Tokyo Journal of Mathematics

Maximal Operator and its Commutators on Generalized Weighted Orlicz-Morrey Spaces

Fatih DERINGOZ, Vagif S. GULIYEV, and Sabir G. HASANOV

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


In the present paper, we shall give necessary and sufficient conditions for the boundedness of the Hardy-Littlewood maximal operator and its commutators on generalized weighted Orlicz-Morrey spaces $M^{\Phi,\varphi}_{w}({\mathbb R}^n)$. The main advance in comparison with the existing results is that we manage to obtain conditions for the boundedness not in integral terms but in less restrictive terms of supremal operators and we do not need $\Delta_2$-condition for the boundedness of the maximal operator. We also consider the vector-valued boundedness of the Hardy-Littlewood maximal operator.

Article information

Tokyo J. Math., Volume 41, Number 2 (2018), 347-369.

First available in Project Euclid: 18 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.)
Secondary: 42B25: Maximal functions, Littlewood-Paley theory 42B35: Function spaces arising in harmonic analysis 46E30: Spaces of measurable functions (Lp-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)


DERINGOZ, Fatih; GULIYEV, Vagif S.; HASANOV, Sabir G. Maximal Operator and its Commutators on Generalized Weighted Orlicz-Morrey Spaces. Tokyo J. Math. 41 (2018), no. 2, 347--369. doi:10.3836/tjm/1502179260.

Export citation


  • M. Agcayazi, A. Gogatishvili, K. Koca and R. Mustafayev, A note on maximal commutators and commutators of maximal functions, J. Math. Soc. Japan. 67 (2) (2015), 581–593.
  • A. Akbulut, V. S. Guliyev and R. Mustafayev, On the boundedness of the maximal operator and singular integral operators in generalized Morrey spaces, Math. Bohem. 137 (1) (2012), 27–43.
  • A. M. Alphonse, An end point estimate for maximal commutators, J. Fourier Anal. Appl. 6 (4) (2000), 449–456.
  • K. Andersen and R. John, Weighted inequalities for vector-valued maximal functions and singular integrals, Studia Math. 69 (1980), 19–31.
  • V. Burenkov, A. Gogatishvili, V. S. Guliyev and R. Mustafayev, Boundedness of the fractional maximal operator in local Morrey-type spaces, Complex Var. Elliptic Equ. 55 (8–10) (2010), 739–758.
  • D. Cruz-Uribe, J. Martell and C. Perez, Weights, extrapolation and the theory of Rubio de Francia. Operator Theory: Advances and Applications, 215. Birkhauser/Springer Basel, 2011.
  • 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 (2014), 139–158.
  • J. Duoandikoetxea, Fourier analysis. Translated and revised from the 1995 Spanish original by David Cruz-Uribe. Graduate Studies in Mathematics, 29. American Mathematical Society, Providence, 2001.
  • C. Fefferman and E. Stein, Some maximal inequalities, Amer. J. Math. 93 (1971), 107–115.
  • S. Gala, Y. Sawano and H. Tanaka, A remark on two generalized Orlicz-Morrey spaces, J. Approx. Theory 98 (2015), 1–9.
  • J. Garcia-Cuerva, E. Harboure, C. Segovia and J. L. Torrea, Weighted norm inequalities for commutators of strongly singular integrals, Indiana Univ. Math. J. 40 (4) (1991), 1397–1420.
  • I. Genebashvili, A. Gogatishvili, V. Kokilashvili and M. Krbec, Weight theory for integral transforms on spaces of homogeneous type, Longman, Harlow, 1998.
  • A. Gogatishvili and V. Kokilashvili, Criteria of weighted inequalities in Orlicz classes for maximal functions defined on homogeneous type spaces, Georgian Math. J. 1 (6) (1994), 641–673.
  • 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).
  • V. S. Guliyev, Generalized weighted Morrey spaces and higher order commutators of sublinear operators, Eurasian Math. J. 3 (3) (2012), 33–61.
  • V. S. Guliyev, Generalized local Morrey spaces and fractional integral operators with rough kernel, J. Math. Sci. (N. Y.) 193 (2) (2013), 211–227.
  • V. S. Guliyev, F. Deringoz and J. J. Hasanov, $\Phi$-admissible singular operators and their commutators on vanishing generalized Orlicz-Morrey spaces, J. Inequal. Appl. (2014), 2014:143, 18 pp.
  • V. S. Guliyev, M. N. Omarova and Y. Sawano, Boundedness of intrinsic square functions and their commutators on generalized weighted Orlicz-Morrey spaces, Banach J. Math. Anal. 9 (2) (2015), 44–62.
  • 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 (2016), 133–174.
  • D. I. Hakim, E. Nakai and Y. Sawano, Generalized fractional maximal operators and vector-valued inequalities on generalized Orlicz-Morrey spaces, Rev. Mat. Complut. 29 (1) (2016), 59–90.
  • K.-P. Ho, Littlewood-Paley spaces, Math. Scand. 108 (2011), 77–102.
  • K.-P. Ho, Wavelet bases in Littlewood-Paley spaces, East J. Approx. 17 (2011), 333–345.
  • K.-P. Ho, Characterizations of BMO by $A_p$ weights and $p$-convexity, Hiroshima Math. J. 41 (2) (2011), 153–165.
  • K.-P. Ho, Vector-valued maximal inequalities on weighted Orlicz-Morrey spaces, Tokyo J. Math. 36 (2) (2013), 499–512.
  • R. A. Kerman and A. Torchinsky, Integral inequalities with weights for the Hardy maximal function, Studia Math. 71 (3) (1981/82), 277–284.
  • J. Kokilashvilli and M. Krbec, Weighted Inequalities in Lorentz and Orlicz spaces, World Scientific, 1991.
  • Y. Komori and S. Shirai, Weighted Morrey spaces and a singular integral operator, Math. Nachr. 282 (2) (2009), 219–231.
  • D. Li, G. Hu and X. Shi, Weighted norm inequalities for the maximal commutators of singular integral operators, J. Math. Anal. Appl. 319 (3) (2006), 509–521.
  • 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) (2014), 221–268.
  • A. Mazzucato, Decomposition of Besov-Morrey spaces, Contemp. Math. 320 (2003), Amer. Math. Soc. 279–294.
  • T. Mizuhara, Boundedness of some classical operators on generalized Morrey spaces, Harmonic Analysis (S. Igari, Editor), ICM 90 Satellite Proceedings, Springer - Verlag, Tokyo (1991), 183–189.
  • C. B. Morrey, On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc. 43 (1938), 126–166.
  • E. Nakai, Hardy–Littlewood maximal operator, singular integral operators and Riesz potentials on generalized Morrey spaces, Math. Nachr. 166 (1994), 95–103.
  • E. Nakai, Generalized fractional integrals on Orlicz-Morrey spaces, In: Banach and Function Spaces. (Kitakyushu, 2003), Yokohama Publishers, Yokohama, 323–333 (2004).
  • W. Orlicz, Über eine gewisse Klasse von Räumen vom Typus B, Bull. Acad. Polon. A (1932), 207–220. ; reprinted in: Collected Papers, PWN, Warszawa (1988), 217–230.
  • W. Orlicz, Über Räume ($L^M$), Bull. Acad. Polon. A (1936), 93–107. ; reprinted in: Collected Papers, PWN, Warszawa (1988), 345–359.
  • J. Poelhuis and A. Torchinsky, Weighted local estimates for singular integral operators, Trans. Amer. Math. Soc. 367 (11) (2015), 7957–7998.
  • M. M. Rao and Z. D. Ren, Theory of Orlicz Spaces, M. Dekker, Inc., New York, 1991.
  • Y. Sawano and H. Tanaka, Morrey spaces for non-doubling measures. Acta Math. Sin. (Engl. Ser.) 21 (6) (2005), 1535–1544.
  • Y. Sawano and H. Tanaka, Decompositions of Besov-Morrey spaces and Triebel-Lizorkin-Morrey spaces, Math. Z. 257 (2007), 871–905.
  • Y. Sawano, Generalized Morrey spaces for non-doubling measures, NoDEA Nonlinear Differential Equations Appl. 15 (2008), 413–425.
  • Y. Sawano, S. Sugano and H. Tanaka, Orlicz-Morrey spaces and fractional operators, Potential Anal. 36 (4) (2012), 517–556.
  • C. Segovia and J. L. Torrea, Weighted inequalities for commutators of fractional and singular integrals, Publ. Mat. 35 (1) (1991), 209–235. Conference on Mathematical Analysis (El Escorial, 1989)
  • C. Segovia and J. L. Torrea, Higher order commutators for vector-valued Calderón-Zygmund operators, Trans. Amer. Math. Soc. 336 (2) (1993), 537–556.
  • L. Tang and J. Xu, Some properties of Morrey type Besov-Triebel spaces, Math. Nachr. 278 (2005), 904–917.
  • H. Wang, Decomposition for Morrey type Besov-Triebel spaces, Math. Nachr. 282 (2009), 774–787.