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Trudy Mat. Inst. Steklova, 2003, Volume 243, Pages 289–312 (Mi tm434)  

This article is cited in 29 scientific papers (total in 29 papers)

Weighted Estimates for the Riemann–Liouville Operators and Applications

D. V. Prokhorova, V. D. Stepanovb

a Computer Centre Far-Eastern Branch of RAS
b Department of Mathematics, Korea Advanced Institute of Science and Technology

Abstract: Necessary and sufficient conditions for the weighted boundedness and compactness of the Riemann–Liouville operators are obtained. Applications to the solvability of the Abel nonlinear integral equations and to the embeddings of Besov-type spaces into weighted Lebesgue spaces on the semiaxis are given.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2003, 243, 278–301

Bibliographic databases:
UDC: 517.51
Received in May 2002

Citation: D. V. Prokhorov, V. D. Stepanov, “Weighted Estimates for the Riemann–Liouville Operators and Applications”, Function spaces, approximations, and differential equations, Collected papers. Dedicated to the 70th birthday of Oleg Vladimirovich Besov, corresponding member of RAS, Trudy Mat. Inst. Steklova, 243, Nauka, MAIK Nauka/Inteperiodika, M., 2003, 289–312; Proc. Steklov Inst. Math., 243 (2003), 278–301

Citation in format AMSBIB
\by D.~V.~Prokhorov, V.~D.~Stepanov
\paper Weighted Estimates for the Riemann--Liouville Operators and Applications
\inbook Function spaces, approximations, and differential equations
\bookinfo Collected papers. Dedicated to the 70th birthday of Oleg Vladimirovich Besov, corresponding member of RAS
\serial Trudy Mat. Inst. Steklova
\yr 2003
\vol 243
\pages 289--312
\publ Nauka, MAIK Nauka/Inteperiodika
\publaddr M.
\jour Proc. Steklov Inst. Math.
\yr 2003
\vol 243
\pages 278--301

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    This publication is cited in the following articles:
    1. V. D. Stepanov, E. P. Ushakova, “Weighted estimates for the integral operators with monotone kernel on a half-axis”, Siberian Math. J., 45:6 (2004), 1124–1134  mathnet  crossref  mathscinet  zmath  isi
    2. Persson L.E., Stepanov V.D., Ushakova E.P., “On integral operators with monotone kernels”, Dokl. Math., 72:1 (2005), 491–494  mathnet  mathscinet  zmath  isi  elib  elib
    3. Lomakina E.N., Stepanov V.D., “Asymptotic estimates for approximative and entropy numbers of the one-weight Riemann–Liouville operator”, Dokl. Math., 72:1 (2005), 574–575  mathnet  mathscinet  zmath  isi  elib  elib
    4. E. N. Lomakina, V. D. Stepanov, “Asymptotic Estimates for the Approximation and Entropy Numbers of a One-Weight Riemann–Liouville Operator”, Siberian Adv. Math., 17:1 (2007), 1–36  mathnet  crossref  mathscinet
    5. Neiman-zade M.I., Shkalikov A.A., “Strongly elliptic operators with singular coefficients”, Russ. J. Math. Phys., 13:1 (2006), 70–78  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    6. N. A. Rautian, “On the boundedness of a class of fractional type integral operators”, Sb. Math., 200:12 (2009), 1807–1832  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    7. Stepanov V.D., Ushakova E.P., “Alternative criteria for the boundedness of Volterra integral operators in Lebesgue spaces”, Math. Inequal. Appl., 12:4 (2009), 873–889  mathscinet  zmath  isi  elib
    8. Prokhorov D.V., “Inequalities for Riemann-Liouville operator involving suprema”, Collect Math, 61:3 (2010), 263–276  crossref  mathscinet  zmath  isi  elib  scopus
    9. Vasil'eva A.A., “Kolmogorov Widths of Weighted Sobolev Classes on a Domain for a Special Class of Weights. II”, Russian Journal of Mathematical Physics, 18:4 (2011), 465–504  crossref  mathscinet  zmath  adsnasa  isi  scopus
    10. Vasil'eva A.A., “Kolmogorov widths of weighted Sobolev classes on a domain for a special class of weights”, Russian Journal of Mathematical Physics, 18:3 (2011), 353–385  crossref  mathscinet  zmath  adsnasa  isi  scopus
    11. S. M. Farsani, “On boundedness and compactness of Riemann–Liouville fractional operators”, Siberian Math. J., 54:2 (2013), 368–378  mathnet  crossref  mathscinet  isi
    12. 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  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    13. D. V. Prokhorov, V. D. Stepanov, “On weighted Hardy inequalities in mixed norms”, Proc. Steklov Inst. Math., 283 (2013), 149–164  mathnet  crossref  crossref  mathscinet  isi  elib
    14. Farsani S.M., “On the Boundedness and Compactness of a Certain Integral Operator”, Banach J. Math. Anal., 7:2 (2013), 86–102  crossref  mathscinet  zmath  isi  elib  scopus
    15. F. G. Avkhadiev, “A geometric description of domains whose Hardy constant is equal to 1/4”, Izv. Math., 78:5 (2014), 855–876  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    16. F. G. Avkhadiev, “Hardy type $L_p$-inequalities in $r$-close-to-convex domains”, Russian Math. (Iz. VUZ), 59:1 (2015), 71–74  mathnet  crossref
    17. F. G. Avkhadiev, “Sharp constants in Hardy type inequalities”, Russian Math. (Iz. VUZ), 59:10 (2015), 53–56  mathnet  crossref
    18. M. G. Nasyrova, E. P. Ushakova, “Hardy–Steklov operators and Sobolev-type embedding inequalities”, Proc. Steklov Inst. Math., 293 (2016), 228–254  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    19. R. Oinarov, “Boundedness and compactness of a class of convolution integral operators of fractional integration type”, Proc. Steklov Inst. Math., 293 (2016), 255–271  mathnet  crossref  crossref  mathscinet  isi  elib
    20. 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
    21. D. V. Prokhorov, V. D. Stepanov, E. P. Ushakova, “Hardy–Steklov Integral Operators”, Proc. Steklov Inst. Math., 300, suppl. 2 (2018), 1–112  mathnet  crossref  crossref  zmath  isi  elib
    22. Abylayeva A.M. Oinarov R. Persson L.-E., “Boundedness and compactness of a class of Hardy type operators”, J. Inequal. Appl., 2016, 324  crossref  mathscinet  zmath  isi  elib  scopus
    23. A. M. Abylaeva, “Boundedness, compactness for a class of fractional integration operators of Weyl type”, Eurasian Math. J., 7:1 (2016), 9–27  mathnet
    24. Roohian H., Farsani S.M., “On the Behavior At Infinity of Certain Integral Operator With Positive Kernel”, Adv. Oper. Theory, 2:3 (2017), 228–236  crossref  mathscinet  zmath  isi
    25. Nasibullin R., “Hardy Type Inequalities For Fractional Integrals and Derivatives of Riemann-Liouville”, Lobachevskii J. Math., 38:4, SI (2017), 709–718  crossref  mathscinet  zmath  isi  scopus
    26. Abylayeva A.M., Persson L.-E., “Hardy Type Inequalities and Compactness of a Class of Integral Operators With Logarithmic Singularities”, Math. Inequal. Appl., 21:1 (2018), 201–215  crossref  mathscinet  zmath  isi  scopus
    27. Luor D.-Ch., “On the Equivalence of Boundedness For Multiple Hardy-Littlewood Averages and Related Operators”, Math. Scand., 123:2 (2018), 273–296  crossref  mathscinet  zmath  isi  scopus
    28. 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
    29. A. Kalybay, R. Oinarov, “Boundedness of Riemann–Liouville operator from weighted Sobolev space to weighted Lebesgue space”, Eurasian Math. J., 12:1 (2021), 39–48  mathnet  crossref
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