Skip to main content
SpringerLink
Log in

The Hardy-Littlewood Maximal Operator

  • Chapter
  • First Online:
Variable Lebesgue Spaces

Part of the book series: Applied and Numerical Harmonic Analysis ((ANHA))

  • 1848 Accesses

Abstract

In this chapter we begin the study of harmonic analysis on variable Lebesgue spaces. Our goal is to determine the behavior of some of the classical operators of harmonic analysis—approximate identities, singular integrals, and Riesz potentials. The foundation for this is the Hardy-Littlewood maximal operator, which is the subject of Chaps. 3 and 4.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Bibliography

  1. M. Aguilar Cañestro and P. Ortega Salvador. Weighted weak type inequalities with variable exponents for Hardy and maximal operators. Proc. Japan Acad. Ser. A Math. Sci., 82(8):126–130, 2006.

    Article  MathSciNet  MATH  Google Scholar 

  2. C. Bennett and R. Sharpley. Interpolation of Operators, volume 129 of Pure and Applied Mathematics. Academic Press Inc., Boston, MA, 1988.

    MATH  Google Scholar 

  3. A. L. Bernardis, A. Gogatishvili, F. J. Martín-Reyes, P. Ortega Salvador, and L. Pick. The one-sided A p conditions and local maximal operator. Proc. Edinb. Math. Soc. (2), 55(1):79–104, 2012.

    Google Scholar 

  4. S. Boza and J. Soria. Weighted Hardy modular inequalities in variable L p spaces for decreasing functions. J. Math. Anal. Appl., 348(1):383–388, 2008.

    Article  MathSciNet  MATH  Google Scholar 

  5. A. P. Calderón and A. Zygmund. On the existence of certain singular integrals. Acta Math., 88:85–139, 1952.

    Article  MathSciNet  MATH  Google Scholar 

  6. C. Capone, D. Cruz-Uribe, and A. Fiorenza. The fractional maximal operator and fractional integrals on variable L p spaces. Rev. Mat. Iberoam., 23(3):743–770, 2007.

    Article  MathSciNet  MATH  Google Scholar 

  7. G. Chiti. Rearrangements of functions and convergence in Orlicz spaces. Applicable Anal., 9(1):23–27, 1979.

    Article  MathSciNet  MATH  Google Scholar 

  8. D. Cruz-Uribe. The class \(A_{\infty }^{+}(g)\) and the one-sided reverse Hölder inequality. Canad. Math. Bull., 40(2):169–173, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  9. D. Cruz-Uribe. The Hardy-Littlewood maximal operator on variable-L p spaces. In Seminar of Mathematical Analysis (Malaga/Seville, 2002/2003), volume 64 of Colecc. Abierta, pages 147–156. Univ. Sevilla Secr. Publ., Seville, 2003.

    Google Scholar 

  10. D. Cruz-Uribe. Interpolation of positive operators on variable Lebesgue spaces. Math. Ineq. Appl., 15(3):639–644, 2012.

    MathSciNet  MATH  Google Scholar 

  11. D. Cruz-Uribe, L. Diening, and A. Fiorenza. A new proof of the boundedness of maximal operators on variable Lebesgue spaces. Boll. Unione Mat. Ital. (9), 2(1):151–173, 2009.

    Google Scholar 

  12. D. Cruz-Uribe and A. Fiorenza. Approximate identities in variable L p spaces. Math. Nachr., 280(3):256–270, 2007.

    Article  MathSciNet  MATH  Google Scholar 

  13. D. Cruz-Uribe and A. Fiorenza. LlogL results for the maximal operator in variable L p spaces. Trans. Amer. Math. Soc., 361(5):2631–2647, 2009.

    Article  MathSciNet  MATH  Google Scholar 

  14. D. Cruz-Uribe, A. Fiorenza, and C. J. Neugebauer. The maximal function on variable L p spaces. Ann. Acad. Sci. Fenn. Math., 28(1):223–238, 2003. See also errata [63].

    Google Scholar 

  15. D. Cruz-Uribe, A. Fiorenza, and C. J. Neugebauer. Corrections to: “The maximal function on variable L p spaces” [Ann. Acad. Sci. Fenn. Math. 28 (2003), no. 1, 223–238; ]. Ann. Acad. Sci. Fenn. Math., 29(1):247–249, 2004.

    Google Scholar 

  16. D. Cruz-Uribe and C. J. Neugebauer. The structure of the reverse Hölder classes. Trans. Amer. Math. Soc., 347(8):2941–2960, 1995.

    MathSciNet  MATH  Google Scholar 

  17. D. Cruz-Uribe and C. J. Neugebauer. Weighted norm inequalities for the centered maximal operator on R  + . Ricerche Mat., 48(2):225–241 (2000), 1999.

    Google Scholar 

  18. A. de la Torre. On the adjoint of the maximal function. In Function spaces, differential operators and nonlinear analysis (Paseky nad Jizerou, 1995), pages 189–194. Prometheus, Prague, 1996.

    Google Scholar 

  19. L. Diening. Maximal function on generalized Lebesgue spaces L p(⋅). Math. Inequal. Appl., 7(2):245–253, 2004.

    MathSciNet  MATH  Google Scholar 

  20. L. Diening. Lebesgue and Sobolev Spaces with Variable Exponent. Habilitation, Universität Freiburg, 2007.

    Google Scholar 

  21. L. Diening, P. Harjulehto, P. Hästö, Y. Mizuta, and T. Shimomura. Maximal functions in variable exponent spaces: limiting cases of the exponent. Ann. Acad. Sci. Fenn. Math., 34(2):503–522, 2009.

    MathSciNet  MATH  Google Scholar 

  22. L. Diening, P. Harjulehto, P. Hästö, and M. R˚užička. Lebesgue and Sobolev spaces with Variable Exponents, volume 2017 of Lecture Notes in Mathematics. Springer, Heidelberg, 2011.

    Google Scholar 

  23. L. Diening, P. Hästö, and A. Nekvinda. Open problems in variable exponent Lebesgue and Sobolev spaces. In FSDONA04 Proceedings (Drabek and Rakosnik (eds.); Milovy, Czech Republic, pages 38–58. Academy of Sciences of the Czech Republic, Prague, 2005.

    Google Scholar 

  24. L. Diening and S. Samko. Hardy inequality in variable exponent Lebesgue spaces. Fract. Calc. Appl. Anal., 10(1):1–18, 2007.

    MathSciNet  MATH  Google Scholar 

  25. J. Duoandikoetxea. Fourier Analysis, volume 29 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2001.

    MATH  Google Scholar 

  26. D. E. Edmunds, V. Kokilashvili, and A. Meskhi. On the boundedness and compactness of weighted Hardy operators in spaces L p(x). Georgian Math. J., 12(1):27–44, 2005.

    MathSciNet  MATH  Google Scholar 

  27. D. E. Edmunds, V. Kokilashvili, and A. Meskhi. One-sided operators in L p(x) spaces. Math. Nachr., 281(11):1525–1548, 2008.

    Article  MathSciNet  MATH  Google Scholar 

  28. D. E. Edmunds, J. Lang, and A. Nekvinda. On L p(x) norms. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 455(1981):219–225, 1999.

    Article  MathSciNet  MATH  Google Scholar 

  29. L. Ephremidze, V. Kokilashvili, and S. Samko. Fractional, maximal and singular operators in variable exponent Lorentz spaces. Fract. Calc. Appl. Anal., 11(4):407–420, 2008.

    MathSciNet  MATH  Google Scholar 

  30. A. Fiorenza. The maximal operator and the Sobolev theorem on variable exponent spaces. In Proceedings of the Renato Caccioppoli Centenary Conference, Napoli, September 23–25, 2004, volume 54(2) of Richerche Mat., pages 571–578, 2005.

    Google Scholar 

  31. A. Fiorenza and M. Krbec. On the domain and range of the maximal operator. Nagoya Math. J., 158:43–61, 2000.

    MathSciNet  MATH  Google Scholar 

  32. A. Fiorenza and J. M. Rakotoson. Relative rearrangement and lebesgue spaces L p(⋅) with variable exponent. J. Math. Pures Appl. (9), 88(6):506–521, 2007.

    Google Scholar 

  33. T. Futamura and Y. Mizuta. Maximal functions for Lebesgue spaces with variable exponent approaching 1. Hiroshima Math. J., 36(1):23–28, 2006.

    MathSciNet  MATH  Google Scholar 

  34. T. Futamura, Y. Mizuta, and T. Shimomura. Integrability of maximal functions and Riesz potentials in Orlicz spaces of variable exponent. J. Math. Anal. Appl., 366(2):391–417, 2010.

    Article  MathSciNet  MATH  Google Scholar 

  35. J. García-Cuerva and J. L. Rubio de Francia. Weighted Norm Inequalities and Related Topics, volume 116 of North-Holland Mathematics Studies. North-Holland Publishing Co., Amsterdam, 1985.

    MATH  Google Scholar 

  36. L. Grafakos. Classical Fourier Analysis, volume 249 of Graduate Texts in Mathematics. Springer, New York, 2nd edition, 2008.

    MATH  Google Scholar 

  37. G. H. Hardy and J. E. Littlewood. A maximal theorem with function-theoretic applications. Acta Math., 54(1):81–116, 1930.

    Article  MathSciNet  MATH  Google Scholar 

  38. P. Harjulehto and P. Hästö. Sobolev inequalities with variable exponent attaining the values 1 and n. Publ. Mat., 52(2):347–363, 2008.

    Google Scholar 

  39. P. Harjulehto, P. Hästö, and M. Koskenoja. Hardy’s inequality in a variable exponent Sobolev space. Georgian Math. J., 12(3):431–442, 2005.

    MathSciNet  MATH  Google Scholar 

  40. P. Harjulehto, P. Hästö, Y. Mizuta, and T. Shimomura. Iterated maximal functions in variable exponent Lebesgue spaces. Manuscripta Math., 135(3–4):381–399, 2011.

    Article  MathSciNet  MATH  Google Scholar 

  41. A. Harman. On necessary condition for the variable exponent Hardy inequality. J. Funct. Spaces Appl., 2012, 2012. Article ID 385925.

    Google Scholar 

  42. A. Harman and F. Mamedov. On boundedness of weighted Hardy operator in L p(⋅) and regularity condition. J. Inequal. Appl., pages Art. ID 837951, 14, 2010.

    Google Scholar 

  43. P. Hästö. The maximal operator in Lebesgue spaces with variable exponent near 1. Math. Nachr., 280(1–2):74–82, 2007.

    Article  MathSciNet  MATH  Google Scholar 

  44. P Hästö, Y Mizuta, T Ohno, and T Shimomura. Sobolev inequalities for Orlicz spaces of two variable exponents. Glasgow Math. J., pages 1–14, 2009.

    Google Scholar 

  45. A. Yu. Karlovich and A. K. Lerner. Commutators of singular integrals on generalized L p spaces with variable exponent. Publ. Mat., 49(1):111–125, 2005.

    Google Scholar 

  46. V. Kokilashvili and M. Krbec. Weighted Inequalities in Lorentz and Orlicz Spaces. World Scientific Publishing Co. Inc., River Edge, NJ, 1991.

    Book  MATH  Google Scholar 

  47. V. Kokilashvili and A. Meskhi. Weighted criteria for generalized fractional maximal functions and potentials in Lebesgue spaces with variable exponent. Integral Transforms Spec. Funct., 18(9–10):609–628, 2007.

    Article  MathSciNet  MATH  Google Scholar 

  48. V. Kokilashvili and A. Meskhi. Two-weighted norm inequalities for the double Hardy transforms and strong fractional maximal functions in variable exponent Lebesgue space. In Spectral theory, function spaces and inequalities, volume 219 of Oper. Theory Adv. Appl., pages 105–124. Birkhäuser/Springer Basel AG, Basel, 2012.

    Google Scholar 

  49. V. Kokilashvili and S. Samko. On Sobolev theorem for Riesz-type potentials in Lebesgue spaces with variable exponent. Z. Anal. Anwendungen, 22(4):899–910, 2003.

    Article  MathSciNet  Google Scholar 

  50. V. Kokilashvili and S. Samko. Singular integrals and potentials in some Banach function spaces with variable exponent. J. Funct. Spaces Appl., 1(1):45–59, 2003.

    Article  MathSciNet  MATH  Google Scholar 

  51. V. Kokilashvili and S. Samko. Maximal and fractional operators in weighted L p(x) spaces. Rev. Mat. Iberoamericana, 20(2):493–515, 2004.

    Article  MathSciNet  MATH  Google Scholar 

  52. T. Kopaliani. A note on strong maximal operator in \({L}^{p(\cdot )}({\mathbb{R}}^{n})\) spaces. Proc. A. Razmadze Math. Inst., 145:43–46, 2007.

    MathSciNet  MATH  Google Scholar 

  53. T. Kopaliani. Interpolation theorems for variable exponent Lebesgue spaces. J. Funct. Anal., 257(11):3541–3551, 2009.

    Article  MathSciNet  MATH  Google Scholar 

  54. S. G. Kreĭn, Yu. Ī. Petunı̄n, and E. M. Semënov. Interpolation of Linear Operators, volume 54 of Translations of Mathematical Monographs. American Mathematical Society, Providence, R.I., 1982. Translated from the Russian by J. Szűcs.

    Google Scholar 

  55. A. Kufner, L. Maligranda, and L.-E. Persson. The Hardy Inequality. Vydavatelský Servis, Plzeň, 2007. About its history and some related results.

    Google Scholar 

  56. A. K. Lerner. On modular inequalities in variable L p spaces. Arch. Math. (Basel), 85(6):538–543, 2005.

    Google Scholar 

  57. A. K. Lerner and C. Pérez. A new characterization of the Muckenhoupt A p weights through an extension of the Lorentz-Shimogaki theorem. Indiana Univ. Math. J., 56(6):2697–2722, 2007.

    Article  MathSciNet  MATH  Google Scholar 

  58. E. H. Lieb and M. Loss. Analysis, volume 14 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, second edition, 2001.

    MATH  Google Scholar 

  59. L. Maligranda. Józef Marcinkiewicz (1910–1940) – On the centenary of his birth. In M. Nawrocki and W. Wnuk, editors, Marcinkiewicz Centenary Volume, volume 95 of Banach Center Publications, pages 133–234. Institute of Mathematics, Polish Academy of Sciences, 2011.

    Google Scholar 

  60. F. Mamedov and A. Harman. On a weighted inequality of Hardy type in spaces L p(⋅). J. Math. Anal. Appl., 353(2):521–530, 2009.

    Article  MathSciNet  MATH  Google Scholar 

  61. F. J. Martín-Reyes. New proofs of weighted inequalities for the one-sided Hardy-Littlewood maximal functions. Proc. Amer. Math. Soc., 117(3):691–698, 1993.

    MathSciNet  MATH  Google Scholar 

  62. F. J. Martín-Reyes, P. Ortega Salvador, and A. de la Torre. Weighted inequalities for one-sided maximal functions. Trans. Amer. Math. Soc., 319(2):517–534, 1990.

    MathSciNet  MATH  Google Scholar 

  63. R. A. Mashiyev, B. Çekiç, F. Mamedov, and S. Ogras. Hardy’s inequality in power-type weighted L p(⋅)(0, ∞) spaces. J. Math. Anal. Appl., 334(1):289–298, 2007.

    Article  MathSciNet  MATH  Google Scholar 

  64. R. A. Mashiyev, B. Çekiç, and S. Ogras. On Hardy’s inequality in L p(x)(0, ∞). JIPAM. J. Inequal. Pure Appl. Math., 7(3):Article 106, 5 pp. (electronic), 2006.

    Google Scholar 

  65. Y. Mizuta, T. Ohno, and T. Shimomura. Integrability of maximal functions for generalized Lebesgue spaces with variable exponent. Math. Nachr., 281(3):386–395, 2008.

    Article  MathSciNet  MATH  Google Scholar 

  66. Y. Mizuta, T. Ohno, and T. Shimomura. Sobolev’s inequalities and vanishing integrability for Riesz potentials of functions in the generalized Lebesgue space \({L}^{p(\cdot )}{(\log L)}^{q(\cdot )}\). J. Math. Anal. Appl., 345(1):70–85, 2008.

    Article  MathSciNet  MATH  Google Scholar 

  67. B. Muckenhoupt and R. L. Wheeden. Weighted norm inequalities for fractional integrals. Trans. Amer. Math. Soc., 192:261–274, 1974.

    Article  MathSciNet  MATH  Google Scholar 

  68. J. Musielak. Orlicz Spaces and Modular Spaces, volume 1034 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1983.

    MATH  Google Scholar 

  69. P. Nägele. Monotone operators in spaces with variable exponents. Master’s thesis, Universität Freiburg, 2009.

    Google Scholar 

  70. A. Nekvinda. Hardy-Littlewood maximal operator on \({L}^{p(x)}({\mathbb{R}}^{n})\). Math. Inequal. Appl., 7(2):255–265, 2004.

    MathSciNet  MATH  Google Scholar 

  71. A. Nekvinda. A note on one-sided maximal operator in \({L}^{p(\cdot )}(\mathbb{R})\). Math. Inequal. Appl., to appear.

    Google Scholar 

  72. C. J. Neugebauer. Weighted variable L p integral inequalities for the maximal operator on non-increasing functions. Studia Math., 192(1):51–60, 2009.

    Article  MathSciNet  MATH  Google Scholar 

  73. B. Opic and A. Kufner. Hardy-type Inequalities, volume 219 of Pitman Research Notes in Mathematics Series. Longman Scientific & Technical, Harlow, 1990.

    MATH  Google Scholar 

  74. H. Rafeiro and S. Samko. Hardy type inequality in variable Lebesgue spaces. Ann. Acad. Sci. Fenn. Math., 34(1):279–289, 2009.

    MathSciNet  MATH  Google Scholar 

  75. H. Rafeiro and S. Samko. Corrigendum to “Hardy type inequality in variable Lebesgue spaces” [mr2489027]. Ann. Acad. Sci. Fenn. Math., 35(2):679–680, 2010.

    Article  MathSciNet  Google Scholar 

  76. J. M. Rakotoson. Réarrangement Relatif, volume 64 of Mathématiques & Applications (Berlin) [Mathematics & Applications]. Springer, Berlin, 2008.

    Book  MATH  Google Scholar 

  77. J. M. Rakotoson. Lipschitz properties in variable exponent problems via relative rearrangement. Chin. Ann. Math., 31B(6):991–1006, 2010.

    Article  MathSciNet  Google Scholar 

  78. W. Rudin. Real and Complex Analysis. McGraw-Hill Book Co., New York, third edition, 1987.

    MATH  Google Scholar 

  79. Y. Sakai. Spectral orders and differences. Hokkaido Math. J., 13(3):271–276, 1984.

    MathSciNet  MATH  Google Scholar 

  80. S. Samko. Hardy inequality in the generalized Lebesgue spaces. Fract. Calc. Appl. Anal., 6(4):355–362, 2003.

    MathSciNet  MATH  Google Scholar 

  81. S. Samko. Hardy-Littlewood-Stein-Weiss inequality in the Lebesgue spaces with variable exponent. Fract. Calc. Appl. Anal., 6(4):421–440, 2003.

    MathSciNet  MATH  Google Scholar 

  82. E. T. Sawyer. A characterization of a two-weight norm inequality for maximal operators. Studia Math., 75(1):1–11, 1982.

    MathSciNet  MATH  Google Scholar 

  83. E. T. Sawyer. Weighted inequalities for the one-sided Hardy-Littlewood maximal functions. Trans. Amer. Math. Soc., 297(1):53–61, 1986.

    Article  MathSciNet  MATH  Google Scholar 

  84. G. Sinnamon. Four questions related to Hardy’s inequality. In Function spaces and applications (Delhi, 1997), pages 255–266. Narosa, New Delhi, 2000.

    Google Scholar 

  85. E. M. Stein. Note on the class L log L. Studia Math., 32:305–310, 1969.

    Google Scholar 

  86. E. M. Stein. Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series, No. 30. Princeton University Press, Princeton, N.J., 1970.

    Google Scholar 

  87. E. M. Stein and G. Weiss. Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press, Princeton, N.J., 1971. Princeton Mathematical Series, No. 32.

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Trinity College, Hartford, Connecticut, USA

    David V. Cruz-Uribe

  2. Dipartimento di Architettura, Università di Napoli “Federico II”, Napoli, Italy

    Alberto Fiorenza

Authors
  1. David V. Cruz-Uribe
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alberto Fiorenza
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer Basel

About this chapter

Cite this chapter

Cruz-Uribe, D.V., Fiorenza, A. (2013). The Hardy-Littlewood Maximal Operator. In: Variable Lebesgue Spaces. Applied and Numerical Harmonic Analysis. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0548-3_3

Download citation

Publish with us

Policies and ethics

Search

Navigation