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Lp-theory of quasiregular mappings

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Quasiconformal Space Mappings

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References

  1. Acerbi, E. and Fusco, N., Semicontinuity problems in the calculus of variations, Arch. Rational Mech. Anal. 86 (1984), 125–145.

    Article  MathSciNet  MATH  Google Scholar 

  2. Ahlfors, L.V., Lectures on quasiconformal mappings, D. van Nostrand Company, Inc., New York, (1966).

    MATH  Google Scholar 

  3. D’Apuzzo, L. and Sbordone, C., Reverse Hölder inequalities. A sharp result. Rendiconti di Matematica, Ser. VII, 10, (1990), 357–366.

    MathSciNet  MATH  Google Scholar 

  4. Baernstein, A., Some sharp inequalities for conjugate functions, Proc. Symp. Pure Math. 35 (1979), 409–416.

    Article  MathSciNet  MATH  Google Scholar 

  5. Baernstein, A. and Manfredi, J., Topics in quasiconformal mappings, Topics in modern harmonic analysis, Istituto Nazionale di Alta Mathematica, Roma (1983), 849–862.

    Google Scholar 

  6. Ball, J.M., Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rat. Mech. Anal. 63 (1977), 337–403.

    Article  MathSciNet  MATH  Google Scholar 

  7. Ball, J.M., Discontinuous equilibrium solutions and cavitation in nonlinear elasticity, Phil. Trans. Roy. Soc. London (A) 306 (1982), 557–612.

    Article  MathSciNet  MATH  Google Scholar 

  8. Ball, J.M. and Murat, F., W 1,p-quasiconvexity and variational problems for multiple integrals, J. Funct. Anal. 58 (1984), 225–253.

    Article  MathSciNet  MATH  Google Scholar 

  9. Beckner, W., Inequalities in Fourier analysis, Annals of Mathematics, 102 (1975), 159–182.

    Article  MathSciNet  MATH  Google Scholar 

  10. Boccardo, L., An L s-estimate for the gradient of solutions of some strongly nonlinear unilateral problems, Ann. Math. Pura Appl.

    Google Scholar 

  11. Bojarski, B., Homeomorphic solutions of Beltrami system, Dokl. Akad.Nauk. SSSR 102 (1955), 661–664.

    MathSciNet  Google Scholar 

  12. Bojarski, B., Generalized solutions of a system of differential equations of first order and elliptic type with discontinuous coefficients, Math. Sb. No. 43 (85), (1957), 451–503.

    MathSciNet  Google Scholar 

  13. Bojarski, B., Remarks on stability of reverse Hölder inequalities and quasiconformal mappings, Ann. Acad. Sci. Fenn. Ser. A.I. (1985), 291–296.

    Google Scholar 

  14. Bojarski, B. and Iwaniec, T., Analytical foundations of the theory of quasiconformal mappings in R n, Ann. Acad. Sci. Fenn. Ser. A.I., 8 (1983), 257–324.

    MathSciNet  MATH  Google Scholar 

  15. Burkholder, D.L., Boundary value problems and sharp inequalities for martingale transforms, The Annals of Probability, vol. 12, No. 3 (1984), 647–702.

    Article  MathSciNet  MATH  Google Scholar 

  16. Coiffman, R.R. and Fefferman, C., Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241–250.

    MathSciNet  MATH  Google Scholar 

  17. Coiffman, R. and Weiss, G., Transference methods in analysis, Amer. Math. Soc. CBMS No. 31.

    Google Scholar 

  18. Dacorogna, B. and Marcellini, P., Semicontinuitè pour des integrandes polyconvexes sans continuitè des determinants, C.R. Acad. Sci. Paris t 311, ser. 1, (1990), 393–396.

    MathSciNet  MATH  Google Scholar 

  19. David, G., Solutions De L’equation De Beltrami Avec ‖μ‖∞=1, Ann. Acad. Sci. Fenn. Ser. A I Math. 13 (1988), 25–70.

    Article  MathSciNet  Google Scholar 

  20. Davis, B., On the weak type (1.1) inequality for conjugate functions and circular symmetrization, Proc. Amer. Math. Soc. 44 (1974), 307–311.

    MathSciNet  Google Scholar 

  21. Donaldson, S.K. and Sullivan, D.P., Quasiconformal 4-manifolds, Acta Math., 163 (1989), 181–252.

    Article  MathSciNet  MATH  Google Scholar 

  22. Fefferman, C. and Stein, E.M., H p-spaces of several variables, Acta Math. 129 (1972), 137–193.

    Article  MathSciNet  MATH  Google Scholar 

  23. Fehlman, R. and Vuroinen, M., Mori’s theorem for n-dimensional quasiconformal mappings, Ann. Acad. Sci. Fenn. Ser. A.I. 13 (1988), 111–124.

    MathSciNet  Google Scholar 

  24. Figiel, T., Iwaniec, T. and Pelczynski, A., Computing norms and critical exponents of some operators in L p-spaces, Studia Math. 79 (1984), 227–274.

    MathSciNet  MATH  Google Scholar 

  25. Fiorenza, A., On some reverse integral inequalities, Atti sem. Math. Fis. Univ. Modena, XXXVIII (1990), 481–491.

    MathSciNet  MATH  Google Scholar 

  26. Franciosi, M. and Moscariello, G., Higher integrability results, Manuscripta Math. 52 (1985), 151–170.

    Article  MathSciNet  MATH  Google Scholar 

  27. Fusco, N. and Sbordone, C., Higher integrability of the gradient of minimizers of functionals with nonstandard growth conditions, Comm. Pure Applied Math., vol. XLIII (1990), 673–683.

    Article  MathSciNet  MATH  Google Scholar 

  28. Gehring, F.W., Rings and quasiconformal mappings in space, Trans. Amer. Math. Soc. 103 (1962), 353–393.

    Article  MathSciNet  MATH  Google Scholar 

  29. Gehring, F.W., Open problems, Proc. Romanian-Finnish Seminar on Teichmüller Spaces and Quasiconformal Mappings, Romania 1969, page 306.

    Google Scholar 

  30. Gehring, F.W., The L p-integrability of the partial derivatives of a quasiconformal mapping, Acta Math. 130 (1973), 265–277.

    Article  MathSciNet  MATH  Google Scholar 

  31. Gehring, F.W., Topics in quasiconformal mappings, Proceedings of the ICM Berkeley (1986), 62–80.

    Google Scholar 

  32. Gehring, F.W. and Reich, E., Area distortion under quasiconformal mappings, Ann. Acad. Sci. Fenn. Ser. A.I. 388 (1966), 1–14.

    MathSciNet  MATH  Google Scholar 

  33. Giaquinta, M., Multiple integrals in the Calculus of Variations and nonlinear elliptic systems, Annals of Math. Stud. No. 105, Princeton Univ. Press 1983.

    Google Scholar 

  34. Giaquinta, M. and Modica, G., Regularity results for some classes of higher order nonlinear elliptic systems, J. reine angew. Math., 311/312 (1979), 145–169.

    MathSciNet  MATH  Google Scholar 

  35. Granlund, S., Lindqvist, P. and Martio, O., Conformally invariant variational integrals, Trans. Amer. Math. Soc., 277 (1983), 43–73.

    Article  MathSciNet  MATH  Google Scholar 

  36. Gurov, L.G. and Reshetnyak, Yu.G., An analogue of functions with bounded mean oscillation, Sibirsk. Math. J. 17, 3 (1976), 540–546.

    Google Scholar 

  37. de Guzmán, M., Real Variable Methods in Fourier Analysis, Mathematics Studies 46, North-Holland, Amsterdam, New York and Oxford (1981).

    Book  MATH  Google Scholar 

  38. Iwaniec, T., Regularity theorems for the solutions of p.d.e. related to quasiregular mappings in several variables, Preprint of Polish Acad. Sciences (Habilitation Thesis), (1978), 1–45.

    Google Scholar 

  39. Iwaniec, T., Gehring’s reverse maximal function inequality, Proc. International Conference on Approximation and Function Spaces, August 1979, Gdańsk. Edited by Z. Ciesielski, 294–305.

    Google Scholar 

  40. Iwaniec, T., Extremal inequalities in Sobolev spaces and quasiconformal mappings, Zeitschrift für Analysis und ihre Anwedungen Bd. 1 (6), (1982), 1–16.

    MathSciNet  MATH  Google Scholar 

  41. Iwaniec, T., On L p-integrability in p.d.e. and quasiregular mappings for large exponents, Ann. Acad. Sci. Fenn. Ser. A.I. 7, (1982), 301–322.

    MathSciNet  MATH  Google Scholar 

  42. Iwaniec, T., Projections onto gradient fields and L p-estimates for degenerated elliptic operators, Studia Math., 75, (1983), 293–312.

    MathSciNet  MATH  Google Scholar 

  43. Iwaniec, T., Some aspects of p.d.e. and quasiregular mappings, Proceedings of the ICM Warsaw (1983), 1193–1208.

    Google Scholar 

  44. Iwaniec, T., The best constant in a BMO-inequality for the Beurling-Ahlfors transform, Michigan Math. J. 33 (1986), 387–394.

    Article  MathSciNet  MATH  Google Scholar 

  45. Iwaniec, T., Hilbert transform in the complex plane and area inequalities for certain quadratic differentials, Michigan Math. J. 34 (1987), 407–434.

    Article  MathSciNet  MATH  Google Scholar 

  46. Iwaniec, T., Lectures on quasiconformal mappings at Syracuse University, Notes (1987–88), 1–426.

    Google Scholar 

  47. Iwaniec, T., p-harmonic tensors and quasiregular mappings, Ann. Math. (to appear).

    Google Scholar 

  48. Iwaniec, T. and Kosecki, R., Sharp estimates for complex potentials and quasiconformal mappings, preprint of Syracuse University, 1–69.

    Google Scholar 

  49. Iwaniec, T. and Martin, G., Quasiregular mappings in even dimensions, Mittag-Leffler Institute, Report No. 19, 1989/90, Acta Math. (to appear).

    Google Scholar 

  50. Iwaniec, T. and Martin, G., The Beurling-Ahlfors transform in ℝn and related singular inegrals, Preprint of Inst. Hautes Etudes Sci., 1990.

    Google Scholar 

  51. Iwaniec, T. and Martin, G., Quasiconformal mappings and capacity, Indiana Univ. Math. J., 40 No. 1 (1991), 101–122.

    Article  MathSciNet  MATH  Google Scholar 

  52. Iwaniec, T. and Manfredi. J., Regularity of p-Harmonic functions on the plane, Revista Mathemática Iberoamericana, Vol. 5, No. 1 (1989), 1–19.

    Article  MathSciNet  MATH  Google Scholar 

  53. Iwaniec, T. and Nolder, C., The Hardy-Littlewood inequality for quasiregular mapping in certain domains in ℝn, Ann. Acad. Sci. Fenn. Ser. A.I. vol. 10 (1985), 267–282.

    MathSciNet  MATH  Google Scholar 

  54. Iwaniec, T. and Sbordone, C., On the integrability of the Jacobian under minimal hypotheses, Arch. Rat. Mech. Anal. (to appear).

    Google Scholar 

  55. Iwaniec, T. and Sbordone, C., Weak minima of variational integrals, in preparation.

    Google Scholar 

  56. Iwaniec, T. and Sverák, V., On mappings with integrable dilatation, (to appear).

    Google Scholar 

  57. Järvi, P. and Vuorinen, M., Self-similar Cantor sets and quasiregular mappings, J. reine angew. Math. 424 (1992), 31–45.

    MathSciNet  MATH  Google Scholar 

  58. Koskela, P. and Martio, O., Removability theorems for quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A.I. vol. 15 (1990), 381–399.

    MathSciNet  MATH  Google Scholar 

  59. Lehto, O., Remarks on the integrability of the derivatives of quasiconformal mappings, Ann. Acad. Sci. Fenn. Ser. A.I. 371 (1965), 1–8.

    MathSciNet  MATH  Google Scholar 

  60. Lehto, O., Quasiconformal mappings and singular integrals, Symposia Mathematica, vol. XVIII, Academic Press London (1976), 429–453.

    Google Scholar 

  61. Lehto, O. and Virtanen, K., Quasiconformal mappings in the plane, Second Edition, Springer-Verlag, New York — Heidelberg 1973.

    Book  MATH  Google Scholar 

  62. Marcellini, P. and Sbordone, C., On the existence of minima of multiple integrals of the Calculus of Variations, J. Math. Pures Appl. 62 (1983), 1–9.

    MathSciNet  MATH  Google Scholar 

  63. Martio, O., On the integrability of the derivatives of a quasiregular mapping, Math. Scand. 35 (1974), 43–48.

    MathSciNet  MATH  Google Scholar 

  64. Martio, O., Rickman, S. and Väisälä, J., Distortion and singularities of quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A.I. 465 (1970), 1–13.

    MathSciNet  MATH  Google Scholar 

  65. Meyers, N. and Elcrat, A., Some results on regularity for solutions of nonlinear elliptic systems and quasiregular functions, Duke Math. J., vol. 42 (1) (1975), 121–136.

    Article  MathSciNet  MATH  Google Scholar 

  66. Migliaccio, L., Reverse Hölder from reverse Jensen inequalities, An International Workshop, Capri, Sept. 17–20, 1990, 129–134.

    Google Scholar 

  67. Muckenhoupt, B., Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207–226.

    Article  MathSciNet  MATH  Google Scholar 

  68. Müller, S., Higher integrability of determinants and weak convergence in L 1, J. reine angew. Math. 412 (1990), 20–34.

    MathSciNet  MATH  Google Scholar 

  69. Pelczynski, A., Norms of classical operators in function spaces, Colloque Laurent Schwartz, Astérisque 131 (1985), 137–162.

    MathSciNet  MATH  Google Scholar 

  70. Pichorides, S.K., On the best values of the constants in the theorems of M. Riesz, Zygmund and Kolmogorov, Studia Math. 44 (1972), 165–179.

    MathSciNet  MATH  Google Scholar 

  71. Reich, E., Some estimates for the two-dimensional Hilbert transform, J. Analyse Math., vol. XVIII (1967), 279–293.

    Article  MathSciNet  MATH  Google Scholar 

  72. Reshetnyak, Yu.G., On the stability of conformal mappings in multidimensional spaces, Siberian Math. J. 8 (1967), 65–85.

    Google Scholar 

  73. Reshetnyak, Yu.G., On extremal properties of mappings with bounded distortion, Sib. Math. J. 10 (1969), 1300–1310.

    MathSciNet  Google Scholar 

  74. Reshetnyak, Yu.G., Stability estimates in Liouville’s theorem and the L p-integrability of the derivatives of quasiconformal mappings, Sib. Math. J. 17 (1976), 868–896.

    Article  Google Scholar 

  75. Reshetnyak, Yu.G., Differentiability properties of quasiconformal mappings and conformal mappings of Riemannian spaces, Sibirsk. Math. J. 19 (1978), 1166–1183.

    MathSciNet  Google Scholar 

  76. Reshetnyak, Yu.G., Space mappings with bounded distortion, Trans. Math. Monographs, Amer. Math. Soc., Vol. 73, 1989.

    Google Scholar 

  77. Rickman, S., Nonremovable Cantor sets for bounded quasiregular mappings, Mittag-Leffler Institute, Report No. 42, 1989/90.

    Google Scholar 

  78. Rickman, S., Quasiregular Mappings, Springer-Verlag, (to appear).

    Google Scholar 

  79. Sbordone, C., Rearrangement of functions and reverse Hölder inequalities, Ennio De Giorgi Colloquium Res. Notes in Math., Pitman, 125 (1985), 139–148.

    MathSciNet  Google Scholar 

  80. Sbordone, C., On some integral inequalities and their applications to the Calculus of Variations, Boll. Unione Mat. Ital. (6), 5 (1986), 73–94.

    MathSciNet  MATH  Google Scholar 

  81. Sbordone, C., Rearrangement of functions and reverse Jensen inequalities, Proc. of Symposia in Pure Math. vol. 45 (1986), Part 2, 325–329.

    Article  MathSciNet  MATH  Google Scholar 

  82. Sibner, L.M. and Sibner, R.B., A non-linear Hodge de Rham theorem, Acta Math., 125, (1970), 57–73.

    Article  MathSciNet  MATH  Google Scholar 

  83. Stein, E.M., Note on the class L log L, Studia Math. 32 (1969), 305–310.

    MathSciNet  MATH  Google Scholar 

  84. Stein, E.M., Singular integrals and differentiability properties of functions, Princeton Univ. Press, Princeton, N.J., 1970.

    MATH  Google Scholar 

  85. Stredulinsky, E.W., Higher integrability from reverse Hölder inequalities, Indiana Univ. Math. J. 29, 3 (1980), 408–417.

    Article  MathSciNet  MATH  Google Scholar 

  86. Uchiyama, A., On the compactness of operators of Hankel types, Tôhoku Math. J. 30, (1978), 163–171.

    Article  MathSciNet  MATH  Google Scholar 

  87. Uhlenbeck, K., Regularity for a class of non-linear elliptic systems, Acta Math., 138 (1977), 219–240.

    Article  MathSciNet  MATH  Google Scholar 

  88. Väisälä, J., Lectures on n-dimensional quasiconformal mappings, Lecture Notes in Mathematics, 229, Springer-Verlag 1971.

    Google Scholar 

  89. Vuorinen, M., Conformal geometry and quasiregular mappings, Lecture Notes in Mathematics, 1319, Springer-Verlag 1988.

    Google Scholar 

  90. Wiener, N., The ergodic theorem, Duke Math. J., 5, (1939), 1–18.

    Article  MathSciNet  MATH  Google Scholar 

  91. Wik, I., A comparison for the integrability of f and Mf with that of f #, Preprint Series No. 2 (1983), Dept. Math., University of Umeå, Sweden.

    Google Scholar 

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Iwaniec, T. (1992). Lp-theory of quasiregular mappings. In: Vuorinen, M. (eds) Quasiconformal Space Mappings. Lecture Notes in Mathematics, vol 1508. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0094237

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