Representability of cones in weighted Lebesgue spaces and extrapolation operators on cones are discussed. Estimates of operators on some cones in spaces rather than on the entire spaces have become very popular. A reduction of estimating operators on cones to estimating them on new spaces is suggested. such reduction makes it possible to apply the whole apparatus developed for obtaining exact estimates on weith Lebesgue spaces to obtain exact estimates of operators on cones. Using the reduction, it is proved that a new extrapolation theorem for a certain class of operator defined on cones in Lebesgue spaces.

Two-sided Hardy-type inequalities for monotone functions are extended. The cases of positive and negative parameter values are also studied. The discrete result of theorem is extended to an arbitrary positive σ-finite Borel measure. The notation A ≪ B means that A≤ cB, where c is a constant depending only on the summation parameter. Different theorems proposed prove the new characterization of the discrete Hardy inequality.