Mean values of convexly arranged numbers and monotone rearrangements in reverse integral inequalities

Werner Clemens

Bollettino dell'Unione Matematica Italiana (2005)

  • Volume: 8-B, Issue: 3, page 737-764
  • ISSN: 0392-4041

Abstract

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We analyse mean values of functions with values in the boundary of a convex two-dimensional set. As an application, reverse integral inequalities imply exactly the same inequalities for the monotone rearrangement. Sharp versions of the classical Gehring lemma, the Gurov-Resetnyak theorem and the Muckenhoupt theorem are obtained.

How to cite

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Clemens, Werner. "Mean values of convexly arranged numbers and monotone rearrangements in reverse integral inequalities." Bollettino dell'Unione Matematica Italiana 8-B.3 (2005): 737-764. <http://eudml.org/doc/196051>.

@article{Clemens2005,
abstract = {We analyse mean values of functions with values in the boundary of a convex two-dimensional set. As an application, reverse integral inequalities imply exactly the same inequalities for the monotone rearrangement. Sharp versions of the classical Gehring lemma, the Gurov-Resetnyak theorem and the Muckenhoupt theorem are obtained.},
author = {Clemens, Werner},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {10},
number = {3},
pages = {737-764},
publisher = {Unione Matematica Italiana},
title = {Mean values of convexly arranged numbers and monotone rearrangements in reverse integral inequalities},
url = {http://eudml.org/doc/196051},
volume = {8-B},
year = {2005},
}

TY - JOUR
AU - Clemens, Werner
TI - Mean values of convexly arranged numbers and monotone rearrangements in reverse integral inequalities
JO - Bollettino dell'Unione Matematica Italiana
DA - 2005/10//
PB - Unione Matematica Italiana
VL - 8-B
IS - 3
SP - 737
EP - 764
AB - We analyse mean values of functions with values in the boundary of a convex two-dimensional set. As an application, reverse integral inequalities imply exactly the same inequalities for the monotone rearrangement. Sharp versions of the classical Gehring lemma, the Gurov-Resetnyak theorem and the Muckenhoupt theorem are obtained.
LA - eng
UR - http://eudml.org/doc/196051
ER -

References

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  1. D'APUZZO, L. - SBORDONE, C., Reverse Hölder Inequalities - A sharp Result, Rend. di Matematica, (Ser. 7), 10 (1990), 357-366. Zbl0711.42027MR1076164
  2. BENNETT, C. - Sharpley, R., Interpolation of Operators, Academic Press1988, Pure and Applied Mathematics, 129. Zbl0647.46057MR928802
  3. BOJARSKI, B., Remarks on the stability of reverse Hölder inequalities and quasiconformal mappings, Ann. Acad. Sci. Fenn. Ser. A.I. Math., 10 (1985), 89- 94. Zbl0582.30016MR802470
  4. BOJARSKI, B. - IWANIEC, T., Analytical foundation of the theory of quasiconformal mappings in R n , Ann. Acad. Sci. Fenn. Ser. A. I. Math., 8 (1983), 257-324. Zbl0548.30016MR731786
  5. FIORENZA, A., Regularity Results for Minimizers of Certain One-Dimensional Lagrange Problems of Calculus of Variations, Bollettino U.M.I., (7) 10B (1996), 943-962, Zbl0909.49025MR1430161
  6. FRANCIOSI, M. - MOSCARIELLO, G., Higher integrability results, Manuscripta Math., 52 (1985), 151-170. Zbl0576.42022MR790797
  7. GARCIA-CUERVA, C. - RUBIO DE FRANCIA, J. L., Weighted norm inequalities and related topics, North-Holland Math. Studies, 116 (1985). Zbl0578.46046MR807149
  8. GEHRING, F.W., The L p -integrability of the partial derivates of a quasiconformal mapping, Acta Math., 130 (1973), 265-277. Zbl0258.30021MR402038
  9. GIAQUINTA, M., Multiple integrals in the calculus of variations and nonlinear elliptic systems, Ann. of Math. Study, 105, Princeton Univ. Press (1983). Zbl0516.49003MR717034
  10. GIAQUINTA, M., Introduction to regularity theory for nonlinear elliptic systemsBirkhäuser Verlag, Basel, Boston, Berlin1993. Zbl0786.35001MR1239172
  11. GUROV, L. G. - RESETNYAK, YU. K., A certain analogue of the concept of a function with bounded mean oscillationSibirsk., Math. Zh., 17, 3 (1976), 540- 546. Zbl0341.26006MR427565
  12. IWANIEC, T., On L p -integrability in PDE's and quasiregular mappings for large exponents, Ann. Acad. Sci. Fenn. Ser. A. I., 7 (1982), 301-322. Zbl0505.30011MR686647
  13. IWANIEC, T., The Gehring Lemma, Proc. of the int. Symp., Ann Arbor, MI, USA (1998), 181-204. Zbl0888.30017MR1488451
  14. IWANIEC, T. - MARTIN, G., Geometric function theory and non-linear analysis, Oxford Math. Mono., Oxford Univ. Press, Oxford, 2001. Zbl1045.30011MR1859913
  15. KLEMES, I., A mean oscillation inequality, Proc. Am. Math. Soc., 93, Nr. 3 (1985), 497-500. Zbl0572.46025MR774010
  16. Kinnunen, J., Sharp results on reverse Hölder inequalities, Ann. Acad. Sci. Fenn., Ser. A, 1. Mathematica, Dissert., 95 (1994), 4-34. Zbl0816.26008MR1283432
  17. KOLYADA, V.I., Rearrangements of functions and imbedding theorems, Russ. Math. Surv., 44, No. 5 (1989), 73-117. Zbl0715.41050MR1040269
  18. KORENOVSKIJ, A. A., The exact continuation of a reverse Hölder inequality and Muckenhoupt's conditions, Math. notes, 52, No. 6 (1992), 1192-1201. Zbl0807.42015MR1208001
  19. KORENOVSKIJ, A. A., On the connection between mean value oscillation and exact integrability classes of functions, Math. USSR Sbornik, 71, No. 2 (1992), 561-567; transl. from Mat. Sb., 181, No. 12 (1990), 1721-1727. Zbl0776.30025MR1099524
  20. MUCKENHOUPT, B., Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc., 165 (1972), 207-226. Zbl0236.26016MR293384
  21. NANIA, L., On some reverse integral inequalities, J. Austral. Math. Soc. (Ser. A) 49 (1990), 319-326. Zbl0715.26010MR1061052
  22. SAGAN, H., Space-filling curves, Springer Verlag, New York, 1994. Zbl0806.01019MR1299533
  23. SBORDONE, C., On some integral inequalities and their applications to the calculus of variations, Boll. Un. Mat. Ital., Ana. Func. Appl., 5-10(6,1) (1986), 73-94. Zbl0678.49008MR897186
  24. STEIN, E. M., Harmonic analysis: Real-variable methods, orthogonality and oscillatory integrals. Princeton Math. Series, 43, Princeton, New Jersey1993. Zbl0821.42001MR1232192
  25. WIK, I., Reverse Hölder inequalities with constant close to 1, Ric. Mat., 39, No. 1 (1990), 151-157. Zbl0746.30022MR1101311

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