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

Abstract

We study weak-type modular inequalities for the Hardy operator restricted to non-increasing functions on weighted Lp(⋅) 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 Bp(⋅). 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.