Order-sharp estimates for Hardy-type operators on the cones of functions with properties of monotonicity

Two-sided estimates are established for two types of generalized Hardy operators on the cones of functions in weighted Lebesgue spaces with some properties of monotonicity.
We prove the results announced in [7], and present some other equivalent forms for the criterion of boundedness. Some other equivalent descriptions, particular cases, and results in the case of degenerate measures will be given in our next paper.