Analysis and Mathematical Physics

, Volume 9, Issue 4, pp 2281–2310 | Cite as

Characterizations of Hardy spaces associated with Laplace–Bessel operators

  • Cansu KeskinEmail author
  • Ismail Ekincioglu
  • Vagif S. Guliyev


In this paper, we obtain a characterization of \(H^{p}_{\varDelta _{\nu }}({\mathbb {R}}^{n}_{+})\) Hardy spaces by using atoms associated with the radial maximal function, the nontangential maximal function and the grand maximal function related to \(\varDelta _{\nu }\) Laplace–Bessel operator for \(\nu >0\) and \(1<p<\infty \). As an application, we further establish an atomic characterization of Hardy spaces \(H^{p}_{\varDelta _{\nu }}({\mathbb {R}}^{n}_{+})\) in terms of the high order Riesz–Bessel transform for \(0<p\le 1\).


Atomic decomposition Fourier–Bessel transform Generalized shift operator Hardy space Riesz–Bessel transform 

Mathematics Subject Classification

42B30 42B20 42B10 42B25 42B35 



The research of V.S. Guliyev was partially supported by the Grant of the \(1\hbox {st}\) Azerbaijan–Russia Joint Grant Competition (Agreement Number No. EIF-BGM-4-RFTF-1/201721/01/1) and by the Ministry of Education and Science of the Russian Federation (Agreement Number No. 02.a03.21.0008).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interests.


  1. 1.
    Aliev, I.A.: Riesz transforms generated by a generalized translation operator. Izv. Acad. Nauk Azerbaijan. SSR Ser. Fiz. Tekhn. Mat. Nauk 8 1, 7–13 (1987)Google Scholar
  2. 2.
    Betancor, J.J., Castro, A.J., Curbelo, J.: Spectral multipliers for multidimensional Bessel operators. J. Fourier Anal. Appl. 17, 932–975 (2011)Google Scholar
  3. 3.
    Betancor, J.J., Castro, A.J., Curbelo, J.: Harmonic analysis operators associated with multidimensional Bessel operators. Proc. R. Soc. Edinb. Sect. A 142, 945–974 (2012)Google Scholar
  4. 4.
    Betancor, J.J., Castro, A.J., Nowak, A.: Calderon-Zygmund operators in the Bessel setting. Monatsh. Math. 167, 375–403 (2012)Google Scholar
  5. 5.
    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, 310–326 (2010)Google Scholar
  6. 6.
    Betancor, J.J., Dziubanski, J., Torrea, J.L.: On Hardy spaces associated with Bessel operators. J. Anal. Math. 107, 195–219 (2009)Google Scholar
  7. 7.
    Betancor, J.J., Fariña, J.C., Buraczewski, D., Martínez, T., Torrea, J.L.: Riesz transform related to Bessel operators. Proc. R. Soc. Edinb. Sect. A 137, 701–725 (2007)Google Scholar
  8. 8.
    Betancor, J.J., Fariña, J.C., Sanabria, A.: On Littlewood–Paley functions associated with Bessel operators. Glasg. Math. J. 51, 55–70 (2009)Google Scholar
  9. 9.
    Castro, A.J., Szarek, T.Z.: Calderon–Zygmund operators in the Bessel setting for all possible type indices. Acta Math. Sin. (Engl. Ser.) 30, 637–648 (2014)Google Scholar
  10. 10.
    Coifman, R.R.: A real variable characterization of \(H^p\). Studia Math. 51, 269–274 (1974)Google Scholar
  11. 11.
    Coifman, R.R., Weiss, G.: Analyse harmonique non-commutative sur certains espaces homog‘enes. Lecture Notes in Math, vol. 242. Springer, Berlin (1971)Google Scholar
  12. 12.
    Coifman, R.R., Weiss, G.: Extensions of Hardy spaces and their use on analysis. Bull. Am. Math. Soc. 83, 569–645 (1977)Google Scholar
  13. 13.
    Ekincioglu, I.: The boundedness of high order Riesz–Bessel transformations generated by the generalized shift operator in weighted \(L_{p, w,\gamma }\)-spaces with general weights. Acta Appl. Math. 109, 591–598 (2010)Google Scholar
  14. 14.
    Fefferman, C., Stein, E.M.: Hardy spaces of several variables. Acta Math. 129, 137–193 (1972)Google Scholar
  15. 15.
    Grafakos, L.: Modern Fourier Analysis. Graduate Texts in Mathematics. Springer, New York (2008)Google Scholar
  16. 16.
    Guliyev, V.S.: Sobolev theorems for the Riesz \(B\)-potentials. Dokl. RAN. (Russian) 358(4), 450–451 (1998)Google Scholar
  17. 17.
    Guliyev, V.S.: On maximal function and fractional integral associated with the Bessel differential operator. Math. Ineq. Appl. 6(2), 317–330 (2003)Google Scholar
  18. 18.
    Guliyev, V.S., Hasanov, J.J.: The Sobolev–Morrey type inequality for Riesz potentials, associated with the Laplace–Bessel differential operator. Frac. Calc. Appl. Anal. 9(1), 17–32 (2006)Google Scholar
  19. 19.
    Guliyev, V.S., Serbetci, A., Akbulut, A., Mammadov, Y.Y.: Nikol’skii–Besov and Lizorkin–Triebel spaces constructed on the base of the multidimensional Fourier–Bessel transform. Eurasian Math. J. 2(3), 42–66 (2011)Google Scholar
  20. 20.
    Kipriyanov, I.A.: Fourier–Bessel transformations and imbedding theorems. Trudy Math. Inst. Steklov 89, 130–213 (1967)Google Scholar
  21. 21.
    Latter, R.: A characterization of \(H^p({\mathbb{R}}^n)\) in terms of atoms. Studia Math. 62, 93–101 (1978)Google Scholar
  22. 22.
    Lee, M.Y., Lin, C.C.: The molecular characterization of weighted Hardy spaces. J. Funct. Anal. 188, 442–460 (2002)Google Scholar
  23. 23.
    Levitan, B.M.: Bessel function expansions in series and Fourier integrals. Uspekhi Mat. Nauk. (Russian) 6(2), 102–143 (1951)Google Scholar
  24. 24.
    Lyakhov, L.N.: Multipliers of the mixed Fourier–Bessel transform. Proc. Steklov Inst. Math. 214, 234–249 (1997)Google Scholar
  25. 25.
    Muckenhoupt, B., Stein, E.M.: Classical expansions and their relation to conjugate harmonic functions. Trans. Am. Math. Soc. 118, 17–92 (1965)Google Scholar
  26. 26.
    Stein, E.M., Weiss, G.: On the theory of harmonic functions of several variables. I. The theory of \(H_p\)-spaces. Acta Math. 103, 25–62 (1960)Google Scholar
  27. 27.
    Stein, E.M., Weiss, G.: Introduction to Fourier analysis on Euclidean spaces. Princeton Univ. Press, Princeton (1971)Google Scholar
  28. 28.
    Stein, E.M.: Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals. Princeton Mathematical Series, 43. Monographs in Harmonic Analysis, III. Princeton University Press, Princeton (1993)Google Scholar
  29. 29.
    Stempak, K.: La theorie de Littlewood–Paley pour la transformation de Fourier–Bessel. C. R. Acad. Sci. Paris Ser. I Math. 303, 15–18 (1986)Google Scholar
  30. 30.
    Uyhan, S.: On Poisson semigroup generated by the generalized B-translation. In: Dynamical Systems and Applications Proceedings, pp. 679–686 (2004)Google Scholar
  31. 31.
    Yang, D., Yang, D.: Real-variable characterizations of Hardy spaces associated with Bessel operators. Anal. Appl. (Singapore) 9(3), 345–368 (2011)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Cansu Keskin
    • 1
    Email author
  • Ismail Ekincioglu
    • 1
  • Vagif S. Guliyev
    • 1
    • 2
    • 3
  1. 1.Department of MathematicsKutahya Dumlupınar UniversityKutahyaTurkey
  2. 2.S.M. Nikolskii Institute of Mathematics at RUDN UniversityMoscowRussia
  3. 3.Institute of Mathematics and Mechanics of NASBakuAzerbaijan

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