Skip to main content
SpringerLink
Log in
Book cover

Variable Lebesgue Spaces pp 1–12Cite as

Birkhäuser

Introduction

  • Chapter
  • First Online:

  • 1774 Accesses

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

Abstract

The variable Lebesgue spaces, as their name implies, are a generalization of the classical Lebesgue spaces, replacing the constant exponent p with a variable exponent function p(⋅). The resulting Banach function spaces L p(⋅) have many properties similar to the L p spaces, but they also differ in surprising and subtle ways. For this reason the variable Lebesgue spaces have an intrinsic interest, but they are also very important for their applications to partial differential equations and variational integrals with non-standard growth conditions. The past 20 years, and especially the past decade, have witnessed an explosive growth in the study of these and related spaces.

This is a preview of subscription content, 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

Learn about institutional subscriptions

Bibliography

  1. R. Aboulaich, S. Boujena, and E. El Guarmah. Sur un modèle non-linéaire pour le débruitage de l’image. C. R. Math. Acad. Sci. Paris, 345(8):425–429, 2007.

    Article  MathSciNet  MATH  Google Scholar 

  2. R. Aboulaich, D. Meskine, and A. Souissi. New diffusion models in image processing. Comput. Math. Appl., 56(4):874–882, 2008.

    Article  MathSciNet  MATH  Google Scholar 

  3. E. Acerbi and G. Mingione. Regularity results for electrorheological fluids: the stationary case. C. R. Math. Acad. Sci. Paris, 334(9):817–822, 2002.

    Article  MathSciNet  MATH  Google Scholar 

  4. E. Acerbi and G. Mingione. Gradient estimates for the p(x)-Laplacean system. J. Reine Angew. Math., 584:117–148, 2005.

    Article  MathSciNet  MATH  Google Scholar 

  5. R. A. Adams and J. J. F. Fournier. Sobolev Spaces, volume 140 of Pure and Applied Mathematics (Amsterdam). Elsevier/Academic Press, Amsterdam, second edition, 2003.

    MATH  Google Scholar 

  6. B. Amaziane, L. Pankratov, and A. Piatnitski. Nonlinear flow through double porosity media in variable exponent Sobolev spaces. Nonlinear Anal. Real World Appl., 10(4):2521–2530, 2009.

    Article  MathSciNet  MATH  Google Scholar 

  7. S. N. Antontsev and S. I. Shmarev. A model porous medium equation with variable exponent of nonlinearity: existence, uniqueness and localization properties of solutions. Nonlinear Anal., 60(3):515–545, 2005.

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  9. Z. W. Birnbaum and W. Orlicz. Über die Verallgemeinerung des Begriffes der zueinander konjugierten Potenzen. Stud. Math., 3:1–67, 1931.

    Google Scholar 

  10. P. Blomgren, T. Chan, P. Mulet, and C. K. Wong. Total variation image restoration: numerical methods and extensions. In Proceedings of the 1997 IEEE International Conference on Image Processing, volume III, pages 384–387, 1997.

    Google Scholar 

  11. E. M. Bollt, R. Chartrand, S. Esedoḡlu, P. Schultz, and K. R. Vixie. Graduated adaptive image denoising: local compromise between total variation and isotropic diffusion. Adv. Comput. Math., 31(1–3):61–85, 2009.

    Article  MathSciNet  MATH  Google Scholar 

  12. H. Brezis. Analyse fonctionnelle. Collection Mathématiques Appliquées pour la Maî trise. [Collection of Applied Mathematics for the Master’s Degree]. Masson, Paris, 1983. Théorie et applications. [Theory and applications].

    Google Scholar 

  13. 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 

  14. B. Çekiç, A. V. Kalinin, R. A. Mashiyev, and M. Avci. l p(x)(ω)-estimates for vector fields and some applications to magnetostatics problems. J. Math. Anal. Appl., 389(2):838–851, 2012.

    Google Scholar 

  15. Y. Chen, W. Guo, Q. Zeng, and Y. Liu. A nonstandard smoothing in reconstruction of apparent diffusion coefficient profiles from diffusion weighted images. Inverse Probl. Imaging, 2(2):205–224, 2008.

    Article  MathSciNet  MATH  Google Scholar 

  16. Y. Chen, S. Levine, and M. Rao. Variable exponent, linear growth functionals in image restoration. SIAM J. Appl. Math., 66(4):1383–1406 (electronic), 2006.

    Google Scholar 

  17. 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 

  18. D. Cruz-Uribe, A. Fiorenza, J. M. Martell, and C. Pérez. The boundedness of classical operators on variable L p spaces. Ann. Acad. Sci. Fenn. Math., 31(1):239–264, 2006.

    MathSciNet  Google Scholar 

  19. 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 

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

    MathSciNet  MATH  Google Scholar 

  21. L. Diening. Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces. Bull. Sci. Math., 129(8):657–700, 2005.

    Article  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 and M. R˚užička. Calderón-Zygmund operators on generalized Lebesgue spaces L p(⋅) and problems related to fluid dynamics. J. Reine Angew. Math., 563:197–220, 2003.

    Google Scholar 

  24. L. Diening and M. R˚užička. Non-Newtonian fluids and function spaces. In NAFSA 8—Nonlinear analysis, function spaces and applications. Vol. 8, pages 94–143. Czech. Acad. Sci., Prague, 2007.

    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 and A. Meskhi. Potential-type operators in L p(x) spaces. Z. Anal. Anwendungen, 21(3):681–690, 2002.

    MathSciNet  MATH  Google Scholar 

  27. X. Fan. The regularity of Lagrangians f(x, ξ) =  | ξ | α(x) with Hölder exponents α(x). Acta Math. Sinica (N.S.), 12(3):254–261, 1996. A Chinese summary appears in Acta Math. Sinica 40 (1997), no. 1, 158.

    Google Scholar 

  28. X. Fan. Regularity of nonstandard Lagrangians f(x, ξ). Nonlinear Anal., 27(6):669–678, 1996.

    Article  MathSciNet  MATH  Google Scholar 

  29. X. Fan. p(x)-Laplacian equations. In Topological methods, variational methods and their applications (Taiyuan, 2002), pages 117–123. World Sci. Publ., River Edge, NJ, 2003.

    Google Scholar 

  30. X. Fan and D. Zhao. Regularity of minimizers of variational integrals with continuous p(x)-growth conditions. Chinese J. Contemp. Math., 17(4):327–336, 1996.

    MathSciNet  Google Scholar 

  31. X. Fan and D. Zhao. Regularity of minimum points of variational integrals with continuous p(x)-growth conditions. Chinese Ann. Math. Ser. A, 17(5):557–564, 1996.

    MathSciNet  MATH  Google Scholar 

  32. 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 

  33. D. Gilbarg and N. S. Trudinger. Elliptic Partial Differential Equations of Second Order. Classics in Mathematics. Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition.

    Google Scholar 

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

    MATH  Google Scholar 

  35. T. C. Halsey. Electrorheological fluids. Science, New Series, 258(5083):761–766, Oct. 30 1992.

    Google Scholar 

  36. P. Harjulehto, P. Hästö, V. Latvala, and O. Toivanen. Critical variable exponent functionals in image restoration. Appl. Math. Let., to appear.

    Google Scholar 

  37. P. Harjulehto, P. Hästö, Ú. V. Lê, and M. Nuortio. Overview of differential equations with non-standard growth. Nonlinear Anal., 72(12):4551–4574, 2010.

    Article  MathSciNet  MATH  Google Scholar 

  38. H. Hudzik. A generalization of Sobolev spaces. I. Funct. Approximatio Comment. Math., 2:67–73, 1976.

    Google Scholar 

  39. H. Hudzik. The problems of separability, duality, reflexivity and of comparison for generalized Orlicz-Sobolev spaces \({W}^{k})_{M}(\Omega )\). Comment. Math. Prace Mat., 21(2):315–324, 1980.

    MathSciNet  Google Scholar 

  40. A. N. Kolmogorov. Zur Normierbarkeit eines allgemeinen topologischen linearen Räumes. Studia Math., 5:29–33, 1934.

    MATH  Google Scholar 

  41. T. Kopaliani. On the Muckenchaupt condition in variable Lebesgue spaces. Proc. A. Razmadze Math. Inst., 148:29–33, 2008.

    MathSciNet  MATH  Google Scholar 

  42. O. Kováčik and J. Rákosník. On spaces L p(x) and W k, p(x). Czechoslovak Math. J., 41(116)(4):592–618, 1991.

    Google Scholar 

  43. S. M. Kozlov. Duality of a type of functionals. Funktsional. Anal. i Prilozhen., 17(3):9–14, 1983.

    MathSciNet  Google Scholar 

  44. M. A. Krasnosel′skiĭ and Ja. B. Rutickiĭ. Convex Functions and Orlicz Spaces. Translated from the first Russian edition by Leo F. Boron. P. Noordhoff Ltd., Groningen, 1961.

    Google Scholar 

  45. A. K. Lerner. On some questions related to the maximal operator on variable L p spaces. Trans. Amer. Math. Soc., 362(8):4229–4242, 2010.

    Article  MathSciNet  MATH  Google Scholar 

  46. S. Levine. An adaptive variational model for image decomposition. In A. Rangarajan et al., editor, Energy Minimization Methods in Computer Vision and Pattern Recognition, volume 3757 of Lecture Notes in Computer Science, pages 382–397. Springer-Verlag, Berlin, 2006.

    Google Scholar 

  47. F. Li, Z. Li, and L. Pi. Variable exponent functionals in image restoration. Appl. Math. Comput., 216(3):870–882, 2010.

    Article  MathSciNet  MATH  Google Scholar 

  48. P. Marcellini. Regularity and existence of solutions of elliptic equations with p, q-growth conditions. J. Differential Equations, 90(1):1–30, 1991.

    Article  MathSciNet  MATH  Google Scholar 

  49. P. Marcellini. Regularity for elliptic equations with general growth conditions. J. Differential Equations, 105(2):296–333, 1993.

    Article  MathSciNet  MATH  Google Scholar 

  50. V. G. Maz’ja. Sobolev Spaces. Springer Series in Soviet Mathematics. Springer-Verlag, Berlin, 1985. Translated from the Russian by T. O. Shaposhnikova.

    Google Scholar 

  51. G. Mingione. Regularity of minima: an invitation to the dark side of the calculus of variations. Appl. Math., 51(4):355–426, 2006.

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  53. J. Musielak and W. Orlicz. On modular spaces. Studia Math., 18:49–65, 1959.

    MathSciNet  MATH  Google Scholar 

  54. H. Nakano. Modulared Semi-Ordered Linear Spaces. Maruzen Co. Ltd., Tokyo, 1950.

    MATH  Google Scholar 

  55. H. Nakano. Topology and Linear Topological Spaces. Maruzen Co. Ltd., Tokyo, 1951.

    Google Scholar 

  56. 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 

  57. A. Nekvinda. Maximal operator on variable Lebesgue spaces for almost monotone radial exponent. J. Math. Anal. Appl., 337(2):1345–1365, 2008.

    Article  MathSciNet  MATH  Google Scholar 

  58. W. Orlicz. Über konjugierte Exponentenfolgen. Stud. Math., 3:200–211, 1931.

    Google Scholar 

  59. L. Pick and M. Růžička. An example of a space L p(x) on which the Hardy-Littlewood maximal operator is not bounded. Expo. Math., 19(4):369–371, 2001.

    Article  MathSciNet  MATH  Google Scholar 

  60. V. R. Portnov. Certain properties of the Orlicz spaces generated by the functions M(x, w). Dokl. Akad. Nauk SSSR, 170:1269–1272, 1966.

    MathSciNet  Google Scholar 

  61. V. R. Portnov. On the theory of Orlicz spaces which are generated by variable N-functions. Dokl. Akad. Nauk SSSR, 175:296–299, 1967.

    MathSciNet  Google Scholar 

  62. B. Ross and S. Samko. Fractional integration operator of variable order in the Hölder spaces H λ(x). Internat. J. Math. Math. Sci., 18(4):777–788, 1995.

    Article  MathSciNet  MATH  Google Scholar 

  63. H. L. Royden. Real Analysis. Macmillan Publishing Company, New York, third edition, 1988.

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  65. M. Růžička. Electrorheological Fluids: Modeling and Mathematical Theory, volume 1748 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2000.

    MATH  Google Scholar 

  66. M. Růžička. Modeling, mathematical and numerical analysis of electrorheological fluids. Appl. Math., 49(6):565–609, 2004.

    Article  MathSciNet  MATH  Google Scholar 

  67. S. Samko. Fractional integration and differentiation of variable order. Anal. Math., 21(3):213–236, 1995.

    Article  MathSciNet  MATH  Google Scholar 

  68. S. Samko. Convolution and potential type operators in \({L}^{p(x)}({\mathbf{R}}^{n})\). Integral Transform. Spec. Funct., 7(3–4):261–284, 1998.

    Article  MathSciNet  MATH  Google Scholar 

  69. S. Samko. Convolution type operators in L p(x). Integral Transform. Spec. Funct., 7(1–2):123–144, 1998.

    Article  MathSciNet  MATH  Google Scholar 

  70. S. Samko. Differentiation and integration of variable order and the spaces L p(x). In Operator theory for complex and hypercomplex analysis (Mexico City, 1994), volume 212 of Contemp. Math., pages 203–219. Amer. Math. Soc., Providence, RI, 1998.

    Google Scholar 

  71. S. Samko and B. Ross. Integration and differentiation to a variable fractional order. Integral Transform. Spec. Funct., 1(4):277–300, 1993.

    Article  MathSciNet  MATH  Google Scholar 

  72. I. I. Sharapudinov. The topology of the space \({\mathcal{L}}^{p(t)}([0,\,1])\). Mat. Zametki, 26(4):613–632, 655, 1979.

    Google Scholar 

  73. I. I. Sharapudinov. Approximation of functions in the metric of the space \({\mathcal{L}}^{p(t)}([a,\,b])\) and quadrature formulas. In Constructive function theory ’81 (Varna, 1981), pages 189–193. Publ. House Bulgar. Acad. Sci., Sofia, 1983.

    Google Scholar 

  74. I. I. Sharapudinov. The basis property of the Haar system in the space \({\mathcal{L}}^{p(t)}([0,1])\) and the principle of localization in the mean. Mat. Sb. (N.S.), 130(172)(2):275–283, 286, 1986.

    Google Scholar 

  75. I. I. Sharapudinov. On the uniform boundedness in L p (p = p(x)) of some families of convolution operators. Mat. Zametki, 59(2):291–302, 320, 1996.

    Google Scholar 

  76. J. E. Stangroom. Electrorheological fluids. Physics in Technology, 14:290–296, 1983.

    Article  Google Scholar 

  77. L. Tartar. An Introduction to Sobolev Spaces and Interpolation Spaces, volume 3 of Lecture Notes of the Unione Matematica Italiana. Springer, Berlin, 2007.

    MATH  Google Scholar 

  78. I. V. Tsenov. Generalization of the problem of best approximation of a function in the space L s. Uch. Zap. Dagestan. Gos. Univ., 7:25–37, 1961.

    Google Scholar 

  79. T. Wunderli. On time flows of minimizers of general convex functionals of linear growth with variable exponent in BV space and stability of pseudosolutions. J. Math. Anal. Appl., 364(2):591–598, 2010.

    Article  MathSciNet  MATH  Google Scholar 

  80. S. Yamamuro. On conjugate spaces of Nakano spaces. Trans. Amer. Math. Soc., 90:291–311, 1959.

    Article  MathSciNet  MATH  Google Scholar 

  81. V. V. Zhikov. Problems of convergence, duality, and averaging for a class of functionals of the calculus of variations. Dokl. Akad. Nauk SSSR, 267(3):524–528, 1982.

    MathSciNet  Google Scholar 

  82. V. V. Zhikov. The Lavrent′ ev effect and averaging of nonlinear variational problems. Differentsial′ nye Uravneniya, 27(1):42–50, 180, 1991.

    Google Scholar 

  83. V. V. Zhikov. On the homogenization of nonlinear variational problems in perforated domains. Russian J. Math. Phys., 2(3):393–408, 1994.

    MathSciNet  MATH  Google Scholar 

  84. V. V. Zhikov. Meyer-type estimates for solving the nonlinear Stokes system. Differ. Uravn., 33(1):107–114, 143, 1997.

    Google Scholar 

  85. V. V. Zhikov. On some variational problems. Russian J. Math. Phys., 5(1):105–116 (1998), 1997.

    Google Scholar 

  86. W. P. Ziemer. Weakly Differentiable Functions, volume 120 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1989.

    Book  MATH  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). Introduction. In: Variable Lebesgue Spaces. Applied and Numerical Harmonic Analysis. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0548-3_1

Download citation

Publish with us

Policies and ethics

Search

Navigation