Hardy–Steklov Integral Operators: Part II

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


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  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. Maligranda, and L.–E. Persson, The Hardy Inequality. About Its History and Some Related Results (VydavatelskýServis, 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).CrossRefzbMATHGoogle Scholar
  5. 5.
    B. Opic and A. Kufner, Hardy–Type Inequalities (Longman Sci. and 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 W. G. Bade and R.G. Bartle) (Intersci., New York, 1958, Inostrannaya Literatura,Moscow, 1962).Google 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
  33. 33.
    D. V. Prokhorov and V. D. Stepanov, “On inequalities with measures of Sobolev type embedding theorems on open sets of the real axis,” Sib. Math. J. 43 (4), 694–707 (2002).CrossRefzbMATHGoogle Scholar
  34. 34.
    B. Ćurgus and T. T. Read, “Discreteness of the spectrumof second–order differential operators and associated embedding theorems,” J. Differential Equations 184 (2), 526–548 (2002).MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    V. G. Maz’ya, “Conductor inequalities and criteria for Sobolev type two–weight imbeddings,” J. Comput. Appl.Math. 194 (1), 94–114 (2006).MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    R. Oinarov, “Boundedness of integral operators from weighted Sobolev space to weighted Lebesgue space,” Complex Var. Elliptic Eq. 56 (10–11), 1021–1038 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    S. P. Eveson, V. D. Stepanov, and E. P. Ushakova, “A duality principle in weighted Sobolev spaces on the real line,” Math. Nachr. 288 (8), 877–897 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    G. Leoni, A First Course in Sobolev Spaces (Amer.Math. Soc., Providence, RI, 2009).CrossRefzbMATHGoogle Scholar
  39. 39.
    K. T. Mynbaev and M. Otelbaev, Weighted Function Spaces and the Spectrum of Differential Operators (Nauka, Moscow, 1988) [in Russian].zbMATHGoogle Scholar
  40. 40.
    R. Oinarov and M. Otelbaev, “A criterion for the discreteness of the spectrum of the general Sturm–Liouville operator, and embedding theorems connected with it,” Diff. Equations 24 (4), 402–408 (1988).MathSciNetzbMATHGoogle Scholar
  41. 41.
    M. G. Nasyrova and E. P. Ushakova, “Hardy–Steklov operators and Sobolev–type embedding inequalities,” Proc. Steklov Inst. Math. 293, 228–254 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    E. Gagliardo, “Proprieta di alcune classi di funzioni in piu variabili,” Ricerche Mat. 7, 102–137 (1958).MathSciNetzbMATHGoogle Scholar
  43. 43.
    L. N. Slobodetskii, “Generalized Sobolev spaces and their application to boundary value problems for partial differential equations,” Uch. Zap. Leningrad Pedagog. Instit. Im. A. I. Gertzena 197, 54–112 (1958).Google Scholar
  44. 44.
    O. V. Besov, V. P. Il’in, and S. M. Nikol’skii, Integral Representations of Functions and Embedding Theorems (Nauka, Moscow, 1996) [in Russian].zbMATHGoogle Scholar
  45. 45.
    J.–L. Lions and E. Magenes, Non–Homogeneous Boundary Value Problems and Applications (Springer, Berlin, 1972;Mir,Moscow, 1971), Vol. 1.CrossRefzbMATHGoogle Scholar
  46. 46.
    H. P. Heinig and G. Sinnamon, “Mapping properties of integral averaging operators,” Studia Math. 129 (2), 157–177 (1998).MathSciNetzbMATHGoogle Scholar

Copyright information

© 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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