Weighted inequalities for a superposition of the Copson operator and the Hardy operator

Abstract

in which (a, b) is any nontrivial interval, q, r are positive real parameters and p ∈ (0, 1]. A simple change of variables can be used to obtain any weighted L p -norm with p ≥ 1 on the right-hand side. Another simple change of variables can be used to equivalently turn this inequality into the one in which the Hardy and Copson operators swap their positions. We focus on characterizing those triples of weight functions (u, v, w) for which this inequality holds for all nonnegative measurable functions f with a constant independent of f .We use a new type of approach based on an innovative method of discretization which enables us to avoid duality techniques and therefore to remove various restrictions that appear in earlier work.This paper is dedicated to Professor Stefan Samko on the occasion of his 80th birthday.2010 Mathematics Subject Classification. 26D10.

Weighted inequalities involving iteration of two Hardy integral operators

“…Inequality (1.1) is investigated thoroughly. Detailed information on the development and history of this inequality may be found in the recent paper [4].…”

“…weighted Hardy inequality, iterated operators, Copson operator, Hardy operator, inequalities for monotone functions. Recently,in [4], with a new and simpler discretization technique requires neither parameter restrictions nor non-degeneracy conditions, characterization of (1.1) is given. We adapt this approach to the specific demands of the inequality considered in this paper.…”

Another approach to weighted inequalities for a superposition of Copson and Hardy operators

“…The "classical" conditions ensuring the validity of (1.3) was recently presented in [19]. Inequalities (1.2) and (1.4) were recently characterized by using discretization techniques in [6] and [17], respectively.…”