Abstract
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 \(0<p\le 1\).
This is a preview of subscription content, log in via an institution to check access.
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.
References
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)
Betancor, J.J., Castro, A.J., Curbelo, J.: Spectral multipliers for multidimensional Bessel operators. J. Fourier Anal. Appl. 17, 932–975 (2011)
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)
Betancor, J.J., Castro, A.J., Nowak, A.: Calderon-Zygmund operators in the Bessel setting. Monatsh. Math. 167, 375–403 (2012)
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)
Betancor, J.J., Dziubanski, J., Torrea, J.L.: On Hardy spaces associated with Bessel operators. J. Anal. Math. 107, 195–219 (2009)
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)
Betancor, J.J., Fariña, J.C., Sanabria, A.: On Littlewood–Paley functions associated with Bessel operators. Glasg. Math. J. 51, 55–70 (2009)
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)
Coifman, R.R.: A real variable characterization of \(H^p\). Studia Math. 51, 269–274 (1974)
Coifman, R.R., Weiss, G.: Analyse harmonique non-commutative sur certains espaces homog‘enes. Lecture Notes in Math, vol. 242. Springer, Berlin (1971)
Coifman, R.R., Weiss, G.: Extensions of Hardy spaces and their use on analysis. Bull. Am. Math. Soc. 83, 569–645 (1977)
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)
Fefferman, C., Stein, E.M.: Hardy spaces of several variables. Acta Math. 129, 137–193 (1972)
Grafakos, L.: Modern Fourier Analysis. Graduate Texts in Mathematics. Springer, New York (2008)
Guliyev, V.S.: Sobolev theorems for the Riesz \(B\)-potentials. Dokl. RAN. (Russian) 358(4), 450–451 (1998)
Guliyev, V.S.: On maximal function and fractional integral associated with the Bessel differential operator. Math. Ineq. Appl. 6(2), 317–330 (2003)
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)
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)
Kipriyanov, I.A.: Fourier–Bessel transformations and imbedding theorems. Trudy Math. Inst. Steklov 89, 130–213 (1967)
Latter, R.: A characterization of \(H^p({\mathbb{R}}^n)\) in terms of atoms. Studia Math. 62, 93–101 (1978)
Lee, M.Y., Lin, C.C.: The molecular characterization of weighted Hardy spaces. J. Funct. Anal. 188, 442–460 (2002)
Levitan, B.M.: Bessel function expansions in series and Fourier integrals. Uspekhi Mat. Nauk. (Russian) 6(2), 102–143 (1951)
Lyakhov, L.N.: Multipliers of the mixed Fourier–Bessel transform. Proc. Steklov Inst. Math. 214, 234–249 (1997)
Muckenhoupt, B., Stein, E.M.: Classical expansions and their relation to conjugate harmonic functions. Trans. Am. Math. Soc. 118, 17–92 (1965)
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)
Stein, E.M., Weiss, G.: Introduction to Fourier analysis on Euclidean spaces. Princeton Univ. Press, Princeton (1971)
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)
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)
Uyhan, S.: On Poisson semigroup generated by the generalized B-translation. In: Dynamical Systems and Applications Proceedings, pp. 679–686 (2004)
Yang, D., Yang, D.: Real-variable characterizations of Hardy spaces associated with Bessel operators. Anal. Appl. (Singapore) 9(3), 345–368 (2011)
Acknowledgements
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).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interests.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Keskin, C., Ekincioglu, I. & Guliyev, V.S. Characterizations of Hardy spaces associated with Laplace–Bessel operators. Anal.Math.Phys. 9, 2281–2310 (2019). https://doi.org/10.1007/s13324-019-00335-5
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13324-019-00335-5
Keywords
- Atomic decomposition
- Fourier–Bessel transform
- Generalized shift operator
- Hardy space
- Riesz–Bessel transform