Taiwanese Journal of Mathematics

Generalized Fractional Integral Operators and Their Commutators with Functions in Generalized Campanato Spaces on Orlicz Spaces

Minglei Shi, Ryutaro Arai, and Eiichi Nakai

Full-text: Open access


We investigate the commutators $[b,I_{\rho}]$ of generalized fractional integral operators $I_{\rho}$ with functions $b$ in generalized Campanato spaces and give a necessary and sufficient condition for the boundedness of the commutators on Orlicz spaces. To do this we define Orlicz spaces with generalized Young functions and prove the boundedness of generalized fractional maximal operators on the Orlicz spaces.

Article information

Taiwanese J. Math., Volume 23, Number 6 (2019), 1339-1364.

Received: 23 July 2018
Revised: 15 December 2018
Accepted: 18 December 2018
First available in Project Euclid: 3 January 2019

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46E30: Spaces of measurable functions (Lp-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 42B35: Function spaces arising in harmonic analysis

Orlicz space Campanato space fractional integral commutator


Shi, Minglei; Arai, Ryutaro; Nakai, Eiichi. Generalized Fractional Integral Operators and Their Commutators with Functions in Generalized Campanato Spaces on Orlicz Spaces. Taiwanese J. Math. 23 (2019), no. 6, 1339--1364. doi:10.11650/tjm/181211. https://projecteuclid.org/euclid.twjm/1546506192

Export citation


  • R. Arai and E. Nakai, Commutators of Calderón-Zygmund and generalized fractional integral operators on generalized Morrey spaces, Rev. Mat. Complut. 31 (2018), no. 2, 287–331.
  • S. Chanillo, A note on commutators, Indiana Univ. Math. J. 31 (1982), no. 1, 7–16.
  • A. Cianchi, Strong and weak type inequalities for some classical operators in Orlicz spaces, J. London Math. Soc. (2) 60 (1999), no. 1, 187–202.
  • F. Deringoz, V. S. Guliyev, E. Nakai, Y. Sawano and M. Shi, Generalized fractional maximal and integral operators on Orlicz and generalized Orlicz–Morrey spaces of the third kind, Positivity, Online First. https://link.springer.com/article/10.1007/s11117-018-0635-9 https://arxiv.org/abs/1812.03649
  • D. E. Edmunds, P. Gurka and B. Opic, Double exponential integrability of convolution operators in generalized Lorentz-Zygmund spaces, Indiana Univ. Math. J. 44 (1995), no. 1, 19–43.
  • X. Fu, D. Yang and W. Yuan, Generalized fractional integrals and their commutators over non-homogeneous metric measure spaces, Taiwanese J. Math. 18 (2014), no. 2, 509–557.
  • L. Grafakos, Modern Fourier Analysis, Third edition, Graduate Texts in Mathematics 250, Springer, New York, 2014.
  • V. S. Guliyev, F. Deringoz and S. G. Hasanov, Riesz potential and its commutators on Orlicz spaces, J. Inequal. Appl. 2017 (2017), no. 75, 18 pp.
  • L. I. Hedberg, On certain convolution inequalities, Proc. Amer. Math. Soc. 36 (1972), 505–510.
  • S. Janson, Mean oscillation and commutators of singular integral operators, Ark. Mat. 16 (1978), no. 2, 263–270.
  • R. Kawasumi and E. Nakai, Pointwise multipliers on weak Orlicz spaces, preprint.
  • H. Kita, On maximal functions in Orlicz spaces, Proc. Amer. Math. Soc. 124 (1996), no. 10, 3019–3025.
  • ––––, On Hardy-Littlewood maximal functions in Orlicz spaces, Math. Nachr. 183 (1997), 135–155.
  • ––––, Orlicz spaces and their applications (Japanese), Iwanami Shoten, Publishers, Tokyo, 2009.
  • V. Kokilashvili and M. Krbec, Weighted Inequalities in Lorentz and Orlicz Spaces, World Scientific, River Edge, NJ, 1991.
  • M, A. Krasnoselsky and Y. B. Rutitsky, Convex functions and Orlicz spaces, Translated from the first Russian edition by Leo F. Boron. P. Noordhoff Ltd., Groningen, 1961.
  • L. Maligranda, Orlicz Spaces and Interpolation, Seminars in Mathematics 5, Universidade Estadual de Campinas, Departamento de Matemática, Campinas, 1989.
  • Y. Mizuta, E. Nakai, T. Ohno and T. Shimomura, Boundedness of fractional integral operators on Morrey spaces and Sobolev embeddings for generalized Riesz potentials, J. Math. Soc. Japan 62 (2010), no. 3, 707–744.
  • E. Nakai, On generalized fractional integrals in the Orlicz spaces, in: Proceedings of the Second ISAAC Congress, Vol. 1 (Fukuoka, 1999), 75–81, Int. Soc. Anal. Appl. Comput. 7, Kluwer Acad. Publ., Dordrecht, 2000.
  • ––––, On generalized fractional integrals, Taiwanese J. Math. 5 (2001), no. 3, 587–602.
  • ––––, On generalized fractional integrals in the Orlicz spaces on spaces of homogeneous type, Sci. Math. Jpn. 54 (2001), no. 3, 473–487.
  • ––––, On generalized fractional integrals on the weak Orlicz spaces, $\mathrm{BMO}_{\phi}$, the Morrey spaces and the Campanato spaces, in: Function Spaces, Interpolation Theory and Related Topics (Lund, 2000), 389–401, de Gruyter, Berlin, 2002.
  • ––––, Generalized fractional integrals on Orlicz-Morrey spaces, in: Banach and Function Spaces, 323–333, Yokohama Publishers, Yokohama, 2004.
  • ––––, Orlicz-Morrey spaces and the Hardy-Littlewood maximal function, Studia Math. 188 (2008), no. 3, 193–221.
  • ––––, A generalization of Hardy spaces $H^p$ by using atoms, Acta Math. Sin. (Engl. Ser.) 24 (2008), no. 8, 1243–1268.
  • E. Nakai and H. Sumitomo, On generalized Riesz potentials and spaces of some smooth functions, Sci. Math. Jpn. 54 (2001), no. 3, 463–472.
  • R. O'Neil, Fractional integration in Orlicz spaces I, Trans. Amer. Math. Soc. 115 (1965), 300–328.
  • W. Orlicz, Über eine gewisse Klasse von Räumen vom Typus B, Bull. Acad. Polonaise A (1932), 207–220; reprinted in his Collected Papers, PWN, Warszawa 1988, 217–230.
  • ––––, Über Räume $(L^M)$, Bull. Acad. Polonaise A (1936), 93–107; reprinted in his Collected Papers, PWN, Warszawa 1988, 345–359.
  • M. M. Rao and Z. D. Ren, Theory of Orlicz Spaces, Monographs and Textbooks in Pure and Applied Mathematics 146, Marcel Dekker, New York, 1991.
  • Y. Sawano, S. Sugano and H. Tanaka, Generalized fractional integral operators and fractional maximal operators in the framework of Morrey spaces, Trans. Amer. Math. Soc. 363 (2011), no. 12, 6481–6503.
  • R. S. Strichartz, A note on Trudinger's extension of Sobolev's inequalities, Indiana Univ. Math. J. 21 (1972), 841–842.
  • A. Torchinsky, Interpolation of operations and Orlicz classes, Studia Math. 59 (1976), no. 2, 177–207.
  • N. S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech. 17 (1967), 473–483.
  • G. Weiss, A note on Orlicz spaces, Portugal. Math. 15 (1956), 35–47.