Generalized fractional maximal operator on generalized local Morrey spaces

Abstract

In this paper, we study the boundedness of generalized fractional maximal operator  on generalized local Morrey spaces and

generalized Morrey spaces, including weak estimates. Firstly, we prove the Spanne type boundedness of generalized fractional maximal operator  on generalized local Morrey spaces for 1 < p < q < infinity and, on weak generalized local Morrey spaces for p = 1 and 1 < q < infinity. Secondly, we prove the Adams type boundedness of generalized fractional maximal operator  on generalized local Morrey spaces for 1<p<q< \infinity   

and, on  weak generalized local Morrey spaces for p = 1 and 1 < q < \infinity. In all cases the conditions for the boundedness of generalized fractional maximal operators are given in terms of supremal-type integral inequalities on (\varphi_1; \varphi_2; \rho) and (\varphi; \rho), which do not assume any assumption on monotonicity of \varphi_1(x; r), \varphi_2(x; r) and \varphi(x; r) in r.

Keywords

• Generalized fractional maximal operator,generalized local Morrey spaces,generalized Morrey spaces

References

1. Adams, D.R., A note on Riesz potentials, Duke Math. 42 (1975) 765-778.
2. Akbulut, A., Guliyev, V.S. and Mustafayev, R., On the boundedness of the maximal operator and singular integral operators in generalized Morrey spaces, Math. Bohem. 137 (1) (2012), 27-43.
3. Burenkov, V., Guliyev, H.V. and Guliyev, V.S., Necessary and sufficient conditions for boundedness of the Riesz operator in the local Morrey-type spaces, Doklady Ross. Akad. Nauk. Matematika 412 (5) (2007), 585-589 (Russian). English trans. in Acad. Sci. Dokl. Math. 76 (2007).
4. Burenkov, V. and Guliyev, V.S., Necessary and sufficient conditions for the boundedness of the Riesz operator in local Morrey-type spaces, Potential Anal. 30 (2009), no. 3, 211-249.
5. Burenkov, V., Guliyev, H.V. and Guliyev, V.S. Necessary and sufficient conditions for boundedness of the fractional maximal operator in the local Morrey-type spaces, Doklady Ross. Akad. Nauk. Matematika 409 (4) (2006), 443-447 (Russian). English trans. in Acad. Sci. Dokl. Math. 74 (2006).
6. Burenkov, V., Guliyev, H.V. and Guliyev, V.S., Necessary and sufficient conditions for boundedness of the fractional maximal operator in the local Morrey-type spaces, J. Comput. Appl. Math. 208 (2007), no. 1, 280-301.
7. Burenkov, V., Gogatishvili, A., Guliyev, V. and Mustafayev, R.Ch., Boundedness of the Riesz potential in local Morrey-type spaces, Potential Anal. 35 (2011), no. 1, 67-87.
8. Burenkov, V., Gogatishvili, A., Guliyev, V.S. and Mustafayev, R., Boundedness of the fractional maximal operator in local Morrey-type spaces, Complex Var. Elliptic Equ. 55 (8-10) (2010), 739-758.

9. Chiarenza, F. and Frasca, M., Morrey spaces and Hardy-Littlewood maximal function, Rend Mat. 7 (1987), 273-279.
10. Eridani, A., On the boundedness of a generalized fractional integral on generalized Morrey spaces, Tamkang J. Math. 33 (2002), no. 4, 335-340.
11. Eridani, A., Gunawan, H., Nakai, E. and Sawano, Y., Characterizations for the generalized fractional integral operators on Morrey spaces, Math. Inequal. Appl. 17 (2014), no. 2, 761-777.
12. Eridani, A., Gunawan, H. and Nakai, E., On generalized fractional integral operators, Sci. Math. Jpn. 60 (2004), no. 3, 539-550.
13. Guliyev, V.S., Integral operators on function spaces on the homogeneous groups and on domains in Rⁿ. Doctoral degree dissertation, Mat. Inst. Steklov, Moscow, 1994, 329 pp. (in Russian)
14. Guliyev, V.S., Boundedness of the maximal, potential and singular operators in the generalized Morrey spaces, J. Inequal. Appl. 2009, Art. ID 503948, 20 pp.
15. Guliyev, V.S., Generalized local Morrey spaces and fractional integral operators with rough kernel, J. Math. Sci. (N.Y.) 193 (2) (2013), 211-227.
16. Guliyev, V.S. and Shukurov, P.S., On the boundedness of the fractional maximal operator, Riesz potential and their commutators in generalized Morrey spaces. Advances in Harmonic Analysis and Operator Theory, Series: Operator Theory: Advances and Applications, 229 (2013), 175-199.
17. Guliyev, V.S., Ismayilova, A.F., Kucukaslan, A. and Serbetci, A., Generalized fractional integral operators on generalized local Morrey spaces, Journal of Function Spaces, Volume 2015, Article ID 594323, 8 pages.
18. Gunawan, H., A note on the generalized fractional integral operators, J. Indones. Math. Soc. 9 (2003), no. 1, 39-43.
19. Kucukaslan, A., Hasanov, S.G. and Aykol, C., Generalized fractional integral operators on vanishing generalized local Morrey spaces, International Journal of Mathematical Analysis, 11, 2017, no. 6, 277 -291.
20. Mizuhara, T., Boundedness of some classical operators on generalized Morrey spaces, Harmonic Analysis (S. Igari, Editor), ICM 90 Satellite Proceedings, Springer - Verlag, Tokyo (1991) 183-189.
21. Morrey, C.B., On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc. 43 (1938) 126-166.
22. Nakai, E., Hardy--Littlewood maximal operator, singular integral operators and Riesz potentials on generalized Morrey spaces, Math. Nachr. 166 (1994) 95-103.
23. Nakai, E., On generalized fractional integrals, Taiwanese J. Math. 5 (2001), 587-602.
24. Peetre, J., On the theory of M_{p,λ}, J. Funct. Anal. 4 (1969) 71-87.
25. Sawano, Y., Sugano, S. and Tanaka, H., Generalized fractional integral operators and fractional maximal operators in the framework of Morrey spaces, Trans. Amer. Math. Soc. 363 (2011), no. 12, 6481-6503.
26. Sawano, Y., Sugano, S. and Tanaka, H., Orlicz-Morrey spaces and fractional operators, Potential Anal., 36 (4) (2012), 517-556.
27. Stein, E.M., Harmonic Analysis: Real Variable Methods, Orthogonality and Oscillatory Integrals, Princeton Univ. Press, Princeton NJ, 1993.