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Uspekhi Mat. Nauk, 2013, Volume 68, Issue 4(412), Pages 3–68 (Mi umn9538)  

This article is cited in 51 scientific papers (total in 51 papers)

Reduction theorems for weighted integral inequalities on the cone of monotone functions

A. Gogatishvilia, V. D. Stepanovb

a Mathematical Institute, Academy of Sciences of the Czech Republic
b Peoples Friendship University of Russia

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.

Funding Agency Grant Number
Czech Science Foundation 201-08-0383
Institute of Mathematics of the Academy of Sciences of the Czech Republic RVO: 67985840
Russian Foundation for Basic Research 12-01-00524
Far Eastern Branch of the Russian Academy of Sciences 12-I-OMN-01


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English version:
Russian Mathematical Surveys, 2013, 68:4, 597–664

Bibliographic databases:

UDC: 517.51
MSC: Primary 26D15; Secondary 47G10
Received: 02.02.2013

Citation: A. Gogatishvili, V. D. Stepanov, “Reduction theorems for weighted integral inequalities on the cone of monotone functions”, Uspekhi Mat. Nauk, 68:4(412) (2013), 3–68; Russian Math. Surveys, 68:4 (2013), 597–664

Citation in format AMSBIB
\by A.~Gogatishvili, V.~D.~Stepanov
\paper Reduction theorems for weighted integral inequalities on the cone of monotone functions
\jour Uspekhi Mat. Nauk
\yr 2013
\vol 68
\issue 4(412)
\pages 3--68
\jour Russian Math. Surveys
\yr 2013
\vol 68
\issue 4
\pages 597--664

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    2. G. E. Shambilova, “The weighted inequalities for a certain class of quasilinear integral operators on the cone of monotone functions”, Siberian Math. J., 55:4 (2014), 745–767  mathnet  crossref  mathscinet  isi
    3. V. D. Stepanov, G. E. Shambilova, “Weight boundedness of a class of quasilinear operators on the cone of monotone functions”, Dokl. Math., 90:2 (2014), 569–572  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib  scopus
    4. L.-E. Persson, O. V. Popova, V. D. Stepanov, “Weighted Hardy-type inequalities on the cone of quasi-concave functions”, Math. Inequal. Appl., 17:3 (2014), 879–898  crossref  mathscinet  zmath  isi  elib  scopus
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    6. V. D. Stepanov, “On Optimal Banach Spaces Containing a Weight Cone of Monotone or Quasiconcave Functions”, Math. Notes, 98:6 (2015), 957–970  mathnet  crossref  crossref  mathscinet  isi  elib
    7. V. D. Stepanov, “On optimal Banach spaces containing a weighted cone of monotone or quasi-concave functions”, Dokl. Math., 92:2 (2015), 545–547  mathnet  crossref  crossref  zmath  isi  elib  elib  scopus
    8. D. V. Prokhorov, “On the boundedness of a class of sublinear operators”, Dokl. Math., 92:2 (2015), 602–605  crossref  crossref  mathscinet  zmath  isi  elib  elib  scopus
    9. N. Azzouz, V. I. Burenkov, A. Senouci, “A weighted Hardy-type inequality for $0<p<1$ with sharp constant”, Math. Inequal. Appl., 18:2 (2015), 787–799  crossref  mathscinet  zmath  isi  elib  scopus
    10. Y. Mizuta, A. Nekvinda, T. Shimomura, “Optimal estimates for the fractional Hardy operator”, Studia Math., 227:1 (2015), 1–19  crossref  mathscinet  zmath  isi  scopus
    11. E. G. Bakhtigareeva, M. L. Goldman, “Construction of an optimal envelope for a cone of nonnegative functions with monotonicity properties”, Proc. Steklov Inst. Math., 293 (2016), 37–55  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    12. D. V. Prokhorov, “On a class of weighted inequalities containing quasilinear operators”, Proc. Steklov Inst. Math., 293 (2016), 272–287  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    13. D. V. Prokhorov, V. D. Stepanov, “Weighted inequalities for quasilinear integral operators on the semi-axis and applications to Lorentz spaces”, Sb. Math., 207:8 (2016), 1159–1186  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    14. V. D. Stepanov, G. E. Shambilova, “Boundedness of quasilinear integral operators on the cone of monotone functions”, Siberian Math. J., 57:5 (2016), 884–904  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    15. V. D. Stepanov, G. E. Shambilova, “Boundedness of a class of quasilinear operators on the cone of monotone functions”, Dokl. Math., 94:3 (2016), 697–702  mathnet  crossref  mathscinet  zmath  isi  elib  scopus
    16. L.-E. Persson, G. E. Shambilova, V. D. Stepanov, “Weighted Hardy type inequalities for supremum operators on the cones of monotone functions”, J. Inequal. Appl., 2016, 237, 18 pp.  crossref  mathscinet  zmath  isi  scopus
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    18. A. N. Kopezhanova, “Some new inequalities for the Fourier transform for functions in generalized Lorentz spaces”, Eurasian Math. J., 8:1 (2017), 58–66  mathnet
    19. E. I. Berezhnoi, L. Maligranda, “On representation of cones of monotone functions in weighted Lebesgue spaces and extrapolation of operators on these cones”, St. Petersburg Math. J., 29:4 (2018), 545–574  mathnet  crossref  isi  elib
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    21. Dokl. Math., 96:3 (2017), 631–635  mathnet  crossref  crossref  zmath  isi  elib  scopus
    22. A. Gogatishvili, M. Křepela, L. Pick, F. Soudský, “Embeddings of Lorentz-type spaces involving weighted integral means”, J. Funct. Anal., 273:9 (2017), 2939–2980  crossref  mathscinet  zmath  isi  scopus
    23. Dokl. Math., 96:1 (2017), 315–320  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib  scopus
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    25. A. Gogatishvili, R.Ch. Mustafayev, “Weighted iterated Hardy-type inequalities”, Math. Inequal. Appl., 20:3 (2017), 683–728  crossref  mathscinet  zmath  isi  scopus
    26. M. Krepela, “Boundedness of Hardy-type operators with a kernel: integral weighted conditions for the case $0<q<1\le p<\infty$”, Rev. Mat. Complut., 30:3 (2017), 547–587  crossref  mathscinet  zmath  isi  scopus
    27. R. Mustafayev, “On weighted iterated Hardy-type inequalities”, Positivity, 22:1 (2018), 275–299  crossref  mathscinet  zmath  isi  scopus
    28. P. Jain, A. P. Singh, M. Singh, V. D. Stepanov, “Sawyer duality principle in grand Lebesgue spaces”, Dokl. Math., 97:1 (2018), 18–19  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib  scopus
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    31. E. I. Berezhnoi, “Exact Calculation of Sums of Cones in Lorentz Spaces”, Funct. Anal. Appl., 52:2 (2018), 134–138  mathnet  crossref  crossref  mathscinet  isi  elib
    32. V. D. Stepanov, G. È. Shambilova, “Iterated Integral Operators on the Cone of Monotone Functions”, Math. Notes, 104:3 (2018), 443–453  mathnet  crossref  crossref  mathscinet  isi  elib
    33. V. D. Stepanov, G. E. Shambilova, “Reduction of weighted bilinear inequalities with integration operators on the cone of nondecreasing functions”, Siberian Math. J., 59:3 (2018), 505–522  mathnet  crossref  crossref  mathscinet  isi  elib
    34. R. Ch. Mustafayev, N. Bilgicli, “Generalized fractional maximal functions in Lorentz spaces $\Lambda$”, J. Math. Inequal., 12:3 (2018), 827–851  crossref  mathscinet  isi
    35. D. Gorbachev, E. Liflyand, S. Tikhonov, “Weighted norm inequalities for integral transforms”, Indiana Univ. Math. J., 67:5 (2018), 1949–2003  crossref  mathscinet  zmath  isi
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    37. A. A. Kalybay, R. Oinarov, “Bounds for a class of quasilinear integral operators on the set of non-negative and non-negative monotone functions”, Izv. Math., 83:2 (2019), 251–272  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    38. Jain P., Singh A.P., Singh M., Stepanov V.D., “Sawyer'S Duality Principle For Grand Lebesgue Spaces”, Math. Nachr., 292:4 (2019), 841–849  crossref  mathscinet  zmath  isi  scopus
    39. Mizuta Y., Nekvinda A., Shimomura T., “Optimal Estimates For the Fractional Hardy Operator on Variable Exponent Lebesgue Spaces”, Math. Inequal. Appl., 22:2 (2019), 445–462  crossref  mathscinet  isi  scopus
    40. Kalybay A., Oinarov R., Temirkhanova A., “Integral Operators With Two Variable Integration Limits on the Cone of Monotone Functions”, J. Math. Inequal., 13:1 (2019), 1–16  crossref  mathscinet  isi  scopus
    41. A. Senouci, N. Azzouz, “Hardy type inequality with sharp constant for $0 < p < 1$”, Eurasian Math. J., 10:1 (2019), 52–58  mathnet  crossref
    42. D. V. Prokhorov, V. D. Stepanov, E. P. Ushakova, “Characterization of the function spaces associated with weighted Sobolev spaces of the first order on the real line”, Russian Math. Surveys, 74:6 (2019), 1075–1115  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    43. Stepanov V.D., Shambilova G.E., “on Iterated and Bilinear Integral Hardy-Type Operators”, Math. Inequal. Appl., 22:4 (2019), 1505–1533  crossref  isi
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    45. V. D. Stepanov, G. E. Shambilova, “Mnogomernye bilineinye neravenstva Khardi”, Sib. matem. zhurn., 61:4 (2020), 913–931  mathnet  crossref
    46. 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  crossref  mathscinet  zmath  isi
    47. Krepela M., Pick L., “Weighted Inequalities For Iterated Copson Integral Operators”, Studia Math., 253:2 (2020), 163–197  crossref  mathscinet  zmath  isi
    48. Sun Q., Yu X., Li H., “The Supremum-Involving Hardy-Type Operators on Lorentz-Type Spaces”, Port Math., 77:1 (2020), 1–29  crossref  mathscinet  zmath  isi
    49. Nursultanov E., Tikhonov S., “Weighted Fourier Inequalities in Lebesgue and Lorentz Spaces”, J. Fourier Anal. Appl., 26:4 (2020), 57  crossref  mathscinet  zmath  isi
    50. Kerman R., “Construction of Weights For Positive Integral Operators”, Symmetry-Basel, 12:6 (2020), 1004  crossref  isi
    51. Sergey Yu. Tikhonov, “Weighted Fourier Inequalities and Boundedness of Variation”, Proc. Steklov Inst. Math., 312 (2021), 282–300  mathnet  crossref  crossref  isi  elib
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