Some new iterated hardy-type inequalities: The case $\theta = 1$

TL;DR

This paper provides a comprehensive characterization of Hardy-type inequalities involving iterated integrals with specific weight functions, extending the understanding of their boundedness properties in weighted Lebesgue spaces.

Contribution

It introduces new necessary and sufficient conditions for the validity of a class of Hardy-type inequalities in the case =1, applicable to weighted Lebesgue spaces and related operators.

Findings

Characterization of Hardy-type inequalities for =1.

New criteria for boundedness of Hardy-type operators on monotone functions.

Applications to generalized Stieltjes operators.

Abstract

In this paper we characterize the validity of the Hardy-type inequality \begin{equation*} \left\|\left\|\int_s^{\infty}h(z)dz\right\|_{p,u,(0,t)}\right\|_{q,w,\infty}\leq c \,\|h\|_{1,v,\infty} \end{equation*} where $0<p< \infty$, $0<q\leq +\infty$, $u$, $w$ and $v$ are weight functions on $(0,\infty)$. It is pointed out that this characterization can be used to obtain new characterizations for the boundedness between weighted Lebesgue spaces for Hardy-type operators restricted to the cone of monotone functions and for the generalized Stieltjes operator.