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Local Hardy Spaces with Variable Exponents Associated with Non-negative Self-Adjoint Operators Satisfying Gaussian Estimates

  • Víctor Almeida
  • Jorge J. Betancor
  • Estefanía Dalmasso
  • Lourdes Rodríguez-MesaEmail author
Article
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Abstract

In this paper we introduce variable exponent local Hardy spaces \(h_L^{p(\cdot )}({\mathbb {R}}^n)\) associated with a non-negative self-adjoint operator L. We assume that, for every \(t>0\), the operator \(e^{-tL}\) has an integral representation whose kernel satisfies a Gaussian upper bound. We define \(h_L^{p(\cdot )}({\mathbb {R}}^n)\) by using an area square integral involving the semigroup \(\{e^{-tL}\}_{t>0}\). A molecular characterization of \(h_L^{p(\cdot )}({\mathbb {R}}^n)\) is established. As an application of the molecular characterization, we prove that \(h_L^{p(\cdot )}({\mathbb {R}}^n)\) coincides with the (global) Hardy space \(H_L^{p(\cdot )}({\mathbb {R}}^n)\) provided that 0 does not belong to the spectrum of L. Also, we show that \(h_L^{p(\cdot )}({\mathbb {R}}^n)=H_{L+I}^{p(\cdot )}({\mathbb {R}}^{n})\).

Keywords

Hardy spaces Molecules Local Variable exponent 

Mathematics Subject Classification

42B35 42B30 42B25 

Notes

Acknowledgements

The authors thank the referee for his/her corrections and insightful comments, which helped to improve the quality of the manuscript.

References

  1. 1.
    Almeida, V., Betancor, J.J., Rodríguez-Mesa, L.: Anisotropic Hardy-Lorentz spaces with variable exponents. Can. J. Math. 69(6), 1219–1273 (2017)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Alvarado, R., Mitrea, M.: Hardy Spaces on Ahlfors-Regular Quasi Metric Spaces. A Sharp Theory, vol. 2142 of Lecture Notes in Mathematics. Springer, Cham (2015)Google Scholar
  3. 3.
    Amenta, A., Kemppainen, M.: Non-uniformly local tent spaces. Publ. Mat. 59(1), 245–270 (2015)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Auscher, P., Duong, X.T., McIntosh, A.: Boundedness of Banach space valued singular integral and Hardy spaces (2005)(Unplublished Manuscript)Google Scholar
  5. 5.
    Auscher, P., Martell, J.M.: 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)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Auscher, P., McIntosh, A., Russ, E.: Hardy spaces of differential forms on Riemannian manifolds. J. Geom. Anal. 18(1), 192–248 (2008)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Betancor, J.J., Damián, W.: Anisotropic local Hardy spaces. J. Fourier Anal. Appl. 16(5), 658–675 (2010)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Bui, H.-Q., Duong, X.T., Yan, L.: Calderón reproducing formulas and new Besov spaces associated with operators. Adv. Math. 229, 2249–2502 (2012)zbMATHGoogle Scholar
  9. 9.
    Bui, T.A., Cao, J., Ky, L.D., Yang, D., Yang, S.: Weighted Hardy spaces associated with operators satisfying reinforced off-diagonal estimates. Taiwanese J. Math. 17(4), 1127–1166 (2013)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Cao, J., Fu, Z., Jiang, R., Yang, D.: Hardy spaces associated with a pair of commuting operators. Forum Math. 27(5), 2775–2824 (2015)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Cao, J., Mayboroda, S., Yang, D.: Local Hardy spaces associated with inhomogeneous higher order elliptic operators. Anal. Appl. (Singap.) 15(2), 137–224 (2017)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Carbonaro, A., McIntosh, A., Morris, A.J.: Local Hardy spaces of differential forms on Riemannian manifolds. J. Geom. Anal. 23(1), 106–169 (2013)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Coifman, R.R., Weiss, G.: Extensions of Hardy spaces and their use in analysis. Bull. Am. Math. Soc. 83(4), 569–645 (1977)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Coifman, R.R., Meyer, Y., Stein, E.M.: Some new function spaces and their applications to harmonic analysis. J. Funct. Anal. 62(2), 304–335 (1985)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Cruz-Uribe, D.V., Fiorenza, A.: Variable Lebesgue Spaces. Applied and Numerical Harmonic Analysis. Birkhäuser/Springer, Heidelberg (2013). Foundations and harmonic analysiszbMATHGoogle Scholar
  16. 16.
    Cruz-Uribe, D., Wang, L.-A.D.: Variable Hardy spaces. Indiana Univ. Math. J. 63(2), 447–493 (2014)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Cruz-Uribe, D., Fiorenza, A., Neugebauer, C.J.: The maximal function on variable \(L^p\) spaces. Ann. Acad. Sci. Fenn. Math. 28(1), 223–238 (2003)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Cruz-Uribe, D.V., Martell, J., Pérez, C.: Weights, Extrapolation and the Theory of Rubio de Francia. Operator Theory: Advances and Applications, vol. 215. Birkhäuser/Springer, Basel (2011)zbMATHGoogle Scholar
  19. 19.
    Dafni, G., Yue, H.: Some characterizations of local bmo and \(h^1\) on metric measure spaces. Anal. Math. Phys. 2(3), 285–318 (2012)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Diening, L.: Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces. Bull. Sci. Math. 129(8), 657–700 (2005)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Diening, L., Harjulehto, P., Hästö, P., Mizuta, Y., Shimomura, T.: Maximal functions in variable exponent spaces: limiting cases of the exponent. Ann. Acad. Sci. Fenn. Math. 34(2), 503–522 (2009)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Diening, L., Hästö, P., Roudenko, S.: Function spaces of variable smoothness and integrability. J. Funct. Anal. 256(6), 1731–1768 (2009)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Diening, L., Harjulehto, P., Hästö, P., Ružička, M.: Lebesgue and Sobolev Spaces with Variable Exponents, vol. 2017 of Lecture Notes in Mathematics. Springer, Heidelberg (2011)Google Scholar
  24. 24.
    Duong, X.T., Yan, L.: Duality of Hardy and BMO spaces associated with operators with heat kernel bounds. J. Am. Math. Soc. 18(4), 943–973 (2005). (electronic)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Duong, X.T., Yan, L.: New function spaces of BMO type, the John-Nirenberg inequality, interpolation, and applications. Commun. Pure Appl. Math. 58(10), 1375–1420 (2005)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Ephremidze, L., Kokilashvili, V., Samko, S.: Fractional, maximal and singular operators in variable exponent Lorentz spaces. Fract. Calc. Appl. Anal. 11(4), 407–420 (2008)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Fefferman, C., Stein, E.M.: \(H^{p}\) spaces of several variables. Acta Math. 129(3–4), 137–193 (1972)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Goldberg, D.: A local version of real Hardy spaces. Duke Math. J. 46(1), 27–42 (1979)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Gong, R., Yan, L.: Weighted \(L^p\) estimates for the area integral associated to self-adjoint operators. Manuscripta Math. 144(1–2), 25–49 (2014)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Gong, R., Li, J., Yan, L.: A local version of Hardy spaces associated with operators on metric spaces. Sci. China Math. 56(2), 315–330 (2013)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Hästö, P.A.: Local-to-global results in variable exponent spaces. Math. Res. Lett. 16(2), 263–278 (2009)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Hofmann, S., Mayboroda, S.: Hardy and BMO spaces associated to divergence form elliptic operators. Math. Ann. 344(1), 37–116 (2009)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Huang, J., Liu, Y.: Molecular characterization of Hardy spaces associated with twisted convolution. J. Funct. Spaces, Art. ID 326940, 6 (2014)Google Scholar
  34. 34.
    Huang, J., Wang, J.: \({H}^p\)-boundedness of Weyl multipliers. J. Inequal. Appl. 2014(422), 9 (2014)zbMATHGoogle Scholar
  35. 35.
    Hytönen, T., Yang, D., Yang, D.: The Hardy space \(H^1\) on non-homogeneous metric spaces. Math. Proc. Cambridge Philos. Soc. 153(1), 9–31 (2012)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Jiang, R., Yang, D., Zhou, Y.: Localized Hardy spaces associated with operators. Appl. Anal. 88(9), 1409–1427 (2009)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Jiang, R., Yang, D., Yang, D.: Maximal function characterizations of Hardy spaces associated with magnetic Schrödinger operators. Forum Math. 24(3), 471–494 (2012)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Kempka, H., Vybíral, J.: Lorentz spaces with variable exponents. Math. Nachr. 287(8–9), 938–954 (2014)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Kemppainen, M.: A note on local Hardy spaces. Forum Math. 29(4), 941–949 (2017)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Koch, H., Ricci, F.: Spectral projections for the twisted Laplacian. Studia Math. 180(2), 103–110 (2007)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Lerner, A.K.: Some remarks on the Hardy-Littlewood maximal function on variable \(L^p\) spaces. Math. Z. 251(3), 509–521 (2005)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Liang, Y., Yang, D., Yang, S.: Applications of Orlicz spaces associated with operators satisfying Poisson estimates. Sci. China Math. 54(11), 2395–2426 (2011)MathSciNetzbMATHGoogle Scholar
  43. 43.
    Liu, S., Zhao, K., Zhou, S.: Weighted Hardy spaces associated to self-adjoint operators and \(BMO_{L, w}\). Taiwanese J. Math. 18(5), 1663–1678 (2014)MathSciNetzbMATHGoogle Scholar
  44. 44.
    Liu, J., Yang, D., Yuan, W.: Anisotropic variable Hardy-Lorentz spaces and their real interpolation. J. Math. Anal. Appl. 456(1), 356–393 (2017)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Mauceri, G., Picardello, M.A., Ricci, F.: A Hardy space associated with twisted convolution. Adv. Math. 39(3), 270–288 (1981)MathSciNetzbMATHGoogle Scholar
  46. 46.
    McIntosh, A.: Operators, which have an \(H_\infty \) functional calculus. In: Miniconference on Operator Theory and Partial Differential Equations (North Ryde, 1986), vol. 14 of Proc. Centre Math. Anal. Austral. Nat. Univ. Austral. Nat. Univ., Canberra, pp. 210–231 (1986)Google Scholar
  47. 47.
    Nakai, E., Sawano, Y.: Hardy spaces with variable exponents and generalized Campanato spaces. J. Funct. Anal. 262(9), 3665–3748 (2012)MathSciNetzbMATHGoogle Scholar
  48. 48.
    Nekvinda, A.: Hardy-Littlewood maximal operator on \(L^{p(x)}({\mathbb{R}})\). Math. Inequal. Appl. 7(2), 255–265 (2004)MathSciNetzbMATHGoogle Scholar
  49. 49.
    Ouhabaz, E.M.: Analysis of Heat Equations on Domains, vol. 31 of London Mathematical Society Monographs Series. Princeton University Press, Princeton, NJ (2005)Google Scholar
  50. 50.
    Ouhabaz, E.M.: Sharp Gaussian bounds and \(L^p\)-growth of semigroups associated with elliptic and Schrödinger operators. Proc. Am. Math. Soc. 134(12), 3567–3575 (2006)zbMATHGoogle Scholar
  51. 51.
    Rudin, W.: Functional Analysis. McGraw-Hill Inc, Singapore (1991)zbMATHGoogle Scholar
  52. 52.
    Russ, E.: The atomic decomposition for tent spaces on spaces of homogeneous type. In: CMA/AMSI Research Symposium “Asymptotic Geometric Analysis, Harmonic Analysis, and Related Topics”, vol. 42 of Proc. Centre Math. Appl. Austral. Nat. Univ. Austral. Nat. Univ., Canberra, pp. 125–135 (2007)Google Scholar
  53. 53.
    Sawano, Y.: Atomic decompositions of Hardy spaces with variable exponents and its application to bounded linear operators. Integral Equ. Oper. Theory 77(1), 123–148 (2013)MathSciNetzbMATHGoogle Scholar
  54. 54.
    Stempak, K., Zienkiewicz, J.: Twisted convolution and Riesz means. J. Anal. Math. 76, 93–107 (1998)MathSciNetzbMATHGoogle Scholar
  55. 55.
    Strichartz, R.S.: The Hardy space \(H^{1}\) on manifolds and submanifolds. Can. J. Math. 24, 915–925 (1972)zbMATHGoogle Scholar
  56. 56.
    Tan, J.: Atomic decomposition of localized Hardy spaces with variable exponents and applications. J. Geom. Anal. 29(1), 799–827 (2019)MathSciNetzbMATHGoogle Scholar
  57. 57.
    Tang, L.: Weighted local Hardy spaces and their applications. Illinois J. Math. 56(2), 453–495 (2012)MathSciNetzbMATHGoogle Scholar
  58. 58.
    Thangavelu, S.: Some remarks on Bochner-Riesz means. Colloq. Math. 83(2), 217–230 (2000)MathSciNetzbMATHGoogle Scholar
  59. 59.
    Tolsa, X.: The space \(H^1\) for nondoubling measures in terms of a grand maximal operator. Trans. Am. Math. Soc. 355(1), 315–348 (2003)MathSciNetzbMATHGoogle Scholar
  60. 60.
    Vogt, H.: \(L_1\)-estimates for eigenfunctions and heat kernel estimates for semigroups dominated by the free heat semigroup. J. Evol. Equ. 15(4), 879–893 (2015)MathSciNetzbMATHGoogle Scholar
  61. 61.
    Yang, D.: Local Hardy and BMO spaces on non-homogeneous spaces. J. Aust. Math. Soc. 79(2), 149–182 (2005)MathSciNetzbMATHGoogle Scholar
  62. 62.
    Yang, D., Yang, S.: Local Hardy spaces of Musielak-Orlicz type and their applications. Sci. China Math. 55(8), 1677–1720 (2012)MathSciNetzbMATHGoogle Scholar
  63. 63.
    Yang, D., Zhang, J.: Variable Hardy spaces associated with operators satisfying Davies-Gaffney estimates on metric measure spaces of homogeneous type. Ann. Acad. Sci. Fenn. Math. 43(1), 47–87 (2018)MathSciNetzbMATHGoogle Scholar
  64. 64.
    Yang, D., Zhuo, C.: Molecular characterizations and dualities of variable exponent Hardy spaces associated with operators. Ann. Acad. Sci. Fenn. Math. 41(1), 357–398 (2016)MathSciNetzbMATHGoogle Scholar
  65. 65.
    Yang, D., Zhuo, C., Nakai, E.: Characterizations of variable exponent Hardy spaces via Riesz transforms. Rev. Mat. Complut. 29(2), 245–270 (2016)MathSciNetzbMATHGoogle Scholar
  66. 66.
    Yang, D., Zhang, J., Zhuo, C.: Variable Hardy spaces associated with operators satisfying Davies-Gaffney estimates. Proc. Edinb. Math. Soc. (2) 61 3, 759–810 (2018)MathSciNetGoogle Scholar
  67. 67.
    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)MathSciNetzbMATHGoogle Scholar
  68. 68.
    Zhuo, C., Yang, D.: Variable weak Hardy spaces \({WH}_{L}^{p(\cdot )}({\mathbb{R}}^n)\) associated with operators satisfying Davies-Gaffney estimates. Forum Math. (2019).  https://doi.org/10.1515/forum-2018-0125
  69. 69.
    Zhuo, C., Sawano, Y., Yang, D.: Hardy spaces with variable exponents on RD-spaces and applications. Diss. Math. (Rozprawy Mat.) 520, 74 (2016)MathSciNetzbMATHGoogle Scholar
  70. 70.
    Zhuo, C., Yang, D., Liang, Y.: Intrinsic square function characterizations of Hardy spaces with variable exponents. Bull. Malays. Math. Sci. Soc. 39(4), 1541–1577 (2016)MathSciNetzbMATHGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2019

Authors and Affiliations

  1. 1.Departamento de Análisis MatemáticoUniversidad de La LagunaLa Laguna (Sta. Cruz de Tenerife)Spain
  2. 2.Instituto de Matemática Aplicada del Litoral, UNL, CONICET, FCE/FIQSanta FeArgentina

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