Inequalities for Hardy-Type Operators on the Cone of Decreasing Functions in a Weighted Orlicz Space

Modular inequalities and inequalities for the norms of Hardy-type operators on the cone of positive functions and on the cone Omega of positive decreasing functions with common weight and common Young function in a weighted Orlicz space are considered. A reduction theorem for the norm of the Hardy operator on the cone Omega is obtained. It is shown that this norm is equivalent to the norm of a modified operator on the cone of all positive functions in the space under consideration. It is proved that the modified operator is a generalized Hardy-type operator. The equivalence of modular inequalities on the cone Omega and modified modular inequalities on the cone of all positive functions in the Orlicz space is shown. A criterion for the validity of such inequalities in general Orlicz spaces is obtained and refined for weighted Lebesgue spaces.