Erdelyi-Kober fractional Integrals on Hardy space and BMO




Fractional integral, Hardy spaces, Bounded mean oscillation, Erdélyi-Kober fractional integrals, Hypergeometric fractional integrals


The mapping properties of the multi. Erdélyi- Kober fractional integral operators on Hardy space and BMO. In particular, our main result gives the boundedness of the Erdélyi-Kober fractional integrals, the hypergeometric fractional integrals and the two-dimensional Weyl integrals on Hardy space and BMO.

Author Biography

Kwok-Pun Ho, The Education University of Hong Kong.

Dept. of Mathematics and Information Technology.


A. Erdélyi, “On fractional integration and its application to the theory of hankel transforms”, The quarterly journal of mathematics, vol. os-11, no. 1, pp. 293–303, Jan. 1940, doi: 10.1093/qmath/os-11.1.293

L. Grafakos, Modern fourier analysis, 2nd ed. New York, NY: Springer, 2009, doi: 10.1007/978-0-387-09434-2

K.-P. Ho, “Atomic decompositions of weighted Hardy-Morrey spaces”, Hokkaido mathematical journal, vol. 42, no. 1, pp. 131–157, Feb. 2013, doi: 10.14492/hokmj/1362406643

K.-P. Ho, “Atomic decomposition of Hardy-Morrey spaces with variable exponents”, Annales academiæ scientiarum fennicæ mathematica, vol. 40, pp. 31-62, 2015, doi: 10.5186/aasfm.2015.4002

K.-P. Ho, “Atomic decompositions and Hardy’s inequality on weak Hardy-Morrey spaces”, Science China Mathematics, vol. 60, no. 3, pp. 449–468, Jan. 2017, doi: 10.1007/s11425-016-0229-1

K.-P. Ho, “Atomic decompositions of weighted Hardy spaces with variable exponents”, Tohoku mathematical journal, vol. 69, no. 3, pp. 383–413, Sep. 2017, doi: 10.2748/tmj/1505181623

K.-P. Ho, “Sublinear operators on weighted Hardy spaces with variable exponents”, Forum mathematicum, vol. 31, no. 3, pp. 607–617, May 2019, doi: 10.1515/forum-2018-0142

K.-P. Ho, “Integral operators on BMO and Campanato spaces”, Indagationes mathematicae, vol. 30, no. 6, pp. 1023–1035, Nov. 2019., doi: 10.1016/j.indag.2019.05.007

K.-P. Ho, “Sublinear operators on mixed-norm Hardy spaces with variable exponents”, Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Rendiconti Lincei Matematica e Applicazioni, 2020. (To appear).

K.-P. Ho, “Erdélyi-Kober fractional integral operators on ball Banach function spaces”, Rendiconti del Seminario matematico della Università di Padova, 2019 (preprint). [On line]. Available:

K.-P. Ho, “Calderón-Zygmund operators, Bochner-Riesz means and Parametric Marcinkiewicz integrals on Hardy-Morrey spaces with variable exponents”, (accepted).

H. Jia and H. Wang, “Decomposition of Hardy–Morrey spaces”, Journal of mathematical analysis and applications, vol. 354, no. 1, pp. 99–110, Jun. 2009, doi: 10.1016/j.jmaa.2008.12.051

V. Kiryakova, Generalized fractional calculus and applications. Harlow: Longman Scientific & Technical, 1994.

H. Kober, “On fractional integrals and derivatives”, The quarterly journal of mathematics, vol. os-11, no. 1, pp. 193–211, Jan. 1940, doi: 10.1093/qmath/os-11.1.193

E. R. Love, “Some integral equations involving hypergeometric functions”, Proceedings of the Edinburgh Mathematical Society, vol. 15, no. 3, pp. 169–198, Jun. 1967, doi: 10.1017/S0013091500011706

S. Lu and D. Yang, “The local versions of ℍp(ℝn) spaces at the origin”, Studia mathematica, vol. 116, no. 2, pp. 103-131, 1995. [On line]. Available:

S. Lu, D. Ang, and G. Hu, Herz type spaces and their applications. Beijing: China Press, 2008.

A. M. Mathai and R. K. Saxena, Generalized hypergeometric functions with applications in statistics and physical sciences, vol. 348. Berlin: Springer, 1973, doi: 10.1007/BFb0060468

E. Nakai and Y. Sawano, “Hardy spaces with variable exponents and generalized Campanato spaces”, Journal of functional analysis, vol. 262, no. 9, pp. 3665–3748, May 2012, doi: 10.1016/j.jfa.2012.01.004

G. Pagnini, “Erdélyi-Kober fractional diffusion”, Fractional calculus and applied analysis, vol. 15, no. 1, Mar. 2012, doi: 10.2478/s13540-012-0008-1

Ł. Płociniczak, “Approximation of the Erdélyi--Kober operator with application to the time-fractional Porous medium equation”, SIAM journal on applied mathematics, vol. 74, no. 4, pp. 1219–1237, 2014, doi: 10.1137/130942450

Ł. Płociniczak and M. Świtała, “Existence and uniqueness results for a time-fractional nonlinear diffusion equation”, Journal of mathematical analysis and applications, vol. 462, no. 2, pp. 1425–1434, Jun. 2018, doi: 10.1016/j.jmaa.2018.02.050

Ł. Płociniczak, “Numerical method for the time-fractional porous medium equation”, SIAM journal on numerical analysis, vol. 57, no. 2, pp. 638–656, 2019, doi: 10.1137/18M1192561

Y. Sawano, K.-P. Ho, D. Yang, and S. Yang, “Hardy spaces for ball quasi-Banach function spaces”, Dissertationes mathematicae, vol. 525, pp. 1–102, Jul. 2017, doi: 10.4064/dm750-9-2016

I. A. Sneddon, “The use in mathematical physics of Erdlyi-Kober operators and of some of their generalizations”, in Fractional calculus and its applications, vol. 457, B. Ross, Ed. Berlin: Springer, 1975, pp. 37-79, doi: 10.1007/BFb0067097

E. M. Stein, Harmonic analysis: Real-variable methods, orthogonality, and oscillatory integrals. Princeton, NJ: Princeton University Press, 1993.



How to Cite

K.-P. Ho, “Erdelyi-Kober fractional Integrals on Hardy space and BMO”, Proyecciones (Antofagasta, On line), vol. 39, no. 3, pp. 663-677, Jun. 2020.