The Modular Inequalities for Hardy-Type Operators on Monotone Functions in Orlicz Space

Abstract

The purpose of this paper is to study the behaviour of integral operators of Hardy-type on monotone functionin orlicz spacewith general weight.on weighted Orlicz spaces. The result is based on the theorem on the reduction of modular inequalities for positively homogeneous operators on the cone Ω, which enables passing to modular inequalities for modified operators on the cone of all nonnegative functions from an Orlicz space. It is shown that, for the Hardy operator, the modified operator is a generalized Hardy-type operator. This enables us to establish explicit criteria for the validity of modular inequalities.