Czechoslovak Mathematical Journal

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

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



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.MathSciNetCrossRefMATHGoogle 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.MathSciNetCrossRefMATHGoogle 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.MathSciNetMATHGoogle 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.MathSciNetCrossRefMATHGoogle 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.MathSciNetMATHGoogle 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.MathSciNetCrossRefMATHGoogle 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.MathSciNetMATHGoogle 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.MathSciNetCrossRefMATHGoogle 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.MathSciNetCrossRefMATHGoogle 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.MathSciNetMATHGoogle 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.MathSciNetCrossRefMATHGoogle 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.MathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    J. Mateu, P. Mattila, A. Nicolau, J. Orobitg: BMO for nondoubling measures. Duke Math. J. 102 (2000), 533–565.MathSciNetCrossRefMATHGoogle Scholar
  15. [15]
    Y. Meng, D. Yang: Boundedness of commutators with Lipschitz functions in nonhomogeneous spaces. Taiwanese J. Math. 10 (2006), 1443–1464.MathSciNetCrossRefMATHGoogle 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.CrossRefMATHGoogle 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.CrossRefMATHGoogle Scholar
  18. [18]
    C.Pérez: Two weighted inequalities for potential and fractional type maximal operators. Indiana Univ. Math. J. 43 (1994), 663–683.MathSciNetCrossRefMATHGoogle Scholar
  19. [19]
    C.Pérez: Weighted norm inequalities for singular integral operators. J. Lond. Math. Soc., II. Ser. 49 (1994), 296–308.MathSciNetCrossRefMATHGoogle Scholar
  20. [20]
    C.Pérez: Endpoint estimates for commutators of singular integral operators. J. Funct. Anal. 128 (1995), 163–185.MathSciNetCrossRefMATHGoogle 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.CrossRefMATHGoogle 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.MathSciNetCrossRefMATHGoogle Scholar
  23. [23]
    C.Pérez, G. Pradolini: Sharp weighted endpoint estimates for commutators of singular integrals. Mich. Math. J. 49 (2001), 23–37.MathSciNetCrossRefMATHGoogle 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.MathSciNetCrossRefMATHGoogle Scholar
  25. [25]
    G. Pradolini, O. Salinas: Maximal operators on spaces of homogeneous type. Proc. Am. Math. Soc. 132 (2004), 435–441.MathSciNetCrossRefMATHGoogle Scholar
  26. [26]
    X. Tolsa: BMO, H 1, and Calderón-Zygmund operators for non doubling measures. Math. Ann. 319 (2001), 89–149.MathSciNetCrossRefMATHGoogle 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.MathSciNetCrossRefMATHGoogle 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