in modern mapping theory, Springer Monographs in Mathematics, Springer, New York, 2009.

    MATH  Google Scholar 

  38. V. G. Maz’ya, “Weak solutions of the Dirichlet and Neumann problems,” Trudy Moskov. Mat. Ob-va., 20, 137–172 (1969).

    MathSciNet  Google Scholar 

  39. V. Maz’ya, Sobolev spaces: with applications to elliptic partial differential equations, Springer, Berlin/Heidelberg, 2010.

    Google Scholar 

  40. V. Maz’ya and T. O. Shaposhbikovs, Multipliers in Spaces of Differentiable Functions, Leningrad Univ. Press, 1986.

    Google Scholar 

  41. A. V. Menovshchikov, “Composition operators in Orlicz-Sobolev spaces,” Siberian Math. J., 57, 849–859 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  42. A. V. Menovshchikov, “Regularity of the inverse of a homeomorphism of a Sobolev-Orlicz space,” Siberian Math. J., 58, 649–662 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  43. M. Nakai, “Algebraic criterion on quasiconformal equivalence of Riemann surfaces,” Nagoya Math. J., 16, 157–184 (1960).

    Article  MathSciNet  MATH  Google Scholar 

  44. M. Oliva and M. Prats, “Sharp bounds for composition with quasiconformal mappings in Sobolev spaces,” J. Math. Anal. Appl., 451, 1026–1044 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  45. L. E. Payne and H. F. Weinberger, “An optimal Poincaré inequality for convex domains,” Arch. Rat. Mech. Anal., 5, 286–292 (1960).

    Article  MATH  Google Scholar 

  46. V. Ryazanov, S. Srebro, and E. Yakubov, “BMO-quasiconformal mappings,” Journal d’Analyse Mathématique, 83, 1–20 (2001).

  47. E. M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series 43, Monographs in Harmonic AnalysisIII, Princeton University Press, Princeton, NJ, 1993.

    Google Scholar 

  48. L. He, G. H. Cao, and Z. H. He, “Composition operators on Hardy-Sobolev spaces,” Indian J. Pure Appl. Math., 46(3), 255–267 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  49. E. M. Stein and G. Weiss, “On the theory of harmonic functions of several variables. The theory of Hp-spaces,” Acta Math., 103, 25–62 (1960).

    Article  MathSciNet  MATH  Google Scholar 

  50. A. Ukhlov, “On mappings, which induce embeddings of Sobolev spaces,” Siberian Math. J., 34, 185–192 (1993).

    MathSciNet  Google Scholar 

  51. A. Ukhlov and S. K. Vodop’yanov, “Mappings associated with weighted Sobolev spaces,” Complex analysis and dynamical systems III, Contemp. Math., Amer. Math. Soc., Providence, RI, 455, 369–382 (2008).

  52. S. K. Vodop’yanov, Taylor Formula and Function Spaces, Novosibirsk Univ. Press, Novosibirsk, 1988.

    Google Scholar 

  53. S. K. Vodop’yanov, “Composition Operators on Weighted Sobolev Spaces and the Theory of Qp-Homeomorphisms,” Doklady Mathematics, 102, 371–375 (2020).

    Article  Google Scholar 

  54. S. K. Vodop’yanov and V. M. Gol’dshtein, “Structure isomorphisms of spaces \( {W}_n^1 \) and quasiconformal mappings,” Siberian Math. J., 16, 224–246 (1975).

  55. S. K. Vodop’yanov, V. M. Gol’dshtein, and Yu. G. Reshetnyak, “On geometric properties of functions with generalized first derivatives,” Uspekhi Mat. Nauk, 34, 17–65 (1979).

    MathSciNet  MATH  Google Scholar 

  56. S.K. Vodop’yanov and A. D. Ukhlov, “Superposition operators in Sobolev spaces,” Russian Mathematics (Izvestiya VUZ), 46(4), 11–33 (2002).

    MathSciNet  MATH  Google Scholar 

  57. S. K. Vodop’yanov and A. D. Ukhlov, “Set Functions and Their Applications in the Theory of Lebesgue and Sobolev Spaces. I,” Siberian Adv. Math., 14(4), 78–125 (2004).

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Sobolev Institute of Mathematics, Novosibirsk, Russia

    Alexander Menovschikov

  2. Department of Mathematics, Ben-Gurion University of the Negev, Beer Sheva, Israel

    Alexander Ukhlov

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  1. Alexander Menovschikov
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  2. Alexander Ukhlov
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Correspondence to Alexander Menovschikov.

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Dedicated to the 80th anniversary of Professor Vladimir Gutlyanskii

Translated from Ukrains’kiĭ Matematychnyĭ Visnyk, Vol. 18, No. 2, pp. 209–225, April–June, 2021.

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Menovschikov, A., Ukhlov, A. Composition operators on Hardy-Sobolev spaces and BMO-quasiconformal mappings. J Math Sci 258, 313–325 (2021). https://doi.org/10.1007/s10958-021-05549-2

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