Skip to main content
SpringerLink
Log in

Characterizations of Product Hardy Spaces in Bessel Setting

  • Published:
Journal of Fourier Analysis and Applications Aims and scope Submit manuscript

    We’re sorry, something doesn't seem to be working properly.

    Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Abstract

In this paper, we work in the setting of Bessel operators and Bessel Laplace equations studied by Weinstein, Huber, and the harmonic function theory in this setting introduced by Muckenhoupt–Stein, especially the generalised Cauchy–Riemann equations and the conjugate harmonic functions. We provide the equivalent characterizations of product Hardy spaces associated with Bessel operators in terms of the Bessel Riesz transforms, non-tangential and radial maximal functions defined via Poisson and heat semigroups, based on the atomic decomposition, the generalised Cauchy–Riemann equations, the extension of Merryfield’s result which connects the product non-tangential maximal function and area function, and on the grand maximal function technique which connects the product non-tangential and radial maximal function. We then obtain directly the decomposition of the product BMO space associated with Bessel operators.

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. Andersen, K.F., Kerman, R.A.: Weighted norm inequalities for generalized Hankel conjugate transformations. Studia Math. 71(1), 15–26 (1981/82)

  2. Betancor, J.J., Castro, A.J., Nowak, A.: Calderón-Zygmund operators in the Bessel setting. Monatsh. Math. 167(3–4), 375–403 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  3. Betancor, J.J., Chicco Ruiz, A., Fariña, J.C., Rodríguez-Mesa, L.: Maximal operators, Riesz transforms and Littlewood-Paley functions associated with Bessel operators on BMO. J. Math. Anal. Appl. 363(1), 310–326 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Betancor, J.J., Duong, X.T., Li, J., Wick, B.D., Yang, D.: Product Hardy, BMO spaces and iterated commutators associated with Bessel Schrödinger operators. Indiana Univ. Math. J. 68(1), 247–289 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  5. Betancor, J.J., Dziubański, J., Torrea, J.L.: On Hardy spaces associated with Bessel operators. J. Anal. Math. 107, 195–219 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Betancor, J.J., Fariña, J.C., Buraczewski, D., Martínez, T., Torrea, J.L.: Riesz transforms related to Bessel operators. Proc. R. Soc. Edinb. Sect. A 137(4), 701–725 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  7. Betancor, J.J., Fariña, J.C., Sanabria, A.: On Littlewood-Paley functions associated with Bessel operators. Glasg. Math. J. 51(1), 55–70 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Betancor, J.J., Harboure, E., Nowak, A., Viviani, B.: Mapping properties of fundamental operators in harmonic analysis related to Bessel operators. Studia Math. 197(2), 101–140 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bownik, M., Li, B., Yang, D., Zhou, Y.: Weighted anisotropic product Hardy spaces and boundedness of sublinear operators. Math. Nachr. 283(3), 392–442 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Castro, A.J., Szarek, T.Z.: Calderón-Zygmund operators in the Bessel setting for all possible type indices. Acta Math. Sin. (Engl. Ser.) 30(4), 637–648 (2014)

  11. Chang, D.-C., Yang, D., Zhou, Y.: Boundedness of sublinear operators on product Hardy spaces and its application. J. Math. Soc. Japan 62(1), 321–353 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chang, S.-Y.A.: Carleson measure on the bi-disc. Ann. Math. (2) 109(3), 613–620 (1979)

  13. Chang, S.-Y.A., Fefferman, R.: A continuous version of duality of \(H^{1}\) with BMO on the bidisc. Ann. Math. (2) 112(1), 179–201 (1980)

  14. Chang, S.-Y.A., Fefferman, R.: The Calderón-Zygmund decomposition on product domains. Am. J. Math. 104(3), 455–468 (1982)

    Article  MATH  Google Scholar 

  15. Chang, S.-Y.A., Fefferman, R.: Some recent developments in Fourier analysis and \(H^p\)-theory on product domains. Bull. Am. Math. Soc. (N.S.) 12(1), 1–43 (1985)

  16. Chen, P., Duong, X.T., Li, J., Ward, L.A., Yan, L.: Product Hardy spaces associated to operators with heat kernel bounds on spaces of homogeneous type. Math. Z. 282(3–4), 1033–1065 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  17. Coifman, R.R., Weiss, G.: Extensions of Hardy spaces and their use in analysis. Bull. Am. Math. Soc. 83(4), 569–645 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  18. Dalenc, L., Ou, Y.: Upper bound for multi-parameter iterated commutators. Publ. Math. 60(1), 191–220 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  19. David, G., Journé, J.-L., Semmes, S.: Opérateurs de Calderón-Zygmund, fonctions para-accrétives et interpolation. Rev. Mat. Iberoamericana 1(4), 1–56 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  20. Deng, D., Song, L., Tan, C., Yan, L.: Duality of Hardy and BMO spaces associated with operators with heat kernel bounds on product domains. J. Geom. Anal. 17(3), 455–483 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  21. Duoandikoetxea, J.: Fourier Analysis. Graduate Studies in Mathematics, vol. 29. American Mathematical Society, Providence, RI (2001)

  22. Duong, X.T., Li, J., Mao, S., Wu, H., Yang, D.: Compactness of Riesz transform commutator associated with Bessel operators. J. Anal. Math. 135(2), 639–673 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  23. Duong, X.T., Li, J., Ou, Y., Wick, B.D., Yang, D.: Product BMO, little BMO, and Riesz commutators in the Bessel setting. J. Geom. Anal. 28(3), 2558–2601 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  24. Duong, X.T., Li, J., Wick, B.D., Yang, D.: Factorization for Hardy spaces and characterization for BMO spaces via commutators in the Bessel setting. Indiana Univ. Math. J. 66(4), 1081–1106 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  25. Duong, X.T., Li, J., Yan, L.: Endpoint estimates for singular integrals with non-smooth kernels on product spaces. arXiv:1509.07548

  26. Duong, X.T., Yan, L.: Hardy spaces of spaces of homogeneous type. Proc. Am. Math. Soc. 131(10), 3181–3189 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  27. 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)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  29. Fefferman, R.: Bounded mean oscillation on the polydisk. Ann. Math. (2) 110(2), 395–406 (1979)

  30. Fefferman, R.: Multiparameter Fourier analysis. In: Beijing lectures in harmonic analysis (Beijing, 1984), volume 112 of Ann. of Math. Stud., pp. 47–130. Princeton University Press, Princeton, NJ (1986)

  31. Fefferman, R.: Calderón-Zygmund theory for product domains: \(H^p\) spaces. Proc. Natl. Acad. Sci. U.S.A. 83(4), 840–843 (1986)

    Article  MATH  Google Scholar 

  32. Fefferman, R.: Harmonic analysis on product spaces. Ann. of Math. (2) 126(1), 109–130 (1987)

  33. Fefferman, R.: Multiparameter Calderón-Zygmund theory. In: Harmonic analysis and partial differential equations (Chicago, IL, 1996), Chicago Lectures in Math., pp. 207–221. University Chicago Press, Chicago, IL (1999)

  34. Fefferman, R., Stein, E.M.: Singular integrals on product spaces. Adv. Math. 45(2), 117–143 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  35. Ferguson, S.H., Lacey, M.T.: A characterization of product BMO by commutators. Acta Math. 189(2), 143–160 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  36. Ferguson, S.H., Sadosky, C.: Characterizations of bounded mean oscillation on the polydisk in terms of Hankel operators and Carleson measures. J. Anal. Math. 81, 239–267 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  37. Grafakos, L., Liu, L., Yang, D.: Maximal function characterizations of Hardy spaces on RD-spaces and their applications. Sci. China Ser. A 51(12), 2253–2284 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  38. Grafakos, L., Liu, L., Yang, D.: Radial maximal function characterizations for Hardy spaces on RD-spaces. Bull. Soc. Math. France 137(2), 225–251 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  39. Gundy, R.F., Stein, E.M.: \(H^{p}\) theory for the poly-disc. Proc. Natl. Acad. Sci. U.S.A. 76(3), 1026–1029 (1979)

    Article  MATH  Google Scholar 

  40. Haimo, D.T.: Integral equations associated with Hankel convolutions. Trans. Am. Math. Soc. 116, 330–375 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  41. Han, Y., Li, J., Lin, C.-C.: Criterion of the \(L^2\) boundedness and sharp endpoint estimates for singular integral operators on product spaces of homogeneous type. Ann. Sci. Norm. Super. Pisa Cl. Sci. (5) 16(3), 845–907 (2016)

  42. Han, Y., Li, J., Lu, G.: Duality of multiparameter Hardy spaces \(H^p\) on spaces of homogeneous type. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 9(4), 645–685 (2010)

  43. Han, Y., Li, J., Lu, G.: Multiparameter Hardy space theory on Carnot-Carathéodory spaces and product spaces of homogeneous type. Trans. Amer. Math. Soc. 365(1), 319–360 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  44. Han, Y., Li, J., Lu, G., Wang, P.: \(H^p\rightarrow H^p\) boundedness implies \(H^p\rightarrow L^p\) boundedness. Forum Math. 23(4), 729–756 (2011)

    MathSciNet  Google Scholar 

  45. Han, Y., Li, J., Pereyra, C., Ward, L.A.: Atomic decomposition of product hardy spaces via wavelet bases on spaces of homogeneous type. arXiv:1810.03788

  46. Han, Y., Li, J., Ward, L.A.: Hardy space theory on spaces of homogeneous type via orthonormal wavelet bases. Appl. Comput. Harmon. Anal. 45(1), 120–169 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  47. Han, Y., Müller, D., Yang, D.: Littlewood-Paley characterizations for Hardy spaces on spaces of homogeneous type. Math. Nachr. 279(13–14), 1505–1537 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  48. 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.: pages Art. ID 893409, 250 (2008)

  49. Han, Y., Yang, D.: \(H^p\) boundedness of Calderón-Zygmund operators on product spaces. Math. Z. 249(4), 869–881 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  50. 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(1007), vi+78 (2011)

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

    Article  MathSciNet  MATH  Google Scholar 

  52. Huber, A.: On the uniqueness of generalized axially symmetric potentials. Ann. Math. 2(60), 351–358 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  53. Hytönen, T., Martikainen, H.: Non-homogeneous \(T1\) theorem for bi-parameter singular integrals. Adv. Math. 261, 220–273 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  54. Jiang, R., Yang, D.: New Orlicz-Hardy spaces associated with divergence form elliptic operators. J. Funct. Anal. 258(4), 1167–1224 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  55. Journé, J.-L.: Calderón-Zygmund operators on product spaces. Rev. Mat. Iberoamericana 1(3), 55–91 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  56. Journé, J.-L.: A covering lemma for product spaces. Proc. Am. Math. Soc. 96(4), 593–598 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  57. Kairema, A., Li, J., Pereyra, M.C., Ward, L.A.: Haar bases on quasi-metric measure spaces, and dyadic structure theorems for function spaces on product spaces of homogeneous type. J. Funct. Anal. 271(7), 1793–1843 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  58. Kerman, R.A.: Boundedness criteria for generalized Hankel conjugate transformations. Can. J. Math. 30(1), 147–153 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  59. Lacey, M.T.: Lectures on Nehari’s theorem on the polydisk. In: Topics in harmonic analysis and ergodic theory, volume 444 of Contemp. Math., pp. 185–213. Am. Math. Soc., Providence, RI (2007)

  60. Lacey, M.T., Petermichl, S., Pipher, J.C., Wick, B.D.: Multiparameter Riesz commutators. Am. J. Math. 131(3), 731–769 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  61. Lacey, M.T., Petermichl, S., Pipher, J.C., Wick, B.D.: Iterated Riesz commutators: a simple proof of boundedness. In: Harmonic analysis and partial differential equations, volume 505 of Contemp. Math., pp. 171–178. Amer. Math. Soc., Providence, RI (2010)

  62. Lacey, M.T., Terwilleger, E.: Hankel operators in several complex variables and product BMO. Houston J. Math. 35(1), 159–183 (2009)

    MathSciNet  MATH  Google Scholar 

  63. Li, B., Bownik, M., Yang, D.: Littlewood-Paley characterization and duality of weighted anisotropic product Hardy spaces. J. Funct. Anal. 266(5), 2611–2661 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  64. Li, B., Bownik, M., Yang, D., Zhou, Y.: Anisotropic singular integrals in product spaces. Sci. China Math. 53(12), 3163–3178 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  65. Li, J., Pipher, J., Ward, L.A.: Dyadic structure theorems for multiparameter function spaces. Rev. Mat. Iberoam. 31(3), 767–797 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  66. Li, J., Ward, L.A.: Singular integrals on Carleson measure spaces \({\text{ CMO }}^p\) on product spaces of homogeneous type. Proc. Am. Math. Soc. 141(8), 2767–2782 (2013)

    Article  MATH  Google Scholar 

  67. Merryfield, K.G.: On the area integral, Carleson measures and \(H^p\) in the polydisc. Indiana Univ. Math. J. 34(3), 663–685 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  68. Muckenhoupt, B., Stein, E.M.: Classical expansions and their relation to conjugate harmonic functions. Trans. Am. Math. Soc. 118, 17–92 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  69. Ortega-Cerdà, J., Seip, K.: A lower bound in Nehari’s theorem on the polydisc. J. Anal. Math. 118(1), 339–342 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  70. Ou, Y.: A \(T(b)\) theorem on product spaces. Trans. Am. Math. Soc. 367(9), 6159–6197 (2015)

    Article  MATH  Google Scholar 

  71. Ou, Y., Petermichl, S., Strouse, E.: Higher order Journé commutators and characterizations of multi-parameter BMO. Adv. Math. 291, 24–58 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  72. Pipher, J.: Journé’s covering lemma and its extension to higher dimensions. Duke Math. J. 53(3), 683–690 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  73. Song, L., Tan, C., Yan, L.: An atomic decomposition for Hardy spaces associated to Schrödinger operators. J. Aust. Math. Soc. 91(1), 125–144 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  74. 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  MathSciNet  MATH  Google Scholar 

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

  76. Treil, S.: \(H^1\) and dyadic \(H^1\). In: Linear and Complex Analysis, volume 226 of Amer. Math. Soc. Transl. Ser. 2, pp. 179–193. Am. Math. Soc., Providence, RI (2009)

  77. Villani, M.: Riesz transforms associated to Bessel operators. Illinois J. Math. 52(1), 77–89 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  78. Weinstein, A.: Discontinuous integrals and generalized potential theory. Trans. Am. Math. Soc. 63, 342–354 (1948)

    Article  MathSciNet  MATH  Google Scholar 

  79. Wu, H., Yang, D., Zhang, J.: Oscillation and variation for semigroups associated with Bessel operators. J. Math. Anal. Appl. 443(2), 848–867 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  80. Yang, D., Yang, D.: Real-variable characterizations of Hardy spaces associated with Bessel operators. Anal. Appl. (Singap.) 9(3), 345–368 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  81. Yang, D., Zhou, Y.: Radial maximal function characterizations of Hardy spaces on RD-spaces and their applications. Math. Ann. 346(2), 307–333 (2010)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the referee for careful reading and helpful suggestions. X. T. Duong and J. Li are supported by ARC DP 160100153 and Macquarie University Research Seeding Grant. B. D. Wick’s research supported in part by National Science Foundation DMS grants #1603246 and #1560955. D. Yang is supported by the NNSF of China (Grant Nos. 11971402,11871254,11571289).

Author information

Authors and Affiliations

  1. Department of Mathematics & Statistics, Macquarie University, Sydney, NSW, 2019, Australia

    Xuan Thinh Duong & Ji Li

  2. Department of Mathematics & Statistics, Washington University–St. Louis, St. Louis, MO, 63130-4899, USA

    Brett D. Wick

  3. School of Mathematical Sciences, Xiamen University, Xiamen, 361005, China

    Dongyong Yang

Authors
  1. Xuan Thinh Duong
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ji Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Brett D. Wick
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Dongyong Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dongyong Yang.

Additional information

Communicated by Dachun Yang.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Duong, X.T., Li, J., Wick, B.D. et al. Characterizations of Product Hardy Spaces in Bessel Setting. J Fourier Anal Appl 27, 24 (2021). https://doi.org/10.1007/s00041-021-09823-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00041-021-09823-4

Keywords

Mathematics Subject Classification

Search

Navigation