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 K(+)f(x) = (integral(x)(0)vertical bar w(t) integral(x)(t) K(s, t)f(s)ds vertical bar(r) dt)(1/r) from L-p,L-v to L-q,L-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 < infinity, 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 <= p <= q < infinity and 0 < q < p < infinity, p >= 1, and also on the cones of non-negative non-increasing and non-negative non-decreasing functions for 0 < q < infinity and 1 <= p < infinity. Here it is assumed only that K(. , .) >= 0. However, the criteria we obtain involve the norm of a linear integral operator from L-p,L-v to L-r,L-w with kernel K(. , .).