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

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 $0<p<\mathrm{\infty }$, $0<q\le +\mathrm{\infty }$, $u$, $w$ and $v$ are weight functions on $(0,\mathrm{\infty })$. 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.