Bounds for a class of quasilinear integral operators on the set of non-negative and non-negative monotone functions

We consider weighted bounds for quasilinear integral operators of the form
$$ \mathcal{K}^+f(x)=\biggl(\int_{0}^{x}\biggl|w(t)\int_{t}^{x} K(s,t)f(s)\,ds\biggr|^{r}\,dt\biggr)^{{1}/{r}} $$
from $L_{p,v}$ to $L_{q,u}$ on the set on non-negative and non-negative monotone functions $f$, where $u$, $v$ and $w$ are weight functions. Under the assumption that $0<r<\infty$, we obtain necessary and sufficient conditions for the validity of these bounds on the set of non-negative functions for the values of the parameters satisfying the conditions $1\leqslant p\leqslant q<\infty$ and $0<q<p<\infty$, $p\geqslant 1$, and also on the cones of non-negative non-increasing and non-negative non-decreasing functions for $0<q<\infty$ and $1\leqslant p<\infty$. Here it is assumed only that $K{(\,\cdot\,,\cdot\,)}\geqslant 0$. However, the criteria we obtain involve the norm of a linear integral operator from $L_{p,v}$ to $L_{r,w}$ with kernel $K{(\,\cdot\,,\cdot\,)}$.