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

We consider weighted bounds for quasilinear integral operators of the form
$\mathcal{K}^+f(x)=(\int_{0}^{x}|w(t)\int_{t}^{x} K(s,t)f(s) ds|^{r} dt)^{{1}/{r}}$
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 $1\leq p\leq q< \infty$ and $0< q< p< \infty$, $p\geq 1$, and also on the cones of non-negative non-increasing and non-negative non-decreasing functions for $0< q< \infty$ and $1\leq p< \infty$. Here it is assumed only that $K{( \cdot ,\cdot )}\geq 0$. However, the criteria we obtain involve the norm of a linear integral operator from $L_{p,v}$ to $L_{r,w}$ with kernel $K{( \cdot ,\cdot )}$.