## Generalized fractional integral operators and their commutators with functions in generalized Campanato spaces on Orlicz spaces.(English)Zbl 1491.47038

Summary: 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.

### MSC:

 47G10 Integral operators 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 42B35 Function spaces arising in harmonic analysis 26A33 Fractional derivatives and integrals

### Keywords:

Orlicz space; Campanato space; fractional integral; commutator
Full Text:

### References:

 [1] 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. · Zbl 1391.42013 [2] S. Chanillo, A note on commutators, Indiana Univ. Math. J. 31 (1982), no. 1, 7-16. [3] 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. · Zbl 0940.46015 [4] 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 · Zbl 1440.42076 [5] 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. · Zbl 0826.47021 [6] 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. · Zbl 1357.42016 [7] L. Grafakos, Modern Fourier Analysis, Third edition, Graduate Texts in Mathematics 250, Springer, New York, 2014. · Zbl 1304.42002 [8] 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. · Zbl 1364.31005 [9] L. I. Hedberg, On certain convolution inequalities, Proc. Amer. Math. Soc. 36 (1972), 505-510. · Zbl 0283.26003 [10] S. Janson, Mean oscillation and commutators of singular integral operators, Ark. Mat. 16 (1978), no. 2, 263-270. · Zbl 0404.42013 [11] R. Kawasumi and E. Nakai, Pointwise multipliers on weak Orlicz spaces, preprint. [12] H. Kita, On maximal functions in Orlicz spaces, Proc. Amer. Math. Soc. 124 (1996), no. 10, 3019-3025. · Zbl 0845.42008 [13] —-, On Hardy-Littlewood maximal functions in Orlicz spaces, Math. Nachr. 183 (1997), 135-155. · Zbl 0864.42007 [14] —-, Orlicz spaces and their applications (Japanese), Iwanami Shoten, Publishers, Tokyo, 2009. [15] V. Kokilashvili and M. Krbec, Weighted Inequalities in Lorentz and Orlicz Spaces, World Scientific, River Edge, NJ, 1991. · Zbl 0751.46021 [16] 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. [17] L. Maligranda, Orlicz Spaces and Interpolation, Seminars in Mathematics 5, Universidade Estadual de Campinas, Departamento de Matemática, Campinas, 1989. · Zbl 0874.46022 [18] 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. · Zbl 1200.26007 [19] 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. · Zbl 1058.26003 [20] —-, On generalized fractional integrals, Taiwanese J. Math. 5 (2001), no. 3, 587-602. · Zbl 0990.26007 [21] —-, On generalized fractional integrals in the Orlicz spaces on spaces of homogeneous type, Sci. Math. Jpn. 54 (2001), no. 3, 473-487. · Zbl 1007.42013 [22] —-, On generalized fractional integrals on the weak Orlicz spaces, $$\text{BMO}_{\phi}$$, the Morrey spaces and the Campanato spaces, in: Function Spaces, Interpolation Theory and Related Topics (Lund, 2000), 389-401, de Gruyter, Berlin, 2002. [23] —-, Generalized fractional integrals on Orlicz-Morrey spaces, in: Banach and Function Spaces, 323-333, Yokohama Publishers, Yokohama, 2004. · Zbl 1118.42005 [24] —-, Orlicz-Morrey spaces and the Hardy-Littlewood maximal function, Studia Math. 188 (2008), no. 3, 193-221. · Zbl 1163.46020 [25] —-, A generalization of Hardy spaces $$H^p$$ by using atoms, Acta Math. Sin. (Engl. Ser.) 24 (2008), no. 8, 1243-1268. · Zbl 1153.42011 [26] E. Nakai and H. Sumitomo, On generalized Riesz potentials and spaces of some smooth functions, Sci. Math. Jpn. 54 (2001), no. 3, 463-472. · Zbl 0995.43004 [27] R. O’Neil, Fractional integration in Orlicz spaces I, Trans. Amer. Math. Soc. 115 (1965), 300-328. [28] 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. [29] —-, Über Räume $$(L^M)$$, Bull. Acad. Polonaise A (1936), 93-107; reprinted in his Collected Papers, PWN, Warszawa 1988, 345-359. [30] 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. [31] 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. · Zbl 1229.42024 [32] R. S. Strichartz, A note on Trudinger’s extension of Sobolev’s inequalities, Indiana Univ. Math. J. 21 (1972), 841-842. · Zbl 0241.46028 [33] A. Torchinsky, Interpolation of operations and Orlicz classes, Studia Math. 59 (1976), no. 2, 177-207. · Zbl 0348.46027 [34] N. S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech. 17 (1967), 473-483. · Zbl 0163.36402 [35] G. Weiss, A note on Orlicz spaces, Portugal. Math. 15 (1956), 35-47. · Zbl 0071.33001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.