The Gehring Lemma

  • Tadeusz Iwaniec

Abstract

I am honored and pleased to comment on the work of Professor Frederick W. Gehring on the occasion of his 70th birthday. Because of his pioneering ideas, the theory of quasiconformal mappings has enjoyed an extensive development. Reverse Hölder inequalities represent one of Fred’s supreme accomplishments, with enormous repercussions for nonlinear PDEs. I am pleased to acknowledge the immense influence Fred has had on me and my work. His wonderful lemma, in particular, has shaped my own views on the L p -theory of mappings of finite distortion and their governing equations. It is to Fred and this lemma that my article is dedicated.

Keywords

Convolution Stein 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [B1]
    B. Bojarski, Homeomorphic solutions of Beltrami systems, Dokl. Akad. Nauk SSSR 102 (1955), 661–664. (Russian)MathSciNetGoogle Scholar
  2. [B2]
    B. Bojarski, Remarks on stability of reverse Hölder inequalities and qua-siconformal mappings Ann. Acad. Sci. Fenn. Ser. A.I. (1985), 291–296.Google Scholar
  3. [BI]
    B. Bojarski and T. Iwaniec, Analytical foundations of the theory of qua-siconfomal mappings in R n, Ann. Acad. Sci. Fenn. Ser. A.I. 8 (1983),257–324.MathSciNetGoogle Scholar
  4. [BIS]
    L. Budney, T. Iwaniec and B. Stroffolini, Removability of singularities of A-harmonic functions, Advances in Differential Equations, (to appear).Google Scholar
  5. [BK]
    S. Buckley and P. KoskelaSobolev-Poincaré inequalities for p < 1, Indi-ana Univ. Math. J. 43 (1994), 221–240.MathSciNetMATHGoogle Scholar
  6. [BKL]
    S. Buckley, P. Koskela, and G. Lu, Subelliptic Poincaré inequalities: the case p < 1, preprint.Google Scholar
  7. [BMR]
    J. Bastero, M. Milman and F.J. Ruiz, Reverse Hölder inequalities and interpolation,preprint, 1996.Google Scholar
  8. [BP]
    R. Bagby and D. Parson, Orlicz Spaces and rearranged maximal functions Math. Nachr. 132 (1987), 15–27.MathSciNetCrossRefGoogle Scholar
  9. [BR]
    J. Bastero and F.J. Ruiz, Elementary reverse Hölder type inequalities with applications to interpolation of operator theory, Proc. AMS, (to appear).Google Scholar
  10. [Bu]
    S. BuckleyPointwise multipliers for reverse Hölder spaces, Studia Math-ematica 109 (1994), 23–39.MathSciNetMATHGoogle Scholar
  11. [CN]
    D. Cruz-Uribe and C.J. Neugebauer, The structure of the reverse Hölder classes, preprint.Google Scholar
  12. [CNO]
    D. Cruz-Uribe, C.J. Neugebauer and V. Olsen, The one-sided minimal operator and the one-sided reverse Hölder inequality, Studia Mathematica.Google Scholar
  13. [Cr]
    D. Cruz-Uribe, The class A + (g) and the one-sided reverse Hölder in-equality, preprint.Google Scholar
  14. [DS]
    L. D’Apuzzo and C. Sbordone, Reverse Hölder inequalities, a sharp result, Rendiconti di Matematica, Ser. VII 10 (1990), 357–366.MathSciNetMATHGoogle Scholar
  15. [EM]
    A. Elcrat and N. Meyers, Some results on regularity for solutions of nonlinear elliptic systems and quasiregular functions, Duke Math. J. 42 (1) (1975), 121–136.MathSciNetMATHGoogle Scholar
  16. [FM]
    M. Franciosi and G. Moscariello, Higher integrability results, Manuscripta Math 52 (1985), 151–170.MathSciNetMATHCrossRefGoogle Scholar
  17. [FC]
    B. Franchi and F. Serra Cassano, Gehring’s Lemma for metrics and higher integrability of the gradient for minimizers of noncoercive variational functionals, Studia Mathematica, (to appear).Google Scholar
  18. [FS]
    N. Fusco and C. Sbordone, Higher integrability of the gradient of mini-mizers of functionals with nonstandard growth conditions, Comm. Pure Applied Math. XLIII (1990), 673–683.Google Scholar
  19. [G1]
    F.W. Gehring, The L P -integrability of the partial derivatives of a quasi-conformal mapping, Acta Math. 130 (1973), 265–277.MathSciNetMATHCrossRefGoogle Scholar
  20. [G2]
    F.W. Gehring, Topics in quasiconformal mappings, Proceedings of the ICM, Berkeley (1986), 62–80.Google Scholar
  21. [GG]
    M. Giaquinta and E. Giusti, On the regularity of the minima of variational integrals, Acta Math. 148 (1982), 31–46.MathSciNetMATHCrossRefGoogle Scholar
  22. [Gil]
    M. Giaquinta, Multiple integrals in the calculus of variations and non-linear elliptic systems,Annals of Math. Stud. No. 105 (1983), Princeton University Press.Google Scholar
  23. [GI]
    L. Greco and T. Iwaniec, New inequalities for the Jacobian, Ann. Inst. H. Poincaré 11 (1994), 17–35.MathSciNetMATHGoogle Scholar
  24. [GIM]
    L. Greco, T. Iwaniec, and G. MoscarielloLimits of the improved integrability of the volume forms, Indiana Univ. Math. J. 44 (2) (1995), 305–339.MathSciNetMATHGoogle Scholar
  25. [GIS]
    L. Greco, T. Iwaniec, and C. Sbordone, Variational integrals of nearly linear growth,Differential and Integral Equations, (to appear).Google Scholar
  26. [GM]
    M. Giaquinta and G. Modica, Regularity results for some classes of higher order nonlinear elliptic systems, J. reine angew. Math 311/312 (1979), 145–169.Google Scholar
  27. [HKM]
    J.T. Heinonen, T. Kilpeläinen, and O. Martio Nonlinear potential theory of degenerate elliptic equations Oxford Univ. Press, 1993.Google Scholar
  28. [Il]
    T. IwaniecOn L p -integrability in p.d.e and quasiregualr mappings for large exponents Ann. Acad. Sci. Fenn. Ser. A.I. 7 (1982), 301–322.MathSciNetMATHGoogle Scholar
  29. [I2]
    T. Iwaniec Projections onto gradient fields and LP -estimates for degenerated elliptic operators, Studia Mathematica 75 (1983), 293–312.MathSciNetMATHGoogle Scholar
  30. [I3]
    T. Iwaniec, Some aspects of p. d. e. and quasiregular mappings, Proceedings of ICM, Warsaw (1983), 1193–1208.Google Scholar
  31. [I4]
    T. Iwaniec, p-harmonic tensors and quasiregular mappings,Annals of Mathematics 136 (1992), 589–624.MathSciNetMATHCrossRefGoogle Scholar
  32. [I5]
    T. Iwaniec, L p -theory of quasiregular mappings, A collection of surveys 1960–1990, Springer Lecture Notes 1508 (1992), 39–64.MathSciNetGoogle Scholar
  33. [IN]
    T. Iwaniec and C. Nolder, The Hardy-Littlewood inequality for quasireg-ular mappings in certain domains in R n, Ann. Acad. Sci. Fenn. Ser. A.I. 10 (1985), 267–282.MathSciNetMATHGoogle Scholar
  34. [IS]
    T. Iwaniec and C. Sbordone, Weak minima of variational integrals,J. reine angew. Math. 454 (1994), 143–161.MathSciNetMATHGoogle Scholar
  35. [ISS]
    T. Iwaniec, C. Scott and B. Stroffolini, Nonlinear Hodge theory on maniforlds with boundary, preprint.Google Scholar
  36. [Kl]
    J. Kinnunen, Higher integrability with weights, Ann. Acad. Sci. Fenn. Ser.A.I. Math. 19 (1994), 355–366.MathSciNetMATHGoogle Scholar
  37. [K2]
    J. Kinnunen, Sharp results on reverse Hölder inequalities, Ann. Acad.Sci. Fenn. Ser. A.I. Math., Dissertations 95 1994.MATHGoogle Scholar
  38. [LMZ]
    C. Li, A. McIntosh, and K. Zhang, Higher integrability and reverse Hölder inequalities, preprint.Google Scholar
  39. [M]
    B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207–226.MathSciNetMATHCrossRefGoogle Scholar
  40. [Ma]
    O. Martio, On the integrability of the derivative of a quasiregular map-ping, Math. Scand. 3 (1974), 43–48.MathSciNetGoogle Scholar
  41. [Mil]
    M. Milman, A note on Gehring’s Lemma, Ann. Acad. Sci. Fenn. Ser. A.I Math, (to appear).Google Scholar
  42. [Mi2]
    M. Milman, A note on interpolation and higher integrability, Ann. Acad. Sci. Fenn. Ser. A.I Math, (to appear).Google Scholar
  43. [Mi3]
    M. Milman, A note on the reverse Hardy inequalities and Gehring’s Lemma,Comm. in Pure and Appl. Math., (to appear).Google Scholar
  44. [Mu]
    S. Müller, A surprising higher integrability property of mappings with positive determinant, Bull. Amer. Math. Soc. 21 (1989), 245–248.MathSciNetMATHCrossRefGoogle Scholar
  45. [RR]
    M.M. Rao and Z.D. Ren, Theory of Orlicz Spaces, Marcel Dekker, 1991. [S1] E.M. Stein, Note on the class LlogL, Studia Math. 32 (1969), 305–310.MathSciNetGoogle Scholar
  46. [S2]
    E.M. Stein, Harmonic Analysis: real-variable methods, orthogonality,and oscillatory integrals, Chapter V, Princeton Univ. Press, 1993.Google Scholar
  47. [Sbl]
    C. Sbordone, Rearrangement of functions and reverse Hölder inequalities, Ernio De Giorgi Colloquium Re. Notes in Math., Pitman, 124 (1985), 139–148.MathSciNetGoogle Scholar
  48. [Sb2]
    C. Sbordone, Rearrangement of functions and reverse Jensen inequalities, Proc. of Symposia in Pure Math. 45 (1986), 325–329.MathSciNetGoogle Scholar
  49. [Sb3]
    C. Sbordone, On some inequalities and their applications to the calculus of variations, Bolletimo V.M.I. Analisi Funzionale e Applicazioné, Ser. VI V-C (1) (1986), 73–94.MathSciNetGoogle Scholar
  50. [St1]
    E.W. Stredulinsky, Higher integrability from reverse Hölder inequalities, Indiana Univ. Math. J. 29,3 (1980), 408–417.Google Scholar
  51. [St2]
    E.W. Stredulinsky, Weighted inequalities and degenerated elliptic partial differential equations, Lecture Notes in Math. 1074 Springer-Verlag, Berlin-Heidelberg, New York, Tokyo, 1984.Google Scholar
  52. [W]
    N. Wiener, The Fourier Integral and Certain of its Applications, Cam-bridge Univ. Press, 1993.Google Scholar

Copyright information

© Springer Science+Business Media New York 1998

Authors and Affiliations

  • Tadeusz Iwaniec
    • 1
  1. 1.Department of MathematicsSyracuse University SyracuseUSA

Personalised recommendations