  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.
