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 


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© 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

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