New equivalence theorems for weighted inequalities involving the composition of monotone quasilinear operators with the Hardy and Copson operators and their applications

TL;DR

This paper establishes new equivalence theorems for the boundedness of compositions of monotone quasilinear operators with Hardy and Copson operators in weighted Lebesgue spaces, with applications to Hardy-type inequalities.

Contribution

It introduces novel equivalence theorems for the boundedness of composed operators, extending the theory of weighted inequalities involving Hardy and Copson operators.

Findings

New equivalence theorems for operator boundedness

Applications to weighted Hardy-type inequalities

Extensions to iterated Hardy-type inequalities

Abstract

In this paper, new equivalence theorems for the boundedness of the composition of a quasilinear operator $T$ with the Hardy and Copson operators in weighted Lebesgue spaces are proved. The usefulness of the obtained results is illustrated in the case of weighted Hardy-type and weighted iterated Hardy-type inequalities.