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A Survey on Some Variable Function Spaces

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Function Spaces and Inequalities

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 206))

Abstract

This article is devoted to presenting a recapitulative introduction of some recent progresses, obtained by the authors and their collaborators, on the theory of variable function spaces including the variable Hardy spaces (associated with operators), the variable weak Hardy spaces and the variable Besov-type and Triebel-Lizorkin-type spaces.

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References

  1. Acerbi, E., Mingione, G.: Regularity results for stationary electro-rheological fluids. Arch. Ration. Mech. Anal. 164, 213–259 (2002)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  3. Almeida, A., Caetano, A.: On 2-microlocal spaces with all exponents variable. Nonlinear Anal. 135, 97–119 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  4. Almeida, A., Hästö, P.: Besov spaces with variable smoothness and integrability. J. Funct. Anal. 258, 1628–1655 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  5. Almeida, A., Hästö, P.: Interpolation in variable exponent spaces. Rev. Mat. Complut. 27, 657–676 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  6. Almeida, A., Harjulehto, P., Hästö, P., Lukkari, T.: Riesz and Wolff potentials and elliptic equations in variable exponent weak Lebesgue spaces. Ann. Mat. Pura Appl. (4). 194, 405–424 (2015)

    Google Scholar 

  7. Auscher P., Duong X.T., McIntosh A.: Boundedness of Banach space valued singular integral operators and Hardy spaces. Unpublished manuscript, 2005

    Google Scholar 

  8. Baroni, P., Colombo, M., Mingione, G.: Harnack inequalities for double phase functionals. Nonlinear Anal. 121, 206–222 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  9. Baroni, P., Colombo, M., Mingione, G.: Non-autonomous functionals, borderline cases and related function classes. St. Petersburg Math. J. 27, 347–379 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  10. Bergh J., Löfström J.: Interpolation Spaces. An introduction. Grundlehren der Mathematischen Wissenschaften, No. 223. Springer, Berlin-New York (1976)

    Google Scholar 

  11. Birnbaum, Z., Orlicz, W.: Über die Verallgemeinerung des Begriffes der zueinander konjugierten Potenzen. Studia Math. 3, 1–67 (1931)

    Article  MATH  Google Scholar 

  12. Bui, T.A., Cao, J., Ky, L.D., Yang, D., Yang, S.: Musielak-Orlicz-Hardy spaces associated with operators satisfying reinforced off-diagonal estimates. Anal. Geom. Metr. Spaces 1, 69–129 (2013)

    MATH  MathSciNet  Google Scholar 

  13. Bui, H.-Q., Duong, X.T., Yan, L.: Calderón reproducing formulas and new Besov spaces associated with operators. Adv. Math. 229, 2449–2502 (2012)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  15. Coifman R.R., Weiss G.: Analyse Harmonique Non-commutative sur Certains Espaces Homogènes, Étude de certaines intégrales singulières. (French) Lecture Notes in Mathematics, vol. 242. Springer, Berlin-New York (1971)

    Google Scholar 

  16. Coifman, R.R., Weiss, G.: Extensions of Hardy spaces and their use in analysis. Bull. Am. Math. Soc. 83, 569–645 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  17. Colombo, M., Mingione, G.: Calderón-Zygmund estimates and non-uniformly elliptic operators. J. Funct. Anal. 270, 1416–1478 (2016)

    Article  MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

  19. Cruz-Uribe, D.V., Fiorenza, A.: Variable Lebesgue Spaces. Foundations and Harmonic Analysis, Applied and Numerical Harmonic Analysis, Birkhäuser/Springer, Heidelberg (2013)

    Book  MATH  Google Scholar 

  20. Cruz-Uribe, D., Wang, L.-A.D.: Variable Hardy spaces. Indiana Univ. Math. J. 63, 447–493 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  21. Diening, L.: Maximal function on generalized Lebesgue spaces \({L^{p(\cdot )}({\mathbb{R}^n})}\). Math. Inequal. Appl. 7, 245–253 (2004)

    MATH  MathSciNet  Google Scholar 

  22. Diening L., Harjulehto P., Hästö P., R\(\mathring{\rm u}\)žička M.: Lebesgue and Sobolev Spaces with Variable Exponents. Lecture Notes in Mathematics. Springer, Heidelberg (2011)

    Google Scholar 

  23. Diening, L., Hästö, P., Roudenko, S.: Function spaces of variable smoothness and integrability. J. Funct. Anal. 256, 1731–1768 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  24. Drihem, D.: Atomic decomposition of Besov spaces with variable smoothness and integrability. J. Math. Anal. Appl. 389, 15–31 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  25. Drihem, D.: Some properties of variable Besov-type spaces. Funct. Approx. Comment. Math. 52, 193–221 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  26. Drihem, D.: Some characterizations of variable Besov-type spaces. Ann. Funct. Anal. 6, 255–288 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  27. Drihem D.: Variable Triebel-Lizorkin-type spaces. arXiv:1508.05980

  28. Duong, X.T., Yan, L.: New function spaces of BMO type, the John-Nirenberg inequality, interpolation, and applications. Commun. Pure Appl. Math. 58, 1375–1420 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  29. Duong, X.T., Yan, L.: Duality of Hardy and BMO spaces associated with operators with heat kernel bounds. J. Am. Math. Soc. 18, 943–973 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  30. Fan, X.: Global \(C^{1,\alpha }\) regularity for variable exponent elliptic equations in divergence form. J. Differ. Eqns. 235, 397–417 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  31. Fan, X., Deng, S.: Remarks on Ricceri’s variational principle and applications to the \(p(x)\)-Laplacian equations. Nonlinear Anal. 67, 3064–3075 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  32. Fan, X., Zhao, D.: On the spaces \(L^p(x)(\Omega )\) and \(W^{m, p(x)}(\Omega )\). J. Math. Anal. Appl. 263, 424–446 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  33. Fefferman, C., Rivière, N.M., Sagher, Y.: Interpolation between \(H^p\) spaces: the real method. Trans. Am. Math. Soc. 191, 75–81 (1974)

    MATH  MathSciNet  Google Scholar 

  34. Fefferman, R., Soria, F.: The space weak \(H^{1}\). Studia Math. 85, 1–16 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  35. Folland G.B., Stein E.M.: Hardy spaces on homogeneous groups. Mathematical Notes, vol. 28. Princeton University Press, Princeton (1982)

    Google Scholar 

  36. Frazier, M., Jawerth, B.: Decomposition of Besov spaces. Indiana Univ. Math. J. 34, 777–799 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  37. Frazier, M., Jawerth, B.: A discrete transform and decompositions of distribution spaces. J. Funct. Anal. 93, 34–170 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  38. Han Y., Müller D., Yang D.: A Theory of Besov and Triebel-Lizorkin spaces on metric measure spaces modeled on Carnot-Carathéodory spaces. Abstr. Appl. Anal. 2008, Art. ID 893409, 250 pp

    Google Scholar 

  39. Harjulehto P., Hästö P., Koskenoja M., Varonen S.: Variable Sobolev capacity and the assumptions on the exponent. In: Orlicz Centenary Volume II, 51–59, Banach Center Publ., 68, Polish Acad. Sci. Inst. Math., Warsaw (2005)

    Google Scholar 

  40. Harjulehto, P., Hästö, P., Latvala, V.: Minimizers of the variable exponent, non-uniformly convex Dirichlet energy. J. Math. Pures Appl. (9). 89, 174–197 (2008)

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  42. Hofmann S., Lu G., Mitrea D., Mitrea M., Yan L.: Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates. Mem. Am. Math. Soc. 214, no. 1007, vi+78 pp (2011)

    Google Scholar 

  43. Jiang, R., Yang, D.: New Orlicz-Hardy spaces associated with divergence form elliptic operators. J. Funct. Anal. 258, 1167–1224 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  44. Jiang, R., Yang, D., Zhou, Y.: Orlicz-Hardy spaces associated with operators. Sci. China Ser. A 52, 1042–1080 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  45. Kerkyacharian, G., Petrushev, P.: Heat kernel based decomposition of spaces of distributions in the framework of dirichlet spaces. Trans. Am. Math. Soc. 367, 121–189 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  46. Kokilashvili V., Meskhi A., Rafeiro H., Samko S.: Integral Operators in Non-Standard Function Spaces. Volume 1: Variable Exponent Lebesgue and Amalgam Spaces, Operator Theory: Advances and Applications 248, Birkhhaüser, Basel (2016)

    Google Scholar 

  47. Kokilashvili V., Meskhi A., Rafeiro H., Samko S.: Integral Operators in Non-Standard Function Spaces. Volume 2: Variable Exponent Hölder, Morrey-Cam-panato and Grand Spaces, Operator Theory: Advances and Applications, vol. 249, Birkhhaüser, Basel (2016)

    Google Scholar 

  48. Kokilashvili V., Paatashvili V., Samko S.: Boundedness in Lebesgue spaces with variable exponent of the Cauchy singular operator on Carleson curves. In: Modern Operator Theory and Applications, pp. 167–186, Oper. Theory Adv. Appl. 170, Birkhäuser, Basel (2007)

    Google Scholar 

  49. Kováčik, O., Rákosník, J.: On spaces \(L^{p(x)}\) and \(W^{k, p(x)}\). Czechoslovak Math. J. 41(116), 592–618 (1991)

    MATH  MathSciNet  Google Scholar 

  50. Ky, L.D.: New Hardy spaces of Musielak-Orlicz type and boundedness of sublinear operators. Integr. Eqns. Oper. Theory 78, 115–150 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  51. Liang, Y., Yang, D.: Intrinsic square function characterizations of Musielak-Orlicz Hardy spaces. Trans. Am. Math. Soc. 367, 3225–3256 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  52. Liang, Y., Yang, D., Jiang, R.: Weak Musielak-Orlicz Hardy spaces and applications. Math. Nach. 289, 634–677 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  53. Liu, H.: The weak \(H^p\) spaces on homogenous groups. In: Harmonic Analysis (Tianjin, 1988), pp. 113–118, Lecture Notes in Math., vol. 1494. Springer, Berlin (1991)

    Google Scholar 

  54. Liu, L., Yang, D., Yuan, W.: Besov-type and Triebel-Lizorkin-type spaces associated with heat kernels. Collect. Math. 67, 247–310 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  55. Luxemburg, W.: Banach Function Spaces. Thesis, Technische Hogeschool te Delft (1955)

    MATH  Google Scholar 

  56. Musielak, J.: Orlicz Spaces and Modular Spaces. Lecture Notes in Math, vol. 1034. Springer, Berlin (1983)

    Google Scholar 

  57. Nakai, E., Sawano, Y.: Hardy spaces with variable exponents and generalized Campanato spaces. J. Funct. Anal. 262, 3665–3748 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  58. Nakano, H.: Modulared Semi-Ordered Linear Spaces. Maruzen Co., Ltd, Tokyo (1950)

    MATH  Google Scholar 

  59. Nakano, H.: Topology of Linear Topological Spaces. Maruzen Co., Ltd, Tokyo (1951)

    Google Scholar 

  60. Noi, T.: Trace and extension operators for Besov spaces and Triebel-Lizorkin spaces with variable exponents. Rev. Mat. Complut. 29, 341–404 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  61. Noi, T.: Duality of variable exponent Triebel-Lizorkin and Besov spaces. J. Funct. Spaces Appl. Art. ID 361807, 19pp (2012)

    Google Scholar 

  62. Noi, T., Sawano, Y.: Complex interpolation of Besov spaces and Triebel-Lizorkin spaces with variable exponents. J. Math. Anal. Appl. 387, 676–690 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  63. Orlicz, W.: Über eine gewisse Klasse von Räumen vom Typus B. Bull. Int. Acad. Pol. Ser. A 8, 207–220 (1932)

    MATH  Google Scholar 

  64. Rivière, N.M., Sagher, Y.: Interpolation between \(L^{\infty }\) and \(H^1\), the real method. J. Funct. Anal. 14, 401–409 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  65. R\(\mathring{\rm u}\)žička, M.: Electrorheological Fluids: Modeling and Mathematical Theory. Lecture Notes in Mathematics, vol. 1748. Springer, Berlin (2000)

    Google Scholar 

  66. Sanchón, M., Urbano, J.M.: Entropy solutions for the \(p(x)\)-Laplace equation. Trans. Am. Math. Soc. 361, 6387–6405 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  67. Sawano, Y.: Atomic decompositions of Hardy spaces with variable exponents and its application to bounded linear operators. Integr. Eqns. Oper. Theory 77, 123–148 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  68. Song, L., Yan, L.: A maximal function characterization for Hardy spaces associated to nonnegative self-adjoint operators satisfying Gaussian estimates. Adv. Math. 287, 463–484 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  69. Stein E.M.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton Mathematical Series 43, Monographs in Harmonic Analysis III, Princeton University Press, Princeton, NJ (1993)

    Google Scholar 

  70. Strömberg, J.-O., Torchinsky, A.: Weighted Hardy Spaces. Lecture Notes in Math, vol. 1381. Springer, Berlin (1989)

    Google Scholar 

  71. Triebel, H.: Theory of Function Spaces. Birkhäuser Verlag, Basel (1983)

    Book  MATH  Google Scholar 

  72. Xu, J.: Variable Besov and Triebel-Lizorkin spaces. Ann. Acad. Sci. Fenn. Math. 33, 511–522 (2008)

    MATH  MathSciNet  Google Scholar 

  73. Xu, J.: The relation between variable Bessel potential spaces and Triebel-Lizorkin spaces. Integral Transf. Spec. Funct. 19, 599–605 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  74. Wilson, M.: The intrinsic square function. Rev. Mat. Iberoam. 23, 771–791 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  75. Yan, L.: Classes of Hardy spaces associated with operators, duality theorem and applications. Trans. Am. Math. Soc. 360, 4383–4408 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  76. Yan, X., Yang, D., Yuan, W., Zhuo, C.: Variable weak Hardy spaces and their applications. J. Funct. Anal. 271, 2822–2887 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  77. Yang, D., Liang, Y., Ky, L.D.: Real-Variable Theory of Musielak-Orlicz Hardy Spaces. Lecture Notes in Mathematics, vol. 2182. Springer, Cham (2017)

    Google Scholar 

  78. Yang, D., Yang, S.: Maximal function characterizations of Musielak-Orlicz-Hardy spaces associated with non-negative self-adjoint operators satisfying Gaussian estiamte. Commun. Pure Appl. Anal. 15, 2135–2160 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  79. Yang, D., Yuan, W.: A new class of function spaces connecting Triebel-Lizorkin spaces and \(Q\) spaces. J. Funct. Anal. 255, 2760–2809 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  80. Yang, D., Yuan, W.: New Besov-type spaces and Triebel-Lizorkin-type spaces including \(Q\) spaces. Math. Z. 265, 451–480 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  81. Yang, D., Yuan, W., Zhuo, C.: Musielak-Orlicz Besov-type and Triebel-Lizorkin-type spaces. Rev. Mat. Complut. 27, 93–157 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  82. Yang D., Zhang J., Zhuo C.: Variable Hardy spaces associated with operators satisfying Davies-Gaffney estimates. Proc. Edinb. Math. Soc. (2), accepted (2017)

    Google Scholar 

  83. Yang, D., Zhou, Y.: New properties of Besov and Triebel-Lizorkin spaces on RD-spaces. Manuscripta Math. 134, 59–90 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  84. Yang, D., Zhuo, C.: Molecular characterizations and dualities of variable exponent Hardy spaces associated with operators. Ann. Acad. Sci. Fenn. Math. 41, 357–398 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  85. Yang, D., Zhuo, C., Nakai, E.: Characterizations of variable exponent Hardy spaces via Riesz transforms. Rev. Mat. Comput. 29, 245–270 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  86. Yang, D., Zhuo, C., Yuan, W.: Triebel-Lizorkin type spaces with variable exponents. Banach J. Math. Anal. 9, 146–202 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  87. Yang, D., Zhuo, C., Yuan, W.: Besov-type spaces with variable smoothness and integrability. J. Funct. Anal. 269, 1840–1898 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  88. Yuan, W., Sickel, W., Yang, D.: Morrey and Campanato Meet Besov, Lizorkin and Triebel. Lecture Notes in Mathematics, vol. 2005. Springer, Berlin (2010)

    Google Scholar 

  89. Zhuo, C., Sawano, Y., Yang, D.: Hardy spaces with variable exponents on RD-spaces and applications. Dissertationes Math. (Rozprawy Mat.) 520, 1–74 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  90. Zhuo, C., Yang, D.: Maximal function characterizations of variable Hardy spaces associated with non-negative self-adjoint operators satisfying Gaussian estimates. Nonlinear Anal. 141, 16–42 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  91. Zhuo C., Yang D., Yuan W.: Interpolation between \(H^{p(\cdot )}(\mathbb{R}^n)\) and \(L^\infty (\mathbb{R}^n)\): real method, J. Geom. Anal. (2017). doi:10.1007/s12220-017-9904-2

  92. Zhuo, C., Yang, D., Liang, Y.: Intrinsic square function characterizations of Hardy spaces with variable exponents. Bull. Malays. Math. Sci. Soc. 39, 1541–1577 (2016)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgements

The authors would like to express their deep thanks to the referee for his/her very careful reading and several useful comments which improve the presentation of this article. This project is supported by the National Natural Science Foundation of China (Grant Nos. 11571039, 11471042 and 11671185). C. Zhuo is supported by the Construct Program of the Key Discipline in Hunan Province of China.

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Authors and Affiliations

  1. School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing, 100875, People’s Republic of China

    Dachun Yang & Wen Yuan

  2. Key Laboratory of High Performance Computing and Stochastic Information Processing (HPCSIP) (Ministry of Education of China), College of Mathematics and Computer Science, Hunan Normal University, Changsha, Hunan, 410081, People’s Republic of China

    Ciqiang Zhuo

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  1. Dachun Yang
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  2. Wen Yuan
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Correspondence to Dachun Yang .

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  1. Department of Mathematics, South Asian University, New Delhi, Delhi, India

    Pankaj Jain

  2. Faculty of Mathematics & Computer Science, Friedrich Schiller University Jena, Jena, Thüringen, Germany

    Hans-Jürgen Schmeisser

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Yang, D., Yuan, W., Zhuo, C. (2017). A Survey on Some Variable Function Spaces. In: Jain, P., Schmeisser, HJ. (eds) Function Spaces and Inequalities. Springer Proceedings in Mathematics & Statistics, vol 206. Springer, Singapore. https://doi.org/10.1007/978-981-10-6119-6_15

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