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 Lp-Lq 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. © 2013 Elsevier Ltd.