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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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/

[1] G. Bennett, K.-G. Grosse-Erdmann, “Weighted Hardy inequalities for decreasing sequences and functions”, Math. Annalen, 334 (2006), 489–531 | DOI | MR | Zbl

[2] V. I. Burenkov, M. L. Goldman, “Calculation of the norm of positive operator on the cone of monotone functions”, Proc. of the Steklov Inst. Math., 210, 1995, 47–65 | MR | Zbl

[3] M. Carro, A. Gogatishvili, J. Martin, L. Pick, “Weighted Inequalities Involving Two Hardy Operators with Applications to Embeddings of Function Spaces”, J. Operator Theory, 59:2 (2008), 309–332 | MR | Zbl

[4] A. Gogatishvili, M. Johansson, C. A. Okpoti, L.-E. Persson, “Characterization of embeddings in Lorentz spaces using a method of discretisation and anti-discretisation”, Bull. Austral. Math. Soc., 76 (2007), 69–92 | DOI | MR | Zbl

[5] M. L. Goldman, “Sharp Estimates of the Norms of Hardy-Type Operators on the Cone of Quasimonotone Functions”, Proc. of the Steklov Inst. Math., 232, 2001, 109–137 | MR | Zbl

[6] M. L. Goldman, “On equivalent criteria for the boundedness of Hardy type operators on the cone of decreasing functions”, Analysis Mathematica, 37:2 (2011), 83–102 | DOI | MR | Zbl

[7] M. L. Goldman, “Order-sharp Estimates for Hardy-Type Operators on the Cones of Quasimonotone Functions”, Eurasian Mathematical Journal, 2:3 (2011), 143–146 | MR | Zbl

[8] M. L. Goldman, Hardy type inequalities on the cone of quasimonotone functions, Research Report no. 98/31, Russian Acad. Sci., Far-East Branch Computer Center, 1998

[9] M. L. Goldman, “Order-sharp estimates for Hardy-type Operators on the Cones of Functions with Properties of Monotonicity”, Eurasian Mathematical Journal, 3:2 (2012), 53–84 | MR | Zbl