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Science China Mathematics

, Volume 62, Issue 1, pp 73–124 | Cite as

Variable exponent Hardy spaces associated with discrete Laplacians on graphs

  • Víctor Almeida
  • Jorge J. Betancor
  • Alejandro J. Castro
  • Lourdes Rodríguez-MesaEmail author
Articles

Abstract

In this paper we develop the theory of variable exponent Hardy spaces associated with discrete Laplacians on infinite graphs. Our Hardy spaces are defined by square integrals, atomic and molecular decompositions. Also we study boundedness properties of Littlewood-Paley functions, Riesz transforms, and spectral multipliers for discrete Laplacians on variable exponent Hardy spaces.

Keywords

graphs discrete Laplacian Hardy spaces variable exponent square functions spectral multipliers 

MSC(2010)

42B30 42B35 60J10 

Notes

Acknowledgements

The first, second and fourth authors are partially supported by Spanish Government Grant (Grant No. MTM2016-79436-P). The third author is also supported by Nazarbayev University Social Policy Grant. The authors would strongly like to give thanks to Professor Dachun Yang for sending us his paper [64] (jointly with C. Zhuo and Y. Sawano). Also, the authors are grateful to the referees for the careful reading of the manuscript.

References

  1. 1.
    Aboulaich R, Meskine D, Souissi A. New diffusion models in image processing. Comput Math Appl, 2008, 56: 874–882MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Acerbi E, Mingione G. Regularity results for a class of functionals with non-standard growth. Arch Ration Mech Anal, 2001, 156: 121–140MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Adamowicz T, Harjulehto P, Hästö P. Maximal operator in variable exponent Lebesgue spaces on unbounded quasimetric measure spaces. Math Scand, 2015, 116: 5–22MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Alexopoulos G-K. Spectral multipliers on discrete groups. Bull Lond Math Soc, 2001, 33: 417–424MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Alexopoulos G-K. Spectral multipliers for Markov chains. J Math Soc Japan, 2004, 56: 833–852MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Badr N, Martell J-M. Weighted norm inequalities on graphs. J Geom Anal, 2012, 22: 1173–1210MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Badr N, Russ E. Interpolation of Sobolev spaces, Littlewood-Paley inequalities and Riesz transforms on graphs. Publ Mat, 2009, 53: 273–328MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bui T-A. Weighted Hardy spaces associated to discrete Laplacians on graphs and applications. Potential Anal, 2014, 41: 817–848MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bui T-A, Cao J, Ky L-D, et al. Musielak-Orlicz-Hardy spaces associated with operators satisfying reinforced off-diagonal estimates. Anal Geom Metr Spaces, 2013, 1: 69–129MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bui T-A, Duong X-T. Hardy spaces associated to the discrete Laplacians on graphs and boundedness of singular integrals. Trans Amer Math Soc, 2014, 366: 3451–3485MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Bui T-A, Duong X-T. Weighted norm inequalities for spectral multipliers on graphs. Potential Anal, 2016, 44: 263–293MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Bui T-A, Duong X-T, Ly F-K. Maximal function characterizations for new local Hardy type spaces on spaces of homogeneous type. Trans Amer Math Soc, 2018, in pressGoogle Scholar
  13. 13.
    Chen Y, Levine S, Rao M. Variable exponent, linear growth functionals in image restoration. SIAM J Appl Math, 2006, 66: 1383–1406MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Coifman R-R, Meyer Y, Stein E-M. Some new function spaces and their applications to harmonic analysis. J Funct Anal, 1985, 62: 304–335MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Coifman R-R, Weiss G. Analyse Harmonique Non-Commutative sur Certains Espaces Homogènes. Lecture Notes in Mathematics, vol. 242. Berlin: Springer-Verlag, 1971Google Scholar
  16. 16.
    Coifman R-R, Weiss G. Extensions of Hardy spaces and their use in analysis. Bull Amer Math Soc, 1977, 83: 569–645MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Cowling M, Meda S, Setti A-G. Estimates for functions of the Laplace operator on homogeneous trees. Trans Amer Math Soc, 2000, 352: 4271–4293MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Cruz-Uribe D, Fiorenza A. Variable Lebesgue Spaces. Applied and Numerical Harmonic Analysis. Heidelberg: Birkhäuser/Springer, 2013zbMATHGoogle Scholar
  19. 19.
    Cruz-Uribe D, Fiorenza A, Martell J-M, et al. The boundedness of classical operators on variable L p spaces. Ann Acad Sci Fenn Math, 2006, 31: 239–264MathSciNetzbMATHGoogle Scholar
  20. 20.
    Cruz-Uribe D, Fiorenza A, Neugebauer C-J. Corrections to: "The maximal function on variable L p spaces". Ann Acad Sci Fenn Math, 2003, 28: 223–238MathSciNetzbMATHGoogle Scholar
  21. 21.
    Cruz-Uribe D, Wang L A D. Variable Hardy spaces. Indiana Univ Math J, 2014, 63: 447–493MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Diening L. Theoretical and numerical results for electrorheological fluids. PhD Thesis. Germany: University of Freiburg, 1998zbMATHGoogle Scholar
  23. 23.
    Diening L. Maximal function on generalized Lebesgue spaces L p(·). Math Inequal Appl, 2004, 7: 245–253MathSciNetzbMATHGoogle Scholar
  24. 24.
    Diening L, Harjulehto P, Hästö P, et al. Lebesgue and Sobolev Spaces with Variable Exponents. Lecture Notes in Mathematics, vol. 2017. Heidelberg: Springer, 2011Google Scholar
  25. 25.
    Dungey N. A note on time regularity for discrete time heat kernels. Semigroup Forum, 2006, 72: 404–410MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Dziubański J, Preisner M. Hardy spaces for semigroups with Gaussian bounds. Ann Acad Sci Fenn Math, 2017,  https://doi.org/10.1007/s10231-017-0711-y Google Scholar
  27. 27.
    Feneuil J. Littlewood-Paley functionals on graphs. Math Nachr, 2015, 288: 1254–1285MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Feneuil J. Hardy and BMO spaces on graphs, application to Riesz transform. Potential Anal, 2016, 45: 1–54MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Grafakos L, Liu L, Yang D. Maximal function characterizations of Hardy spaces on RD-spaces and their applications. Sci China Ser A, 2008, 51: 2253–2284MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Grafakos L, Liu L, Yang D. Radial maximal function characterizations for Hardy spaces on RD-spaces. Bull Soc Math France, 2009, 137: 225–251MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    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, 2008: 893409CrossRefzbMATHGoogle Scholar
  32. 32.
    Harjulehto P, Hästö P, Pere M. Variable exponent Lebesgue spaces on metric spaces: The Hardy-Littlewood maximal operator. Real Anal Exchange, 2004, 30: 87–103MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Khabazi M. The maximal operator in spaces of homogenous type. Proc A Razmadze Math Inst, 2005, 138: 17–25MathSciNetzbMATHGoogle Scholar
  34. 34.
    Kyrezi I, Marias M. H p-bounds for spectral multipliers on graphs. Trans Amer Math Soc, 2009, 361: 1053–1067MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Kokilashvili V, Samko S. The maximal operator in weighted variable spaces on metric measure spaces. Proc A Razmadze Math Inst, 2007, 144: 137–144MathSciNetzbMATHGoogle Scholar
  36. 36.
    Kováčik O, Rákosník J-M. On spaces L p(x) and W k,p(x). Czechoslovak Math J, 1991, 41: 592–618MathSciNetzbMATHGoogle Scholar
  37. 37.
    Macías R-A, Segovia C. A decomposition into atoms of distributions on spaces of homogeneous type. Adv Math, 1979, 33: 271–309MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Macías R-A, Segovia C. Lipschitz functions on spaces of homogeneous type. Adv Math, 1979, 33: 257–270MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Martell J-M, Prisuelos-Arribas C. Weighted norm inequalities for conical square functions. Part I. Trans Amer Math Soc, 2017, 369: 4193–4233CrossRefzbMATHGoogle Scholar
  40. 40.
    Nakai E, Sawano Y. Hardy spaces with variable exponents and generalized Campanato spaces. J Funct Anal, 2012, 262: 3665–3748MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Nakai E, Yabuta K. Pointwise multipliers for functions of weighted bounded mean oscillation on spaces of homogeneous type. Math Japon, 1997, 46: 15–28MathSciNetzbMATHGoogle Scholar
  42. 42.
    Nakano H. Modulared Semi-Ordered Linear Spaces. Tokyo: Maruzen, 1950zbMATHGoogle Scholar
  43. 43.
    Nekvinda A. A note on maximal operator on l {pn} and L p(x)(ℝ). J Funct Spaces Appl, 2007, 5: 49–88MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Orlicz W. Über konjugierte Exponentenfolgen. Studia Math, 1931, 3: 200–211CrossRefzbMATHGoogle Scholar
  45. 45.
    Pang M-M-H. Heat kernels of graphs. J Lond Math Soc, 1993, 47: 50–64MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Pang M-M-H. The heat kernel of the Laplacian defined on a uniform grid. Semigroup Forum, 2009, 78: 238–252MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Pick L-S, Růžicka M. An example of a space Lp(x) on which the Hardy-Littlewood maximal operator is not bounded. Expo Math, 2001, 19: 369–371MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Rajagopal K-R, Růžička M. On the modeling of electrorheological materials. Mech Res Comm, 1996, 23: 401–407CrossRefzbMATHGoogle Scholar
  49. 49.
    Russ E. Espaces de Hardy et transformées de Riesz sur les graphes et les variétés. PhD Thesis. Pontoise: University Cergy, 1998Google Scholar
  50. 50.
    Russ E. H1-BMO duality on graphs. Colloq Math, 2000, 86: 67–91MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Russ E. Riesz transforms on graphs for 1 ≤ p ≤ 2. Math Scand, 2000, 87: 133–160MathSciNetCrossRefzbMATHGoogle 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". Proceedings of the Centre for Mathematics and its Applications, vol. 42. Canberra: The Australian National University, 2007, 125–135MathSciNetzbMATHGoogle Scholar
  53. 53.
    Růžička M. Electrorheological Fluids: Modeling and Mathematical Theory. Lecture Notes in Mathematics, vol. 1748. Berlin: Springer-Verlag, 2000Google Scholar
  54. 54.
    Sawano Y. Atomic decompositions of Hardy spaces with variable exponents and its application to bounded linear operators. Integral Equations Operator Theory, 2013, 77: 123–148MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    Song L, Yan L. Maximal function characterizations for Hardy spaces associated with non-negative self-adjoint operators on spaces of homogeneous type. J Evol Equ, 2018, 18: 221–243MathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    Uchiyama A. A maximal function characterization of H p on the space of homogeneous type. Trans Amer Math Soc, 1980, 262: 579–592MathSciNetzbMATHGoogle Scholar
  57. 57.
    Yan L. Classes of Hardy spaces associated with operators, duality theorem and applications. Trans Amer Math Soc, 2008, 360: 4383–4408MathSciNetCrossRefzbMATHGoogle Scholar
  58. 58.
    Yang D, Yang S. Musielak-Orlicz-Hardy spaces associated with operators and their applications. J Geom Anal, 2014, 24: 495–570MathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    Yang D, Yang S. Maximal function characterizations of Musielak-Orlicz-Hardy spaces associated to non-negative selfadjoint operators satisfying Gaussian estimates. Commun Pure Appl Anal, 2016, 15: 2135–2160MathSciNetCrossRefzbMATHGoogle Scholar
  60. 60.
    Yang D, Zhang J, Zhuo C. Variable Hardy spaces associated with operators satisfying Davies-Gaffney estimates. Proc Edinb Math Soc (2), arXiv:1601.06358, 2016Google Scholar
  61. 61.
    Yang D, Zhou Y. Localized Hardy spaces H1 related to admisible functions on RD-spaces and applications to Schrödinger operators. Trans Amer Math Soc, 2011, 363: 1197–1239MathSciNetCrossRefzbMATHGoogle Scholar
  62. 62.
    Yang D, Zhuo C. Molecular characterizations and dualities of variable exponent Hardy spaces associated with operators. Ann Acad Sci Fenn Math, 2016, 41: 357–398MathSciNetCrossRefzbMATHGoogle Scholar
  63. 63.
    Yang D, Zhuo C, Nakai E. Characterizations of variable exponent Hardy spaces via Riesz transforms. Rev Mat Complut, 2016, 29: 245–270MathSciNetCrossRefzbMATHGoogle Scholar
  64. 64.
    Zhuo C, Sawano Y, Yang D. Hardy spaces with variable exponents on RD-spaces and applications. Dissertations Math (Rozprawy Mat), 2016, 520: 1–74MathSciNetCrossRefzbMATHGoogle Scholar
  65. 65.
    Zhuo C, Yang D. Maximal function characterizations of variable Hardy spaces associated with non-negative self-adjoint operators satisfying Gaussian estimates. Nonlinear Anal, 2016, 141: 16–42MathSciNetCrossRefzbMATHGoogle Scholar
  66. 66.
    Zhuo C, Yang D, Liang Y. Intrinsic square function characterizations of Hardy spaces with variable exponents. Bull Malays Math Sci Soc (2), 2016, 39: 1541–1577MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Víctor Almeida
    • 1
  • Jorge J. Betancor
    • 1
  • Alejandro J. Castro
    • 2
  • Lourdes Rodríguez-Mesa
    • 1
    Email author
  1. 1.Departamento de Análisis MatemáticoUniversidad de La LagunaLa LagunaSpain
  2. 2.Department of MathematicsNazarbayev UniversityAstanaKazakhstan

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