Abstract:
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.
Citation:
A. Gogatishvili, V. D. Stepanov, “Reduction theorems for weighted integral inequalities on the cone of monotone functions”, Russian Math. Surveys, 68:4 (2013), 597–664
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\paper Reduction theorems for weighted integral inequalities on the cone of monotone functions
\jour Russian Math. Surveys
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\vol 68
\issue 4
\pages 597--664
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Linking options:
https://www.mathnet.ru/eng/rm9538
https://doi.org/10.1070/RM2013v068n04ABEH004849
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This publication is cited in the following 69 articles:
Mikhail Dyachenko, Vladimir D. Stepanov, Sergey Tikhonov, “Fourier inequalities for averaged integral transforms”, Journal of Mathematical Analysis and Applications, 543:2 (2025), 128940
V. D. Stepanov, G. E. Shambilova, “On the Iterated Integral Operators on the Cone of Monotone Functions”, Sib Math J, 66:2 (2025), 345
Rovshan Bandaliyev, Dunya Aliyeva, “A characterization of two-weight norm inequalities for multidimensional Hausdorff operators on Lebesgue spaces”, Positivity, 28:2 (2024)
Amiran Gogatishvili, Tuğçe Ünver, “New characterization of weighted inequalities involving superposition of Hardy integral operators”, Mathematische Nachrichten, 2024
A. T. Beszhanova, A. O. Bayarystanov, “WEIGHTED ESTIMATE OF A MATRIX OPERATOR WITH VARIABLE UPPER LIMIT ON THE CONE OF MONOTONE SEQUENCES”, jour, 21:4 (2024), 136
A. A. Kalybay, A. M. Temirkhanova, “New weighted Hardy-type inequalities for monotone functions”, Eurasian Math. J., 15:4 (2024), 54–65
V. D. Stepanov, E. P. Ushakova, “Strong and weak associativity of weighted Sobolev spaces of the first order”, Russian Math. Surveys, 78:1 (2023), 165–202
Stepanov V.D., “On Cesaro and Copson Type Function Spaces. Reflexivity”, J. Math. Anal. Appl., 507:1 (2022), 125764
V. D. Stepanov, “On Spaces Associated with Weighted Cesàro and Copson Spaces”, Math. Notes, 111:3 (2022), 470–477
Gogatishvili A., Mihula Z., Pick L., Turcinova H., Unver T., “Weighted Inequalities For a Superposition of the Copson Operator and the Hardy Operator”, J. Fourier Anal. Appl., 28:2 (2022), 24
V. I. Burenkov, M. A. Senouci, “ON BOUNDEDNESS OF THE GENERALIZED RIESZ POTENTIAL IN LOCAL MORREY-TYPE SPACES”, J Math Sci, 266:5 (2022), 765
Elijah Liflyand, “HARDY TYPE INEQUALITIES WITH POWER WEIGHTS FOR HAUSDORFF OPERATORS”, J Math Sci, 266:3 (2022), 435
Sergey Yu. Tikhonov, “Weighted Fourier Inequalities and Boundedness of Variation”, Proc. Steklov Inst. Math., 312 (2021), 282–300
Kerman R., Spektor S., “Orlicz-Lorentz Gauge Functional Inequalities For Positive Integral Operators”, Bull. Sci. Math., 173 (2021), 103056
Bandaliyev R., Safarova K., “On Two-Weight Inequalities For Hausdorff Operators of Special Kind in Lebesgue Spaces”, Hacet. J. Math. Stat., 50:5 (2021), 1334–1346
Elijah Liflyand, Pathways in Mathematics, Harmonic Analysis on the Real Line, 2021, 131
Gogatishvili A., Neves J.S., “Weighted Norm Inequalities For Positive Operators Restricted on the Cone of Lambda-Quasiconcave Functions”, Proc. R. Soc. Edinb. Sect. A-Math., 150:1 (2020), 17–39
V. D. Stepanov, G. È. Shambilova, “Multidimensional bilinear hardy inequalities”, Siberian Math. J., 61:4 (2020), 725–742
Mustafayev R., Bilgicli N., “Boundedness of Weighted Iterated Hardy-Type Operators Involving Suprema From Weighted Lebesgue Spaces Into Weighted Cesaro Function Spaces”, Real Anal. Exch., 45:2 (2020), 339–374
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