Geodesic
Some equivalent criteria for the boundedness of Hardy-type operators on the cone of quasimonotone functions
Eurasian mathematical journal, Tome 4 (2013) no. 4, pp. 43-63

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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. In this paper we continue the proofs given in [9] for the main results announced in our paper [7]. Also we present here some other equivalent descriptions, consider some particular cases and establish results in the case of a degenerate measure. 
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     author = {M. L. Goldman},
     title = {Some equivalent criteria for the boundedness of {Hardy-type} operators on the cone of quasimonotone functions},
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M. L. Goldman. Some equivalent criteria for the boundedness of Hardy-type operators on the cone of quasimonotone functions. Eurasian mathematical journal, Tome 4 (2013) no. 4, pp. 43-63. http://geodesic.mathdoc.fr/item/EMJ_2013_4_4_a3/