Reduction theorems for operators on the cones of monotone functions

For a quasilinear operator on the semiaxis a reduction theorem is proved on the cones of monotone functions in the L p − L q setting for 0 < q < ∞ , 1 ≤ p < ∞ . The case 0 < p < 1 is also studied for operators with additional properties. In particular, we obtain criteria for three-weight inequalities for the Hardy-type operators on monotone functions in the case 0 < q < p ≤ 1 .