Boundedness of the maximal operator and the Riesz potential on Musielak-Orlicz-Morrey spaces

Abstract

In this paper, we investigate the strong and weak type boundedness of the maximal operator in Musielak-Orlicz-Morrey spaces. As an application of this boundedness, we give a sufficient condition for the strong and weak Adams type boundedness of the Riesz potential in these spaces.

Keywords

• Musielak-Orlicz-Morrey spaces
• Hardy-Littlewood maximal operator
• Riesz potential

References

1. [1] D.R. Adams, Morrey Spaces, in: Lecture Notes in Applied and Numerical Harmonic Analysis, Birkhäuser, 2015.
2. [2] F. Deringoz, V.S. Guliyev and 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, 139–158, 2014.
3. [3] F. Deringoz, V.S. Guliyev and S.G. Hasanov, Maximal operator and its commutators on generalized weighted Orlicz-Morrey spaces, Tokyo J. Math. 41 (2), 347369, 2018.
4. [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 23 (3), 727757, 2019.
5. [5] F. Deringoz, V.S. Guliyev, M.N Omarova and M.A. Ragusa, Calderón-Zygmund operators and their commutators on generalized weighted Orlicz-Morrey spaces, Bull. Math. Sci. 13 (1), 26 pp, 2023.
6. [6] S. Gala, Y. Sawano and H. Tanaka, A remark on two generalized Orlicz-Morrey spaces, J. Approx. Theory 98, 1-9, 2015.
7. [7] V.S. Guliyev, Boundedness of the maximal, potential and singular operators in the generalized Morrey spaces, J. Inequal. Appl. Art. ID 503948, 20 pp, 2009.
8. [8] V.S. Guliyev and S. Samko, Maximal, potential, and singular operators in the generalized variable exponent Morrey spaces on unbounded sets, J. Math. Sci. (N.Y.) 193 (2), 228248, 2013.

9. [9] V.S. Guliyev and F. Deringoz, On the Riesz potential and its commutators on generalized Orlicz-Morrey spaces, J. Funct. Spaces Art. ID 617414, 11 pp, 2014.
10. [10] V.S. Guliyev, S.G. Hasanov, Y. Sawano and T. Noi, Non-smooth atomic decompositions for generalized Orlicz-Morrey spaces of the third kind, Acta Appl. Math. 145, 133-174, 2016.
11. [11] V.S. Guliyev and F. Deringoz, Riesz potential and its commutators on generalized weighted Orlicz-Morrey spaces, Math. Nachr. 295 (4), 706724, 2022.
12. [12] V.S. Guliyev, M.N Omarova and M.A. Ragusa, Characterizations for the genuine Calder’on-Zygmund operators and commutators on generalized Orlicz-Morrey spaces, Adv. Nonlinear Anal. 12 (1), 16 pp, 2023.
13. [13] P. Harjulehto and P. Hästö, Orlicz spaces and generalized Orlicz spaces, Lecture Notes in Mathematics 2236, Springer, Cham, 2019.
14. [14] P.A. Hästö, The maximal operator on generalized Orlicz spaces, J. Funct. Anal. 269 (12), 4038-4048, 2015.
15. [15] L.I. Hedberg, On certain convolution inequalities, Proc. Amer. Math. Soc. 36, 505- 510, 1972.
16. [16] A. Karppinen, Fractional operators and their commutators on generalized Orlicz spaces, Opuscula Math. 42 (4), 583-604, 2022.
17. [17] Y. Liang, E. Nakai, D. Yang and J. Zhang, Boundedness of intrinsic Littlewood-Paley functions on Musielak-Orlicz Morrey and Campanato spaces, Banach J. Math. Anal. 8 (1), 221-268, 2014.
18. [18] F. Maeda, Y. Mizuta, T. Ohno and T. Shimomura, Boundedness of maximal operators and Sobolevs inequality on Musielak-Orlicz-Morrey spaces, Bull. Sci. Math. 137, 76- 96, 2013.
19. [19] C.B. Morrey, On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc. 43, 126-166, 1938.
20. [20] J. Musielak, Orlicz spaces and modular spaces, Lecture Notes in Mathematics 1034, Springer, Berlin, 1983.
21. [21] E. Nakai, Generalized fractional integrals on Orlicz-Morrey spaces, In: Banach and Function Spaces, (Kitakyushu, 2003), Yokohama Publishers, Yokohama, 323-333, 2004.
22. [22] E. Nakai, Pointwise multipliers on Musielak-Orlicz-Morrey spaces, Function spaces and inequalities, 257281, Springer Proc. Math. Stat., 206, Springer, Singapore, 2017.
23. [23] H. Nakano, Modulared Semi-Ordered Linear Spaces, Maruzen Co. Ltd., Tokyo, 1950.
24. [24] H. Nakano, Topology and Linear Topological Spaces, Maruzen Co. Ltd., Tokyo, 1951.
25. [25] A. Nekvinda, Hardy-Littlewood maximal operator on $L^{p(x)}(\mathbb{R}^n)$, Math. Ineq. Appl. 7, 255-266, 2004.
26. [26] W. Orlicz, Über konjugierte Exponentenfolgen, Stud. Math. 3, 200-211, 1931.
27. [27] H. Rafeiro, N. Samko and S. Samko, MorreyCampanato spaces: an overview, in: Operator Theory, Pseudo-Differential Equations and Mathematical Physics, in: Advances and Applications 228, 293323, Springer, Basel, 2013.
28. [28] Y. Sawano, S. Sugano and H. Tanaka, Orlicz-Morrey spaces and fractional operators, Potential Anal. 36 (4), 517-556, 2012.
29. [29] Y. Sawano, G. Di Fazio and D.I. Hakim, Morrey spacesintroduction and applications to integral operators and PDE’s, Vol. I, Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, FL, 2020.
30. [30] Y. Sawano, G. Di Fazio and D.I. Hakim, Morrey spacesintroduction and applications to integral operators and PDE’s, Vol. II, Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, FL, 2020.
31. [31] D. Yang, Y. Liang and L.D. Ky, Real-variable theory of Musielak-Orlicz Hardy spaces, Lecture Notes in Mathematics 2182, Springer, Cham, 2017.