Glasnik Matematicki, Vol. 53, No. 1 (2018), 143-151.

PROPER INCLUSIONS OF MORREY SPACES

Hendra Gunawan, Denny Ivanal Hakim and Mochammad Idris

Department of Mathematics, Bandung Institute of Technology, Bandung 40132, Indonesia
e-mail: hgunawan@math.itb.ac.id
e-mail: dennyivanalhakim@gmail.com
e-mail: idemath@gmail.com

Abstract.   In this paper, we prove that the inclusions between Morrey spaces, between weak Morrey spaces, and between a Morrey space and a weak Morrey space are all proper. The proper inclusion between a Morrey space and a weak Morrey space is established via the unboundedness of the Hardy-Littlewood maximal operator on Morrey spaces of exponent 1. In addition, we also give a necessary condition for each inclusion. Our results refine previous inclusion properties studied in [4].

2010 Mathematics Subject Classification.   42B35, 46E30.

Key words and phrases.   Morrey spaces, weak Morrey spaces, inclusion properties.


Full text (PDF) (free access)

DOI: 10.3336/gm.53.1.10


References:

  1. F. Chiarenza and M. Frasca, Morrey spaces and Hardy-Littlewood maximal function, Rend. Mat. Appl. (7) 7 (1987), 273-279.
    MathSciNet    

  2. H. Gunawan, A note on the generalized fractional integral operators, J. Indones. Math. Soc. 9 (2003), 39-43.
    MathSciNet    

  3. H. Gunawan, D. I. Hakim, Y. Sawano and I. Sihwaningrum, Weak type inequalities for some integral operators on generalized nonhomogeneous Morrey spaces, J. Funct. Spaces Appl. 2013, Art. ID 809704, 12 pp.
    MathSciNet    

  4. H. Gunawan, D. I. Hakim, K. M. Limanta and A. A. Masta, Inclusion properties of generalized Morrey spaces, Math. Nachr. 290 (2017), 332-340.
    MathSciNet     CrossRef

  5. D. I. Hakim and H. Gunawan, Weak (p,q) inequalities for fractional integral operators on generalized Morrey spaces of non-homogeneous type, Math. Aeterna 3 (2013), 161-168.
    MathSciNet    

  6. D. D. Haroske and L. Skrzypczak, Embeddings of weighted Morrey spaces, Math. Nachr. 290 (2017), 1066-1086.
    MathSciNet     CrossRef

  7. C. B. Morrey, Jr., On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc. 43 (1938), 126-166.
    MathSciNet     CrossRef

  8. E. Nakai, Orlicz-Morrey spaces and the Hardy-Littlewood maximal function, Studia Math. 188 (2008), 193-221.
    MathSciNet     CrossRef

  9. J. Peetre, On the theory of p,Λ spaces, J. Functional Analysis 4 (1969), 71-87.
    MathSciNet    

  10. L. C. Piccinini, Proprietä di Inclusione e Interpolazione tra Spazi di Morrey e loro Generalizzazioni, Tesi di perfezionamento, Scuola Normale Superior Pisa, 1969.

  11. L. C. Piccinini, Inclusioni tra spazi di Morrey, Boll. Un. Mat. Ital. (4) 2 (1969), 95-99.
    MathSciNet    

  12. M. Rosenthal, Morrey-Räume aus der Sicht der Harmonischen Analysis, Master thesis, Friedrich-Schiller-Universität Jena, 2009.

Glasnik Matematicki Home Page