Advertisement

Hardy Spaces with Variable Exponents

  • Víctor Almeida
  • Jorge J. BetancorEmail author
  • Estefanía Dalmasso
  • Lourdes Rodríguez-Mesa
Chapter
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

Abstract

In this paper, we make a survey on some recent developments of the theory of Hardy spaces with variable exponents in different settings.

References

  1. 1.
    E. Acerbi, G. Mingione, Regularity results for a class of functionals with non-standard growth. Arch. Ration. Mech. Anal. 156(2), 121–140 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    V. Almeida, J.J. Betancor, A.J. Castro, L. Rodríguez-Mesa, Variable exponent Hardy spaces associated with discrete Laplacians on graphs. Sci. China Math. 62(1), 73–124 (2019)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    V. Almeida, J.J. Betancor, E. Dalmasso, L. Rodríguez-Mesa, Local Hardy spaces with variable exponents associated to non-negative self-adjoint operators satisfying Gaussian estimates. J. Geom. Anal. (2019). https://doi.org/10.1007/s12220-019-00199-y
  4. 4.
    V. Almeida, J.J. Betancor, L. Rodríguez-Mesa, Anisotropic Hardy-Lorentz spaces with variable exponents. Canad. J. Math. 69(6), 1219–1273 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    S.N. Antontsev, J.F. Rodrigues, On stationary thermo-rheological viscous flows. Ann. Univ. Ferrara Sez. VII Sci. Mat. 52(1), 19–36 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    P. Auscher, X.T. Duong, A. McIntosh, Boundedness of banach space valued singular integral and Hardy spaces. Unplublished Manuscript (2005)Google Scholar
  7. 7.
    P. Auscher, J.M.A. Martell, Weighted norm inequalities, off-diagonal estimates and elliptic operators. II. Off-diagonal estimates on spaces of homogeneous type. J. Evol. Equ. 7, 2, 265–316 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    P. Auscher, A. McIntosh, E. Russ, Hardy spaces of differential forms on Riemannian manifolds. J. Geom. Anal. 18(1), 192–248 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    M. Bownik, Anisotropic Hardy spaces and wavelets. Mem. Amer. Math. Soc. 164, 781, vi+122 (2003)Google Scholar
  10. 10.
    H.Q. Bui, Weighted Hardy spaces. Math. Nachr. 103, 45–62 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    J. Cao, D.-C. Chang, D. Yang, S. Yang, Weighted local Orlicz-Hardy spaces on domains and their applications in inhomogeneous Dirichlet and Neumann problems. Trans. Amer. Math. Soc. 365(9), 4729–4809 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    J. Cao, S. Mayboroda, D. Yang, Local Hardy spaces associated with inhomogeneous higher order elliptic operators. Anal. Appl. (Singap.) 15(2), 137–224 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    A. Carbonaro, A. McIntosh, A.J. Morris, Local Hardy spaces of differential forms on Riemannian manifolds. J. Geom. Anal. 23(1), 106–169 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Y. Chen, S. Levine, M. Rao, Variable exponent, linear growth functionals in image restoration. SIAM J. Appl. Math. 66(4), 1383–1406 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    R.R. Coifman, A real variable characterization of \(H^{p}\). Studia Math. 51, 269–274 (1974)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    D.V. Cruz-Uribe, A. Fiorenza, Variable Lebesgue Spaces: Foundations and Harmonic Analysis. Applied and Numerical Harmonic Analysis (Birkhäuser/Springer, Heidelberg, 2013)zbMATHCrossRefGoogle Scholar
  17. 17.
    D. Cruz-Uribe, A. Fiorenza, J.M. Martell, C. Pérez, The boundedness of classical operators on variable \(L^p\) spaces. Ann. Acad. Sci. Fenn. Math. 31(1), 239–264 (2006)MathSciNetzbMATHGoogle Scholar
  18. 18.
    D. Cruz-Uribe, A. Fiorenza, C.J. Neugebauer, The maximal function on variable \(L^p\) spaces. Ann. Acad. Sci. Fenn. Math. 28(1), 223–238 (2003)MathSciNetzbMATHGoogle Scholar
  19. 19.
    D. Cruz-Uribe, L.-A.D. Wang, Variable Hardy spaces. Indiana Univ. Math. J. 63(2), 447–493 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    D. Cruz-Uribe, L.-A.D. Wang, Extrapolation and weighted norm inequalities in the variable Lebesgue spaces. Trans. Amer. Math. Soc. 369(2), 1205–1235 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    G. Dafni, H. Yue, Some characterizations of local bmo and \(h^1\) on metric measure spaces. Anal. Math. Phys. 2(3), 285–318 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    L. Diening, Theoretical and numerical results for electrorheological fluids. Ph.D. thesis, Univ. Freiburg im Breisgau, Mathematische Fakultät, 156 p. (2002)Google Scholar
  23. 23.
    L. Diening, P. Harjulehto, P. Hästö, M. Ružička, Lebesgue and Sobolev Spaces with Variable Exponents. Lecture Notes in Mathematics, vol. 2017 (Springer, Heidelberg, 2011)zbMATHCrossRefGoogle Scholar
  24. 24.
    L. Diening, Maximal function on generalized Lebesgue spaces \(L^{p(\cdot )}\). Math. Inequal. Appl. 7(2), 245–253 (2004)MathSciNetzbMATHGoogle Scholar
  25. 25.
    L. Diening, P. Harjulehto, P. Hästö, Y. Mizuta, T. Shimomura, Maximal functions in variable exponent spaces: limiting cases of the exponent. Ann. Acad. Sci. Fenn. Math. 34(2), 503–522 (2009)MathSciNetzbMATHGoogle Scholar
  26. 26.
    L. Diening, P. Hästö, S. Roudenko, Function spaces of variable smoothness and integrability. J. Funct. Anal. 256(6), 1731–1768 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    L. Diening, M. Ružička, Calderón-Zygmund operators on generalized Lebesgue spaces \(L^{p(\cdot )}\) and problems related to fluid dynamics. J. Reine Angew. Math. 563, 197–220 (2003)MathSciNetzbMATHGoogle Scholar
  28. 28.
    X.T. Duong, L. Yan, Duality of Hardy and BMO spaces associated with operators with heat kernel bounds. J. Amer. Math. Soc. 18(4), 943–973 (2005) (electronic)Google Scholar
  29. 29.
    P.L. Duren, B.W. Romberg, A.L. Shields, Linear functionals on \(H^{p}\) spaces with \(0<p<1\). J. Reine Angew. Math. 238, 32–60 (1969)MathSciNetzbMATHGoogle Scholar
  30. 30.
    J. Dziubański, M. Preisner, Hardy spaces for semigroups with Gaussian bounds. Ann. Mat. Pura Appl. 197(3), 965–987 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    C. Fefferman, E.M. Stein, \(H^{p}\) spaces of several variables. Acta Math. 129(3–4), 137–193 (1972)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    A. Gogatishvili, A. Danelia, T. Kopaliani, Local Hardy-Littlewood maximal operator in variable Lebesgue spaces. Banach J. Math. Anal. 8(2), 229–244 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    D. Goldberg, A local version of real Hardy spaces. Duke Math. J. 46(1), 27–42 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    L. Grafakos, L. Liu, D. Yang, Radial maximal function characterizations for Hardy spaces on RD-spaces. Bull. Soc. Math. France 137(2), 225–251 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    P. Harjulehto, P. Hästö, V. Latvala, O. Toivanen, Critical variable exponent functionals in image restoration. Appl. Math. Lett. 26(1), 56–60 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    P.A. Hästö, Local-to-global results in variable exponent spaces. Math. Res. Lett. 16(2), 263–278 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    S. Hofmann, G. Lu, D. Mitrea, M. Mitrea, L. Yan, Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates. Memoirs of the Amer. Math. Soc. 214 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    S. Hofmann, S. Mayboroda, Hardy and BMO spaces associated to divergence form elliptic operators. Math. Ann. 344(1), 37–116 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    R. Jiang, D. Yang, D. Yang, Maximal function characterizations of Hardy spaces associated with magnetic Schrödinger operators. Forum Math. 24(3), 471–494 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    R. Jiang, D. Yang, Y. Zhou, Localized Hardy spaces associated with operators. Appl. Anal. 88(9), 1409–1427 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    M. Kemppainen, A note on local Hardy spaces. Forum Math. 29(4), 941–949 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    O. Kováčik, J. Rákosník, On spaces \(L^{p(x)}\) and \(W^{k,p(x)}\). Czechoslovak Math. J. 41(4)(116), 592–618 (1991)Google Scholar
  43. 43.
    R.H. Latter, A characterization of \(H^{p}({ R}^{n})\) in terms of atoms. Studia Math. 62(1), 93–101 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    A.K. Lerner, On modular inequalities in variable \(L^p\) spaces. Arch. Math. (Basel) 85(6), 538–543 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    F. Li, Z. Li, L. Pi, Variable exponent functionals in image restoration. Appl. Math. Comput. 216(3), 870–882 (2010)MathSciNetzbMATHGoogle Scholar
  46. 46.
    J. Liu, D. Yang, W. Yuan, Anisotropic variable Hardy-Lorentz spaces and their real interpolation. J. Math. Anal. Appl. 456(1), 356–393 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    G. Mauceri, M.A. Picardello, F. Ricci, A Hardy space associated with twisted convolution. Adv. in Math. 39(3), 270–288 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    E. Nakai, Y. Sawano, Hardy spaces with variable exponents and generalized Campanato spaces. J. Funct. Anal. 262(9), 3665–3748 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    H. Nakano, Modulared Semi-ordered Linear Spaces (Maruzen Co. Ltd., Tokyo, 1950)zbMATHGoogle Scholar
  50. 50.
    A.S. Nekvinda, Hardy-Littlewood maximal operator on \(L^{p(x)}(\mathbb{R})\). Math. Inequal. Appl. 7(2), 255–265 (2004)Google Scholar
  51. 51.
    W. Orlicz, Über Konjugierte Exponentenfolgen. Studia Math. 3(1), 200–211 (1931)zbMATHCrossRefGoogle Scholar
  52. 52.
    M.M. Peloso, S. Secco, Local Riesz transforms characterization of local Hardy spaces. Collect. Math. 59(3), 299–320 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  53. 53.
    L.S. Pick, M. Ružička, 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)Google Scholar
  54. 54.
    K.R. Rajagopal, M. Ružička, On the modelling of electrorheological materials. Mech. Res. Commun. 23(4), 401–407 (1996)zbMATHCrossRefGoogle Scholar
  55. 55.
    M. Ružička, Electrorheological Fluids: Modeling and Mathematical Theory. Lecture Notes in Mathematics, vol. 1748 (Springer, Berlin, 2000)zbMATHCrossRefGoogle Scholar
  56. 56.
    Y. Sawano, Atomic decompositions of Hardy spaces with variable exponents and its application to bounded linear operators. Int. Equ. Oper. Theory 77(1), 123–148 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  57. 57.
    I.I. Sharapudinov, The topology of the space \({\cal{L}}^{p(t)}([0,\,1])\). Mat. Zametki 26(4), 613–632, 655 (1979)Google Scholar
  58. 58.
    L. Song, L. Yan, Maximal function characterization for Hardy spaces associated with nonnegative self-adjoint operators on spaces of homogeneous type. J. Evol. Equations. 18(1), 221–243 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  59. 59.
    L. Song, L. Yan, A maximal function characterization for Hardy spaces associated to nonnegative self-adjoint operators satisfying Gaussian estimates. Adv. Math. 287, 463–484 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  60. 60.
    E.M. Stein, Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals. Princeton Mathematical Series, vol. 43 (Princeton University Press, Princeton, NJ, 1993)Google Scholar
  61. 61.
    E.M. Stein, G. Weiss, On the theory of harmonic functions of several variables. I. The theory of \(H^{p}\)-spaces. Acta Math. 103, 25–62 (1960)MathSciNetzbMATHCrossRefGoogle Scholar
  62. 62.
    J.-O. Strömberg, A. Torchinsky, Weighted Hardy Spaces. Lecture Notes in Mathematics, vol. 1381 (Springer, Berlin, 1989)zbMATHCrossRefGoogle Scholar
  63. 63.
    L. Tang, Weighted local Hardy spaces and their applications. Illinois J. Math. 56(2), 453–495 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  64. 64.
    H. Triebel, Theory of Function Spaces. Monographs in Mathematics, vol. 78 (Birkhäuser Verlag, Basel, 1983)Google Scholar
  65. 65.
    A. Uchiyama, A maximal function characterization of \(H^{p}\) on the space of homogeneous type. Trans. Amer. Math. Soc. 262(2), 579–592 (1980)MathSciNetzbMATHGoogle Scholar
  66. 66.
    M. Wilson, The intrinsic square function. Rev. Mat. Iberoam. 23(3), 771–791 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  67. 67.
    A.S. Wineman, K.R. Rajagopal, On a constitutive theory for materials undergoing microstructural changes. Arch. Mech. (Arch. Mech. Stos.) 42(1), 53–75 (1990)Google Scholar
  68. 68.
    L. Yan, Classes of Hardy spaces associated with operators, duality theorem and applications. Trans. Amer. Math. Soc. 360(8), 4383–4408 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  69. 69.
    D. Yang, J. Zhang, C. Zhuo, Variable Hardy spaces associated with operators satisfying Davies-Gaffney estimates. Proc. Edinburgh Math. Soc. 61(3), 759–810 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  70. 70.
    D. Yang, S. Yang, Local Hardy spaces of Musielak-Orlicz type and their applications. Sci. China Math. 55(8), 1677–1720 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  71. 71.
    D. Yang, C. Zhuo, Molecular characterizations and dualities of variable exponent Hardy spaces associated with operators. Ann. Acad. Sci. Fenn. Math. 41(1), 357–398 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  72. 72.
    D. Yang, C. Zhuo, E. Nakai, Characterizations of variable exponent Hardy spaces via Riesz transforms. Rev. Mat. Complut. 29(2), 245–270 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  73. 73.
    X. Yan, D. Yang, W. Yuan, C. Zhuo, Variable weak Hardy spaces and their applications. J. Funct. Anal. 271(10), 2822–2887 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  74. 74.
    V.V. Zhikov, Meyer-type estimates for solving the nonlinear Stokes system. Differ. Uravn. 33(1), 107–114, 143 (1997)Google Scholar
  75. 75.
    C. Zhuo, Y. Sawano, D. Yang, Hardy spaces with variable exponents on RD-spaces and applications. Dissertationes Math. (Rozprawy Mat.) 520, 74 (2016)MathSciNetzbMATHGoogle Scholar
  76. 76.
    C. Zhuo, D. Yang, Maximal function characterizations of variable Hardy spaces associated with non-negative self-adjoint operators satisfying Gaussian estimates. Nonlinear Anal. 141, 16–42 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  77. 77.
    C. Zhuo, D. Yang, Y. Liang, Intrinsic square function characterizations of Hardy spaces with variable exponents. Bull. Malays. Math. Sci. Soc. 39(4), 1541–1577 (2016)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Víctor Almeida
    • 1
  • Jorge J. Betancor
    • 1
    Email author
  • Estefanía Dalmasso
    • 2
  • Lourdes Rodríguez-Mesa
    • 1
  1. 1.Departamento de Análisis MatemáticoUniversidad de La LagunaLa Laguna (Santa Cruz de Tenerife)Spain
  2. 2.Instituto de Matemática Aplicada del LitoralUNL, CONICET, FIQSanta FeArgentina

Personalised recommendations