Skip to main content
SpringerLink
Log in

Maximal function characterizations of Hardy spaces associated with both non-negative self-adjoint operators satisfying Gaussian estimates and ball quasi-Banach function spaces

  • Published:
Acta Mathematica Scientia Aims and scope Submit manuscript

Abstract

Assume that L is a non-negative self-adjoint operator on L2(ℝn) with its heat kernels satisfying the so-called Gaussian upper bound estimate and that X is a ball quasi-Banach function space on ℝn satisfying some mild assumptions. Let HX, L(ℝn) be the Hardy space associated with both X and L, which is defined by the Lusin area function related to the semigroup generated by L. In this article, the authors establish various maximal function characterizations of the Hardy space HX,L(ℝn) and then apply these characterizations to obtain the solvability of the related Cauchy problem. These results have a wide range of generality and, in particular, the specific spaces X to which these results can be applied include the weighted space, the variable space, the mixed-norm space, the Orlicz space, the Orlicz-slice space, and the Morrey space. Moreover, the obtained maximal function characterizations of the mixed-norm Hardy space, the Orlicz-slice Hardy space, and the Morrey-Hardy space associated with L are completely new.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adams D R. Morrey Spaces. Lecture Notes in Applied and Numerical Harmonic Analysis. Cham: Birkhäuser/Springer, 2015

    Google Scholar 

  2. Andersen K F, John R T. Weighted inequalities for vector-valued maximal functions and singular integrals. Studia Math, 1980, 69(1): 19–31

    Article  MathSciNet  Google Scholar 

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 Unpublished manuscript, 2005

  4. Auscher P, Mourgoglou M. Representation and uniqueness for boundary value elliptic problems via first order systems. Rev Mat Iberoam, 2019, 35(1): 241–315

    Article  MathSciNet  Google Scholar 

  5. Auscher P, Prisuelos-Arribas C. Tent space boundedness via extrapolation. Math Z, 2017, 286(3): 1575–1604

    Article  MathSciNet  Google Scholar 

  6. Benedek A, Panzone R. The space Lp, with mixed norm. Duke Math J, 1961, 28: 301–324

    Article  MathSciNet  Google Scholar 

  7. Bennett C, Sharpley R. Interpolation of Operators. Boston: Academic Press, 1988

    Google Scholar 

  8. Bui T A. Weighted Hardy spaces associated to discrete Laplacians on graphs and applications. Potential Anal, 2014, 41(3): 817–848

    Article  MathSciNet  Google Scholar 

  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: 69–129

    Article  MathSciNet  Google Scholar 

  10. Bui T A, Cao J, Ky L D, et al. Weighted Hardy spaces associated with operators satisfying reinforced off-diagonal estimates. Taiwanese J Math, 2013, 17(4): 1127–1166

    Article  MathSciNet  Google Scholar 

  11. 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, 370(10): 7229–7292

    Article  MathSciNet  Google Scholar 

  12. Bui T A, Duong X T, Ly F K. Maximal function characterizations for Hardy spaces on spaces of homogeneous type with finite measure and applications. J Funct Anal, 2020, 278(8): Art 108423

  13. Bui T A, Li J. Orlicz-Hardy spaces associated to operators satisfying bounded H∞ functional calculus and Davies-Gaffney estimates. J Math Anal Appl, 2011, 373(2): 485–501

    Article  MathSciNet  Google Scholar 

  14. Chiarenza F, Frasca M. Morrey spaces and Hardy-Littlewood maximal function. Rend Mat Appl (7), 1987, 7(3): 273–279

    MathSciNet  Google Scholar 

  15. Cleanthous G, Georgiadis A G, Nielsen M. Anisotropic mixed-norm Hardy spaces. J Geom Anal, 2017, 27(4): 2758–2787

    Article  MathSciNet  Google Scholar 

  16. Cleanthous G, Georgiadis A G, Nielsen M. Discrete decomposition of homogeneous mixed-norm Besov spaces//Cwikel M, Milman M. Functional Analysis, Harmonic Analysis, and Image Processing: A Collection of Papers in Honor of Björn Jawerth. Contemp Math 693. Providence, RI: Amer Math Soc, 2017: 167–184

    Google Scholar 

  17. Cleanthous G, Georgiadis A G, Nielsen M. Molecular decomposition of anisotropic homogeneous mixed-norm spaces with applications to the boundedness of operators. Appl Comput Harmon Anal, 2019, 47(2): 447–480

    Article  MathSciNet  Google Scholar 

  18. Coifman R R, Lions P L, Meyer Y, Semmes S. Compensated compactness and Hardy spaces. J Math Pures Appl (9), 1993, 72(3): 247–286

    MathSciNet  Google Scholar 

  19. Coulhon T, Duong X T. Maximal regularity and kernel bounds: observations on a theorem by Hieber and Prüss. Adv Differential Equations, 2000, 5(1): 343–368

    MathSciNet  Google Scholar 

  20. Cruz-Uribe D V, Fiorenza A. Variable Lebesgue Spaces, Foundations and Harmonic Analysis. Heidelberg: Birkhäuser/Springer, 2013

    Book  Google Scholar 

  21. Cruz-Uribe D V, Wang L A D. Variable Hardy spaces. Indiana Univ Math J, 2014, 63(2): 447–493

    Article  MathSciNet  Google Scholar 

  22. Del Campo R, Fernández A, Mayoral F, Naranjo F. Orlicz spaces associated to a quasi-Banach function space: applications to vector measures and interpolation. Collect Math, 2021, 72(3): 481–499

    Article  MathSciNet  Google Scholar 

  23. Diening L, Häastoö P, Roudenko S. Function spaces of variable smoothness and integrability. J Funct Anal, 2009, 256(6): 1731–1768

    Article  MathSciNet  Google Scholar 

  24. Duoandikoetxea J. Fourier Analysis. Providence, RI: American Mathematical Society, 2001

    Google Scholar 

  25. Duong X T, Li J. Hardy spaces associated to operators satisfying Davies-Gaffney estimates and bounded holomorphic functional calculus. J Funct Anal, 2013, 264(6): 1409–1437

    Article  MathSciNet  Google Scholar 

  26. Duong X T, Yan L. Duality of Hardy and BMO spaces associated with operators with heat kernel bounds. J Amer Math Soc, 2005, 18(4): 943–973

    Article  MathSciNet  Google Scholar 

  27. Duong X T, Yan L. New function spaces of BMO type, the John-Nirenberg inequality, interpolation, and applications. Comm Pure Appl Math, 2005, 58(10): 1375–1420

    Article  MathSciNet  Google Scholar 

  28. Fefferman C, Stein E M. Hp spaces of several variables. Acta Math, 1972, 129(3): 137–193

    Article  MathSciNet  Google Scholar 

  29. Georgiadis A G, Kerkyacharian G, Kyriazis G, Petrushev P. Atomic and molecular decomposition of homogeneous spaces of distributions associated to non-negative self-adjoint operators. J Fourier Anal Appl, 2019, 25(6): 3259–3309

    Article  MathSciNet  Google Scholar 

  30. Georgiadis A G, Kerkyacharian G, Kyriazis G, Petrushev P. Homogeneous Besov and Triebel-Lizorkin spaces associated to non-negative self-adjoint operators. J Math Anal Appl, 2017, 449(2): 1382–1412

    Article  MathSciNet  Google Scholar 

  31. Grafakos L. Classical Fourier Analysis. Third edition. New York: Springer, 2014

    Book  Google Scholar 

  32. Ho K P. Vector-valued maximal inequalities on weighted Orlicz-Morrey spaces. Tokyo J Math, 2013, 36(2): 49–512

    Article  MathSciNet  Google Scholar 

  33. Ho K P. Atomic decomposition of Hardy-Morrey spaces with variable exponents. Ann Acad Sci Fenn Math, 2015, 40(1): 31–62

    Article  MathSciNet  Google Scholar 

  34. Ho K P. Dilation operators and integral operators on amalgam space (Lp, lq). Ric Mat, 2019, 68(2): 661–677

    Article  MathSciNet  Google Scholar 

  35. Hofmann S, Mayboroda S. Hardy and BMO spaces associated to divergence form elliptic operators. Math Ann, 2009, 344(1): 37–116

    Article  MathSciNet  Google Scholar 

  36. Hofmann S, Mayboroda S, McIntosh A. Second order elliptic operators with complex bounded measurable coefficients in Lp, Sobolev and Hardy spaces. Ann Sci École Norm Sup (4), 2011, 44(5): 723–800

    Article  Google Scholar 

  37. Holland F. Harmonic analysis on amalgams of Lp and lq. J London Math Soc (2), 1975, 10: 295–305

    Article  MathSciNet  Google Scholar 

  38. Hörmander L. Estimates for translation invariant operators in Lp spaces. Acta Math, 1960, 104: 93–140

    Article  MathSciNet  Google Scholar 

  39. Huang L, Liu J, Yang D, Yuan W. Atomic and Littlewood-Paley characterizations of anisotropic mixed-norm Hardy spaces and their applications. J Geom Anal, 2019, 29(3): 1991–2067

    Article  MathSciNet  Google Scholar 

  40. Huang L, Liu J, Yang D, Yuan W. Real-variable characterizations of new anisotropic mixed-norm Hardy spaces. Commun Pure Appl Anal, 2020, 19(6): 3033–3082

    Article  MathSciNet  Google Scholar 

  41. Jia H, Wang H. Decomposition of Hardy-Morrey spaces. J Math Anal Appl, 2009, 354(1): 99–110

    Article  MathSciNet  Google Scholar 

  42. Kikuchi N, Nakai E, Tomita N, et al. Calderón-Zygmund operators on amalgam spaces and in the discrete case. J Math Anal Appl, 2007, 335(1): 198–212

    Article  MathSciNet  Google Scholar 

  43. Kokilashvili V, Krbec M. Weighted Inequalities in Lorentz and Orlicz Spaces. River Edge, NJ: World Scientific Publishing, 1991

    Book  Google Scholar 

  44. Lizorkin P I. Multipliers of Fourier integrals and estimates of convolutions in spaces with mixed norm. Applications. Izv Akad Nauk SSSR Ser Mat, 1970, 34: 218–247

    MathSciNet  Google Scholar 

  45. Morrey C B. On the solutions of quasi-linear elliptic partial differential equations. Trans Amer Math Soc, 1938, 43(1): 126–166

    Article  MathSciNet  Google Scholar 

  46. Nakai E, Sawano Y. Hardy spaces with variable exponents and generalized Campanato sapces. J Funct Anal, 2012, 262(9): 3665–3748

    Article  MathSciNet  Google Scholar 

  47. Nakano H. Modulared Semi-Ordered Linear Spaces. Tokyo: Maruzen Co, 1950

    Google Scholar 

  48. Nakano H. Topology of Linear Topological Spaces. Tokyo: Maruzen Co, 1951

    Google Scholar 

  49. Nogayama T. Mixed Morrey spaces. Positivity, 2019, 23(4): 961–1000

    Article  MathSciNet  Google Scholar 

  50. Nogayama T, Ono T, Salim D, Sawano Y. Atomic decomposition for mixed Morrey spaces. J Geom Anal, 2021, 31(9): 9338–9365

    Article  MathSciNet  Google Scholar 

  51. Ouhabaz E M. Analysis of Heat Equations on Domains. Princeton, NJ: Princeton University Press, 2005

    Google Scholar 

  52. Pazy A. Semigroups of Linear Operators and Applications to Partial Differential Equations. New York: Springer-Verlag, 1983

    Book  Google Scholar 

  53. Rao M M, Ren Z D. Theory of Orlicz Spaces. New York: Marcel Dekker, 1991

    Google Scholar 

  54. Rao M M, Ren Z D. Applications of Orlicz Spaces. New York: Marcel Dekker, 2002

    Book  Google Scholar 

  55. Sawano Y, Di Fazio G, Hakim D. Morrey Spaces: Introduction and Applications to Integral Operators and PDE’s. Vol I. Boca Raton, FL: CRC Press, 2020

    Book  Google Scholar 

  56. Sawano Y, Di Fazio G, Hakim D. Morrey Spaces: Introduction and Applications to Integral Operators and PDE’s. Vol II. Boca Raton, FL: CRC Press, 2020

    Book  Google Scholar 

  57. Sawano Y, Ho K P, Yang D, Yang S. Hardy spaces for ball quasi-Banach function spaces. Dissertationes Math, 2017, 525: 1–102

    Article  MathSciNet  Google Scholar 

  58. Semmes S. A primer on Hardy spaces, and some remarks on a theorem of Evans and Müller. Comm Partial Differential Equations, 1994, 19(1): 277–319

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  60. Song L, Yan L. Maximal function characterizations for Hardy spaces associated with nonnegative self-adjoint operators on spaces of homogeneous type. J Evol Equ, 2018, 18(1): 221–243

    Article  MathSciNet  Google Scholar 

  61. Stein E M, Weiss G. On the theory of harmonic functions of several variables. I. The theory of Hp-spaces. Acta Math, 1960, 103: 25–62

    Article  MathSciNet  Google Scholar 

  62. Wang F, Yang D, Yang S. Applications of Hardy spaces associated with ball quasi-Banach function spaces. Results Math, 2020, 75(1): Art 26

  63. Yan L. Classes of Hardy spaces associated with operators, duality theorem and applications. Trans Amer Math Soc, 2008, 360(8): 4383–4408

    Article  MathSciNet  Google Scholar 

  64. Yan X, He Z, Yang D, Yuan W. Hardy spaces associated with ball quasi-Banach function spaces on spaces of homogeneous type: Littlewood-Paley characterizations with applications to boundedness of Calderon-Zygmund operators. Acta Math Sin (Engl Ser), 2022, 38(7): 1133–1184

    Article  MathSciNet  Google Scholar 

  65. Yan X, He Z, Yang D, Yuan W. Hardy spaces associated with ball quasi-Banach function spaces on spaces of homogeneous type: Characterizations of maximal functions, decompositions, and dual spaces. Math Nachr, 2023, 296(7): 3056–3116

    Article  MathSciNet  Google Scholar 

  66. Yuan W, Sickel W, Yang D. Morrey and Campanato Meet Besov, Lizorkin and Triebel. Lecture Notes in Mathematics, Vol 2005. Berlin: Springer-Verlag, 2010

    Book  Google Scholar 

  67. Yang D, Yang S. Maximal function characterizations of Musielak-Orlicz-Hardy spaces associated to nonnegative self-adjoint operators satisfying Gaussian estimates. Commun Pure Appl Anal, 2016, 15(6): 2135–2160

    Article  MathSciNet  Google Scholar 

  68. Yosida K. Functional Analysis. Berlin: Springer-Verlag, 1995

    Book  Google Scholar 

  69. Zhang Y, Yang D, Yuan W, Wang S. Weak Hardy-type spaces associated with ball quasi-Banach function spaces I: Decompositions with applications to boundedness of Calderón-Zygmund operators. Sci China Math, 2021, 64(9): 2007–2064

    Article  MathSciNet  Google Scholar 

  70. Zhang Y, Yang D, Yuan W, Wang S. Real-variable characterizations of Orlicz-slice Hardy spaces. Anal Appl (Singap), 2019, 17(4): 597–664

    Article  MathSciNet  Google Scholar 

  71. 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: 16–42

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratory of Mathematics and Complex Systems (Ministry of Education of China), School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, China

    Xiaosheng Lin & Dachun Yang

  2. School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou, 730000, China

    Sibei Yang

  3. Laboratory of Mathematics and Complex Systems (Ministry of Education of China), School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, China

    Wen Yuan

Authors
  1. Xiaosheng Lin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dachun Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sibei Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Wen Yuan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dachun Yang.

Ethics declarations

Conflict of Interest The authors declare no conflict of interest.

Additional information

This project was partially supported by the National Key Research and Development Program of China (2020YFA0712900), the National Natural Science Foundation of China (12371093, 12071197, 12122102 and 12071431), the Key Project of Gansu Provincial National Science Foundation (23JRRA1022), the Fundamental Research Funds for the Central Universities (2233300008 and lzujbky-2021-ey18) and the Innovative Groups of Basic Research in Gansu Province (22JR5RA391).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lin, X., Yang, D., Yang, S. et al. Maximal function characterizations of Hardy spaces associated with both non-negative self-adjoint operators satisfying Gaussian estimates and ball quasi-Banach function spaces. Acta Math Sci 44, 484–514 (2024). https://doi.org/10.1007/s10473-024-0207-y

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10473-024-0207-y

Key words

2020 MR Subject Classification

Search

Navigation