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Hardy–Steklov Integral Operators: Part I

  • D. V. ProkhorovEmail author
  • V. D. Stepanov
  • E. P. Ushakova
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References

  1. 1.
    D. E. Edmunds and W. D. Evans, Hardy Operators, Function Spaces and Embeddings (Springer-Verlag, Berlin, 2004).CrossRefzbMATHGoogle Scholar
  2. 2.
    A. Kufner and L.-E. Persson, Weighted Inequalities of Hardy Type (World Sci. Publ., River Edge, NJ, 2003).CrossRefzbMATHGoogle Scholar
  3. 3.
    A. Kufner, L. Maligrand a, and L.-E. Persson, The Hardy Inequality. About Its History and Some Related Results (Vydavatelsky Servis, Pilsen, 2007).zbMATHGoogle Scholar
  4. 4.
    M. A. Lifshits and W. Linde, Approximation and Entropy Numbers of Volterra Operators with Application to Brownian Motion (Amer.Math. Soc., Providence, RI, 2002).zbMATHGoogle Scholar
  5. 5.
    B. Opic and A. Kufner, Hardy-Type Inequalities (Longman Sci. & Tech., Harlow, 1990).zbMATHGoogle Scholar
  6. 6.
    C. Bennett and R. Sharpley, Interpolation of Operators (Academic, Boston,MA, 1988).zbMATHGoogle Scholar
  7. 7.
    G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities (Cambridge Univ. Press, Cambridge, 1934; Inostrannaya Literatura, Moscow, 1948).zbMATHGoogle Scholar
  8. 8.
    N. Dunford and J. T. Schwartz, Linear Operators. Vol. 1: General Theory (with the assistance of William G. Bade and Robert G. Bartle) (Intersci., New York, 1958, Inostrannaya Literatura,Moscow, 1962).zbMATHGoogle Scholar
  9. 9.
    W. Rudin, Real and Complex Analysis (McGraw-Hill, New York, 1987).zbMATHGoogle Scholar
  10. 10.
    R. Oinarov, “Two-sided norm estimates for certain classes of integral operators,” Proc. Steklov Inst. Math. 204, 205–214 (1994).zbMATHGoogle Scholar
  11. 11.
    S. Bloom and R. Kerman, “Weighted norm inequalities for operators of Hardy type,” Proc. Am. Math. Soc. 113, 135–141 (1991).MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    V. D. Stepanov, “Weighted norm inequalities of Hardy type for a class of integral operators,” J. LondonMath. Soc. (2) 50 (1), 105–120 (1994).MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    G. Bennett, “Some elementary inequalities. III,” Quart. J.Math. Oxford Ser. (2) 42 (166), 149–174 (1991).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    K.-G. Grosse-Erdmann, The Blocking Technique, Weighted Mean Operators and Hardy’s Inequality (Springer-Verlag, Berlin, 1998).CrossRefzbMATHGoogle Scholar
  15. 15.
    M. L. Goldman, “Sharp estimates for the norms of Hardy-type operators on the cones of quasimonotone functions,” Proc. Steklov Inst.Math. 232, 109–137 (2001).MathSciNetzbMATHGoogle Scholar
  16. 16.
    E. Lomakina and V. Stepanov, “On the Hardy-type integral operators in Banach function spaces,” Publ.Mat. 42 (1), 165–194 (1998).MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    V. G. Maz’ya, Sobolev Spaces (Izd. Leningrad. Univ., Leningrad, 1985) [in Russian].zbMATHGoogle Scholar
  18. 18.
    G. J. Sinnamon, “Weighted Hardy and Opial-type inequalities,” J. Math. Anal. Appl. 160 (2), 434–445 (1991).MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    G. Sinnamon and V.D. Stepanov, “The weightedHardy inequality: newproofs and the case p = 1,” J. London Math. Soc. (2) 54 (1), 89–101 (1996).MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    D. V. Prokhorov and V. D. Stepanov, “Weighted estimates for the Riemann-Liouville operators and applications,” Proc. Steklov Inst.Math. 243, 278–301 (2003).MathSciNetzbMATHGoogle Scholar
  21. 21.
    L. V. Kantorovich and G. P. Akilov, Functional Analysis (Nauka, Moscow, 1984) [in Russian].zbMATHGoogle Scholar
  22. 22.
    L.-E. Persson and V. D. Stepanov, “Weighted integral inequalities with the geometric mean operator,” J. Inequal. Appl. 7 (5), 727–746 (2002).MathSciNetzbMATHGoogle Scholar
  23. 23.
    V. D. Stepanov and E. P. Ushakova, “Alternative criteria for the boundedness of Volterra integral in Lebesgue spaces,” Math. Inequal. Appl. 12 (4), 873–889 (2009).MathSciNetzbMATHGoogle Scholar
  24. 24.
    P. J. Martin-Reyes and E. T. Sawyer, “Weighted inequalities for Riemann–Liouville fractional integrals of order one and greater,” Proc. Am. Math. Soc. 106 (3), 727–733 (1989).MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    V. D. Stepanov and E. P. Ushakova, “On integral operators with variable limits of integration,” Proc. Steklov Inst.Math. 232, 290–309 (2001).MathSciNetzbMATHGoogle Scholar
  26. 26.
    E. N. Lomakina, “Estimates for the approximation numbers of one class of integral operators. I,” Sib.Math. J. 44 (1), 147–159 (2003).CrossRefzbMATHGoogle Scholar
  27. 27.
    E. N. Lomakina, “Estimates for the approximation numbers of one class of integral operators. II,” Sib. Math. J. 44 (2), 298–310 (2003).CrossRefzbMATHGoogle Scholar
  28. 28.
    V. D. Stepanov and E. P. Ushakova, “Hardy operator with variable limits on monotone functions,” J. Funct. Spaces Appl. 1 (1), 1–15 (2003).MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    V. D. Stepanov and E. P. Ushakova, “On the geometric mean operator with variable limits of integration,” Proc. Steklov Inst. Math. 260, 254–278 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    V. D. Stepanov and E. P. Ushakova, “Kernel operators with variable intervals of integration in Lebesgue spaces and applications,” Math. Inequal. Appl. 13 (3), 449–510 (2010).MathSciNetzbMATHGoogle Scholar
  31. 31.
    E. P. Ushakova, “Estimates for Schatten–von Neumann norms of Hardy–Steklov operators,” J. Approx. Theory 173, 158–175 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    E. P. Ushakova, “Alternative boundedness characteristics for the Hardy–Steklov operator,” Eurasian Math. J. 8 (2), 74–96 (2017).MathSciNetGoogle Scholar

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© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  • D. V. Prokhorov
    • 1
    Email author
  • V. D. Stepanov
    • 1
  • E. P. Ushakova
    • 1
  1. 1.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia

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