Czechoslovak Mathematical Journal

, Volume 68, Issue 1, pp 77–94 | Cite as

Two-weighted estimates for generalized fractional maximal operators on non-homogeneous spaces

  • Gladis Pradolini
  • Jorgelina Recchi


Let μ be a nonnegative Borel measure on R d satisfying that μ(Q) ⩽ l(Q)n for every cube Q ⊂ R n , where l(Q) is the side length of the cube Q and 0 < nd.

We study the class of pairs of weights related to the boundedness of radial maximal operators of fractional type associated to a Young function B in the context of non-homogeneous spaces related to the measure μ. Our results include two-weighted norm and weak type inequalities and pointwise estimates. Particularly, we give an improvement of a two-weighted result for certain fractional maximal operator proved in W.Wang, C. Tan, Z. Lou (2012).


non-homogeneous space generalized fractional operator weight 

MSC 2010



Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    A. Bernardis, E. Dalmasso, G. Pradolini: Generalized maximal functions and related operators on weighted Musielak-Orlicz spaces. Ann. Acad. Sci. Fenn., Math. 39 (2014), 23–50.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    A. Bernardis, S. Hartzstein, G. Pradolini: Weighted inequalities for commutators of fractional integrals on spaces of homogeneous type. J. Math. Anal. Appl. 322 (2006), 825–846.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    A. L. Bernardis, M. Lorente, M. S. Riveros: Weighted inequalities for fractional integral operators with kernel satisfying Hörmander type conditions. Math. Inequal. Appl. 14 (2011), 881–895.MathSciNetzbMATHGoogle Scholar
  4. [4]
    A. L. Bernardis, G. Pradolini, M. Lorente, M. S. Riveros: Composition of fractional Orlicz maximal operators and A 1-weights on spaces of homogeneous type. Acta Math. Sin., Engl. Ser. 26 (2010), 1509–1518.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    D. Cruz-Uribe, A. Fiorenza: The A property for Young functions and weighted norm inequalities. Houston J. Math. 28 (2002), 169–182.MathSciNetzbMATHGoogle Scholar
  6. [6]
    D. Cruz-Uribe, A. Fiorenza: Endpoint estimates and weighted norm inequalities for commutators of fractional integrals. Publ. Mat., Barc. 47 (2003), 103–131.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    D. Cruz-Uribe, C.Pérez: On the two-weight problem for singular integral operators. Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 1 (2002), 821–849.MathSciNetzbMATHGoogle Scholar
  8. [8]
    J. García-Cuerva, J. M. Martell: Two-weight norm inequalities for maximal operators and fractional integrals on non-homogeneous spaces. Indiana Univ. Math. J. 50 (2001), 1241–1280.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    O. Gorosito, G. Pradolini, O. Salinas: Weighted weak-type estimates for multilinear commutators of fractional integrals on spaces of homogeneous type. Acta Math. Sin., Engl. Ser. 23 (2007), 1813–1826.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    O. Gorosito, G. Pradolini, O. Salinas: Boundedness of the fractional maximal operator on variable exponent Lebesgue spaces: a short proof. Rev. Unión Mat. Argent. 53 (2012), 25–27.MathSciNetzbMATHGoogle Scholar
  11. [11]
    G. H. Hardy, J. E. Littlewood, G. Pólya: Inequalities. Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1988.Google Scholar
  12. [12]
    M. Lorente, J. M. Martell, M. S. Riveros, A. de la Torre: Generalized Hörmander’s conditions, commutators and weights. J. Math. Anal. Appl. 342 (2008), 1399–1425.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    M. Lorente, M. S. Riveros, A. de la Torre: Weighted estimates for singular integral operators satisfying Hörmander’s conditions of Young type. J. Fourier Anal. Appl. 11 (2005), 497–509.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    J. Mateu, P. Mattila, A. Nicolau, J. Orobitg: BMO for nondoubling measures. Duke Math. J. 102 (2000), 533–565.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    Y. Meng, D. Yang: Boundedness of commutators with Lipschitz functions in nonhomogeneous spaces. Taiwanese J. Math. 10 (2006), 1443–1464.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    F. Nazarov, S. Treil, A. Volberg: Cauchy integral and Calderón-Zygmund operators on nonhomogeneous spaces. Int. Math. Res. Not. 1997 (1997), 703–726.CrossRefzbMATHGoogle Scholar
  17. [17]
    F. Nazarov, S. Treil, A. Volberg: Weak type estimates and Cotlar inequalities for Calderón-Zygmund operators on nonhomogeneous spaces. Int. Math. Res. Not. 1998 (1998), 463–487.CrossRefzbMATHGoogle Scholar
  18. [18]
    C.Pérez: Two weighted inequalities for potential and fractional type maximal operators. Indiana Univ. Math. J. 43 (1994), 663–683.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    C.Pérez: Weighted norm inequalities for singular integral operators. J. Lond. Math. Soc., II. Ser. 49 (1994), 296–308.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    C.Pérez: Endpoint estimates for commutators of singular integral operators. J. Funct. Anal. 128 (1995), 163–185.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    C.Pérez: On sufficient conditions for the boundedness of the Hardy-Littlewood maximal operator between weighted L p-spaces with different weights. Proc. Lond. Math. Soc., III. Ser. 71 (1995), 135–157.CrossRefzbMATHGoogle Scholar
  22. [22]
    C.Pérez: Sharp estimates for commutators of singular integrals via iterations of the Hardy-Littlewood maximal function. J. Fourier Anal. Appl. 3 (1997), 743–756.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    C.Pérez, G. Pradolini: Sharp weighted endpoint estimates for commutators of singular integrals. Mich. Math. J. 49 (2001), 23–37.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    G. Pradolini: Weighted inequalities and pointwise estimates for the multilinear fractional integral and maximal operators. J. Math. Anal. Appl. 367 (2010), 640–656.MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    G. Pradolini, O. Salinas: Maximal operators on spaces of homogeneous type. Proc. Am. Math. Soc. 132 (2004), 435–441.MathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    X. Tolsa: BMO, H 1, and Calderón-Zygmund operators for non doubling measures. Math. Ann. 319 (2001), 89–149.MathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    D. Yang, D. Yang, G. Hu: The Hardy Space H 1 with Non-doubling Measures and Their Applications. Lecture Notes in Mathematics 2084, Springer, Cham, 2013.Google Scholar
  28. [28]
    W. Wang, C. Tan, Z. Lou: A note on weighted norm inequalities for fractional maximal operators with non-doubling measures. Taiwanese J. Math. 16 (2012), 1409–1422.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2018

Authors and Affiliations

  1. 1.Facultad de Ingeniería Química (CONICET UNL)Santa FeArgentina
  2. 2.Instituto de Matemática Bahía Blanca (CONICET UNS), and Departamento de Matemáticas (UNS)Bahía BlancaArgentina

Personalised recommendations