Fractional oscillatory integral operators and their commutators on generalized Orlicz--Morrey spaces of the third kind

A. Eroglu

Eurasian mathematical journal, Tome 8 (2017) no. 3, pp. 36-47.

Résumé

We deal with the generalized Orlicz–Morrey space $M_{\Phi,\varphi}$ of the third kind and consider the boundedness of the oscillatory integral operators and fractional oscillatory integral operators on $M_{\Phi,\varphi}$. Some integral estimates for generalized Orlicz–Morrey spaces of the third kind are also obtained by using weighted Hardy operators. The corresponding commutators generated by BMO-functions are also considered.

Bibliographie

[1] A. Cianchi, “Strong and weak type inequalities for some classical operators in Orlicz spaces”, J. London Math. Soc., 60:1 (1999), 187–202 | DOI | MR | Zbl

[2] S. Chanillo, M. Christ, “Weak (1,1) bounds for oscillatory singular integral”, Duke Math. J., 55 (1987), 141–155 | DOI | MR | Zbl

[3] R. Coifman, R. Rochberg, G. Weiss, “Factorization theorems for Hardy spaces in several variables”, Ann. of Math., 103:2 (1976), 611–635 | DOI | MR | Zbl

[4] F. Deringoz, V. S. Guliyev, S. Samko, “Boundedness of maximal and singular operators on generalized Orlicz–Morrey spaces”, Operator Theory, Operator Algebras and Applications, Series: Operator Theory: Advances and Applications, 242 (2014), 139–158 | DOI | MR | Zbl

[5] X. Fu, D. Yang, W. Yuan, “Boundedness of multilinear commutators of Calderón-Zygmund operators on Orlicz spaces over non-homogeneous spaces”, Taiwanese J. Math., 16 (2012), 2203–2238 | DOI | MR | Zbl

[6] A. Eroglu, “Boundedness of fractional oscillatory integral operators and their commutators on generalized Morrey spaces”, Bound. Value Probl., 2013:70 (2013), 12 pp. | MR

[7] V. S. Guliyev, “Boundedness of the maximal, potential and singular operators in the generalized Morrey spaces”, J. Inequal. Appl., 2009, 503948 | DOI | MR | Zbl

[8] V. S. Guliyev, “Generalized weighted Morrey spaces and higher order commutators of sublinear operators”, Eurasian Math. J., 3:3 (2012), 33–61 | MR | Zbl

[9] V. S. Guliyev, “Generalized local Morrey spaces and fractional integral operators with rough kernel”, J. Math. Sci., 193:2 (2013), 211–227 | DOI | MR | Zbl

[10] V. S. Guliyev, F. Deringoz, “On the Riesz potential and its commutators on generalized Orlicz–Morrey spaces”, J. Funct. Spaces, 2014, 617414, 11 pp. | MR | Zbl

[11] V. S. Guliyev, F. Deringoz, “Boundedness of fractional maximal operator and its commutators on generalized Orlicz–Morrey spaces”, Complex Anal. Oper. Theory, 9:6 (2015), 1249–1267 | DOI | MR | Zbl

[12] V. S. Guliyev, S. G. Hasanov, Y. Sawano, T. Noi, “Non-smooth atomic decompositions for generalized Orlicz–Morrey spaces of the third kind”, Acta Appl. Math., 145 (2016), 133–174 | DOI | MR | Zbl

[13] Janson S., “Mean oscillation and commutators of singular integral operators”, Ark. Mat., 1978, no. 16, 263–270 | DOI | MR | Zbl

[14] D. I. Hakim, E. Nakai, Y. Sawano, “Generalized fractional maximal operators and vector-valued inequalities on generalized Orlicz–Morrey spaces”, Rev. Mat. Complut., 29:1 (2016), 59–90 | DOI | MR | Zbl

[15] J. J. Hasanov, “$\Phi$-admissible sublinear singular operators and generalized Orlicz–Morrey spaces”, J. Funct. Spaces, 2014, 505237, 7 pp. | MR | Zbl

[16] Ho Kwok-Pun, “Characterization of BMO in terms of rearrangement-invariant Banach function spaces”, Expo. Math., 27 (2009), 363–372 | DOI | MR | Zbl

[17] S. Z. Lu, “A class of oscillatory integrals”, Int. J. Appl. Math. Sci., 2:1 (2005), 42–58 | MR

[18] S. Z. Lu, Y. Ding, D. Y. Yan, Singular integrals and related topics, World Sci. Publ., Singapore, 2007 | Zbl

[19] S. Z. Lu, Y. Zhang, “Criterion on Lp-boundedness for a class of oscillatory singular integrals with rough kernels”, Rev. Math. Iberoam., 8 (1992), 201–219 | MR | Zbl

[20] T. Mizuhara, “Boundedness of some classical operators on generalized Morrey spaces”, Harmonic Analysis, ICM 90 Satellite Proceedings, ed. S. Igari, Springer-Verlag, Tokyo, 1991, 183–189 | MR

[21] C. B. Morrey, “On the solutions of quasi-linear elliptic partial differential equations”, Trans. Amer. Math. Soc., 43 (1938), 126–166 | DOI | MR

[22] E. Nakai, “Hardy–Littlewood maximal operator, singular integral operators and Riesz potentials on generalized Morrey spaces”, Math. Nachr., 166 (1994), 95–103 | DOI | MR | Zbl

[23] E. Nakai, “Generalized fractional integrals on Orlicz-Morrey spaces”, Banach and Function Spaces (Kitakyushu, 2003), Yokohama Publishers, Yokohama, 2004, 323–333 | MR | Zbl

[24] E. Nakai, “Orlicz–Morrey spaces and the Hardy–Littlewood maximal function”, Studia Math., 188:3 (2008), 193–221 | DOI | MR | Zbl

[25] D. H. Phong, E. M. Stein, “Singular integrals related to the Radon transform and boundary value problems”, Proc. Nat. Acad. USA, 80 (1983), 7697–7701 | DOI | MR | Zbl

[26] F. Ricci, E. M. Stein, “Harmonic analysis on Nilpotant groups and singular integrals I: Oscillatory Integrals”, J. Funct. Anal., 73 (1987), 179–194 | DOI | MR | Zbl

[27] Y. Sawano, S. Sugano, H. Tanaka, “A note on generalized fractional integral operators on generalized Morrey spaces”, Bound. Value Probl., 2009, 835865, 18 pp. | MR | Zbl

[28] Y. Sawano, S. Sugano, H. Tanaka, “Orlicz–Morrey spaces and fractional operators”, Potential Anal., 36:4 (2012), 517–556 | DOI | MR | Zbl

[29] L. Softova, “Singular integrals and commutators in generalized Morrey spaces”, Acta Math. Sin. (Engl. Ser.), 22:3 (2006), 757–766 | DOI | MR | Zbl