Revista Matemática Complutense

, Volume 25, Issue 2, pp 459–474 | Cite as

Weighted weak modular and norm inequalities for the Hardy operator in variable L p spaces of monotone functions

  • Santiago Boza
  • Javier Soria


We study weak-type modular inequalities for the Hardy operator restricted to non-increasing functions on weighted L p(⋅) spaces, where p(⋅) is a variable exponent. These new estimates complete the results of Boza and Soria (J. Math. Anal. Appl. 348:383–388, 2008) where we showed some necessary and sufficient conditions on the exponent p(⋅) and on the weights to obtain weighted modular inequalities with variable exponents. For this purpose, we introduced the class of weights B p(⋅). We prove that, for exponents p(x)>1, this is also the class of weights for which the weak modular inequality holds, and a characterization is also given in the case p(x)≤1. Finally, we compare our theory with the results in Neugebauer (Stud. Math. 192(1):51–60, 2009), giving examples for very concrete and simple exponents which show that inequalities in norm hold true in a very general context.


Modular inequalities Hardy operator Bp weights Variable Lp spaces Monotone functions 

Mathematics Subject Classification (2000)

26D10 46E30 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aguilar, M.I., Ortega, P.: Weighted weak type inequalities with variable exponents for Hardy and maximal operators. Proc. Jpn. Acad., Ser. A, Math. Sci. 82, 126–130 (2006) zbMATHCrossRefGoogle Scholar
  2. 2.
    Ariño, M.A., Muckenhoupt, B.: Maximal functions on classical Lorentz spaces and Hardy’s inequality with weights for nonincreasing functions. Trans. Am. Math. Soc. 320, 727–735 (1990) zbMATHGoogle Scholar
  3. 3.
    Boza, S., Soria, J.: Weighted Hardy modular inequalities in variable L p spaces for decreasing functions. J. Math. Anal. Appl. 348, 383–388 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Carro, M.J., Soria, J.: Boundedness of some integral operators. Can. J. Math. 45, 1155–1166 (1993) MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Carro, M.J., Soria, J.: Weighted Lorentz spaces and the Hardy operator. J. Funct. Anal. 112, 480–494 (1993) MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Carro, M.J., Pick, L., Soria, J., Stepanov, V.: On embeddings between classical Lorentz spaces. Math. Inequal. Appl. 4, 397–428 (2001) MathSciNetzbMATHGoogle Scholar
  7. 7.
    Carro, M.J., Raposo, J.A., Soria, J.: Recent Developments in the Theory of Lorentz Spaces and Weighted Inequalities. Mem. Am. Math. Soc., vol. 187, Am. Math. Soc., Providence (2007) Google Scholar
  8. 8.
    Cruz-Uribe, D., Fiorenza, A., Neugebauer, C.J.: The maximal function on variable L p spaces. Ann. Acad. Sci. Fenn. Math. 28, 223–238 (2003) MathSciNetzbMATHGoogle Scholar
  9. 9.
    Cruz-Uribe, D., Fiorenza, A., Martell, J.M., Pérez, C.: The boundedness of classical operators on variable L p spaces. Ann. Acad. Sci. Fenn. Math. 31, 239–264 (2006) MathSciNetGoogle Scholar
  10. 10.
    Diening, L., Hästö, P.: Muckenhoupt weights in variable exponents spaces. Preprint Google Scholar
  11. 11.
    Edmunds, D.E., Kokilashvili, V., Meshki, A.: On the boundedness and compactness of weighted Hardy operators in spaces L p(x). Georgian Math. J. 12, 27–44 (2005) MathSciNetzbMATHGoogle Scholar
  12. 12.
    Kovavcik, O., Rakosnik, J.: On spaces L p(x) and W p(x). Czechoslov. Math. J. 41, 592–618 (1991) Google Scholar
  13. 13.
    Lerner, A.K.: On modular inequalities in variable L p spaces. Arch. Math. 85, 538–543 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Mashiyev, R.A., Çekiç, B., Mamedov, F.I., Ogras, S.: Hardy’s inequality in power-type weighted L p(⋅)(0,∞) spaces. J. Math. Anal. Appl. 334, 289–298 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Neugebauer, C.J.: Weighted norm inequalities for averaging operators of monotone functions. Publ. Mat. 35, 429–447 (1991) MathSciNetzbMATHGoogle Scholar
  16. 16.
    Neugebauer, C.J.: Weighted variable L p integral inequalities for the maximal operator on non-increasing functions. Stud. Math. 192(1), 51–60 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Sawyer, E.: Boundedness of classical operators on classical Lorentz spaces. Stud. Math. 96, 145–158 (1990) MathSciNetzbMATHGoogle Scholar
  18. 18.
    Sinnamon, G.: Four questions related to Hardy’s inequality. In: Function Spaces and Applications, Delhi, 1997, pp. 255–266. Narosa, New Delhi (2000) Google Scholar

Copyright information

© Revista Matemática Complutense 2011

Authors and Affiliations

  1. 1.Department of Applied Mathematics IV, EPSEVGPolytechnical University of CataloniaVilanova i GeltrúSpain
  2. 2.Department of Applied Mathematics and AnalysisUniversity of BarcelonaBarcelonaSpain

Personalised recommendations