On equivalent criteria for the boundedness of Hardy type operators on the cone of decreasing functions

We prove the equivalence of some three-weighted criteria for the boundedness of Hardy type operators on the cone of decreasing functions. These criteria are then applied to realize the direct comparisons of the results obtained in different forms in the papers of several authors (M. Goldman, G. Bennett and K.-G. Grosse-Erdmann, M. Carro, A. Gogatishvili, J. Martin, and L. Pick).