Skip to main content
SpringerLink
Log in

Boundedness and compactness of Volterra type integral operators

  • Published:
Siberian Mathematical Journal Aims and scope Submit manuscript

Abstract

We introduce some nested classes of Volterra type integral operators. For the operators of these classes we establish criteria for boundedness and compactness in Lebesgue spaces.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Stepanov V. D., “Weighted inequalities of Hardy type for higher order derivatives, and their applications,” Sov. Math., Dokl., 38, 389–393 (1989).

    MATH  Google Scholar 

  2. Stepanov V. D., “A weighted inequality of Hardy type for higher order derivatives, and their applications,” Trudy Mat. Inst. Steklov., 187, 178–190 (1989).

    MathSciNet  Google Scholar 

  3. Stepanov V. D., “Two-weighted estimates of Riemann-Liouville integrals,” Izv. Akad. Nauk SSSR Ser. Mat., 54, No. 3, 645–654 (1990).

    MATH  Google Scholar 

  4. Stepanov V. D., “Weighted inequalities of Hardy type for fractional Riemann-Liouville integrals, ” Siberian Math. J., 31, No. 3, 513–522 (1990).

    Article  MATH  MathSciNet  Google Scholar 

  5. Stepanov V. D., “On boundedness and compactness of a class of integral operators,” Sov. Math., Dokl., 41, 468–470 (1990).

    MATH  Google Scholar 

  6. Martin-Reyes F. I. and Sawyer E., “Weighted inequalities for Riemann-Liouville fractional integrals of order one and greater,” Proc. Amer. Math. Soc., 106, No. 3, 727–733 (1989).

    Article  MATH  MathSciNet  Google Scholar 

  7. Bloom S. and Kerman R., “Weighted norm inequalities for operators of Hardy type,” Proc. Amer. Math. Soc., 113, No. 1, 135–141 (1991).

    Article  MATH  MathSciNet  Google Scholar 

  8. Berezhnoi E. I., “Two-weighted estimates for a class of integral operators,” Trudy Mat. Inst. Steklov., 201, 14–26 (1992).

    Google Scholar 

  9. Oinarov R., “Two-sided norm estimates for certain classes of integral operators,” Trudy Mat. Inst. Steklov., 204, 240–250 (1993).

    Google Scholar 

  10. Oinarov R., “Weighted inequalities for one class of integral operators,” Dokl. Akad. Nauk SSSR, 319, No. 5, 1076–1076 (1991).

    Google Scholar 

  11. Edmunds D. E. and Stepanov V. D., “On the singular numbers of certain Volterra integral operators,” J. Funct. Anal., 134, No. 1, 222–246 (1995).

    Article  MATH  MathSciNet  Google Scholar 

  12. Stepanov V. D., “Weighted norm inequalities for integral operators and related topics,” in: Nonlinear Analysis, Function Spaces and Applications: Proceedings of the Spring School Held in Prague, May 23–28, 1994, Prague, 1994, 5, pp. 139–175.

    Google Scholar 

  13. Stepanov V. D., “On the lower bounds for Schatten-von Neumann norms of certain Volterra integral operators,” J. London Math. Soc., 61, No. 2, 905–922 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  14. Stepanov V. D. and Ushakova E. P., “On integral operators with variable integration limits, ” Trudy Mat. Inst. Steklov., 232, 298–317 (2001).

    MathSciNet  Google Scholar 

  15. Baiarystanov A. O., “On one generalization of V. D. Stepanov’s inequality and their applications, ” submitted to KazNIINTI Ka93, 1993, No. 3975.

  16. Baiarystanov A. O., “A two-sided weighted estimate of operators of weighted multiple integration, ” submitted to KazNIINTI Ka96, 1996, No. 7183.

  17. Baideldinov B. L. and Oinarov R., “Two-weighted estimation for operators of fractional integration operators of composition type,” Dokl. NAN RK, No. 6, 16–22 (1996).

  18. Oinarov R. and Sagindykov B. O., “An estimate of a weighted multiple integration operator on the cone of monotone functions (1 < q < p < ∞),” Nauka i Obrazovanie Yuzhnogo Kazakhstana Ser. Mat., Inform., 2, No. 7, 53–60 (1998).

    Google Scholar 

  19. Sagindykov B. O., “An upper estimate for a norm of one class of integral operators,” in: Proceedings of the International Conference “Science and Education,” Shymkent, 1998, pp. 50–54.

  20. Dyn’kin E. M. and Osilenker B. P., “Weighted estimates for singular integrals and some of their applications,” in: Mathematical Analysis [in Russian], VINITI, Moscow, 1983, Vol. 21, pp. 42–129 (Itogi Nauki i Tekhniki).

    Google Scholar 

  21. Krasnosel’skii M. A. et al., Integral Operators in Spaces of Summable Functions [in Russian], Nauka, Moscow (1966).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Gumilev Eurasian National University, Astana, Kazakhstan

    R. Oinarov

Authors
  1. R. Oinarov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to R. Oinarov.

Additional information

__________

Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 48, No. 5, pp. 1100–1115, September–October, 2007.

Original Russian Text Copyright © 2007 Oĭnarov R.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Oinarov, R. Boundedness and compactness of Volterra type integral operators. Sib Math J 48, 884–896 (2007). https://doi.org/10.1007/s11202-007-0091-4

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11202-007-0091-4

Keywords

Search

Navigation