Geodesic
Reduction theorems for weighted integral inequalities on the cone of monotone functions
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 68 (2013) no. 4, pp. 597-664

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This paper surveys results related to the reduction of integral inequalities involving positive operators in weighted Lebesgue spaces on the real semi-axis and valid on the cone of monotone functions, to certain more easily manageable inequalities valid on the cone of non-negative functions. The case of monotone operators is new. As an application, a complete characterization for all possible integrability parameters is obtained for a number of Volterra operators. Bibliography: 118 titles.
Keywords: weighted Lebesgue space, cone of monotone functions, duality principle, weighted integral inequality, bounded operators, reduction theorem. 
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     title = {Reduction theorems for weighted integral inequalities on the cone of monotone functions},
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A. Gogatishvili; V. D. Stepanov. Reduction theorems for weighted integral inequalities on the cone of monotone functions. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 68 (2013) no. 4, pp. 597-664. http://geodesic.mathdoc.fr/item/RM_2013_68_4_a0/