Skip to main content
SpringerLink
Log in

The Images of Integration Operators in Weighted Function Spaces

  • Published:
Siberian Mathematical Journal Aims and scope Submit manuscript

Abstract

We obtain conditions for the validity of norm inequalities on the images and preimages of integration operators of positive integer orders in the Besov and Triebel–Lizorkin spaces with local Muckenhoupt weights on \( {𝕉}^{N} \). As application, we establish a connection between the entropy and approximative numbers of the embedding operators and the same characteristics of the integration operators.

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. Samko S.G., Kilbas A.A., and Marichev O.I., Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Amsterdam (1993).

    MATH  Google Scholar 

  2. Triebel H., Theory of Function Spaces, Birkhäuser, Basel (1983).

    Book  MATH  Google Scholar 

  3. Berkolaiko M.Z. and Novikov I.Ya., “On infinitely smooth compactly supported almost-wavelets,” Math. Notes, vol. 55, no. 5, 877–883 (1994).

    Article  MATH  Google Scholar 

  4. Rychkov V.S., “Littlewood–Paley theory and function spaces with \( A_{p}^{\operatorname{loc}} \) weights,” Math. Nachr., vol. 224, 145–180 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  5. Izuki M. and Sawano Y., “Atomic decomposition for weighted Besov/Triebel–Lizorkin spaces with local class of weights,” Math. Nachr., vol. 285, 103–126 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  6. Triebel H., Tempered Homogeneous Function Spaces, European Math. Soc., Zurich (2015).

    Book  MATH  Google Scholar 

  7. Prokhorov D.V., Stepanov V.D., and Ushakova E.P., “The Hardy–Steklov integral operators,” Contemporary Problems of Mathematics, vol. 22, 3–185 (2016).

    Google Scholar 

  8. Ushakova E.P., “Spline wavelet decomposition in weighted function spaces,” Proc. Steklov Inst. Math., vol. 312, 301–324 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  9. Malecka A., “Haar functions in weighted Besov and Triebel–Lizorkin spaces,” J. Approx. Theory, vol. 200, 1–27 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  10. Wojciechowska A., Multidimensional Wavelet Bases in Besov and Lizorkin–Triebel Spaces. Dissertation, Adam Mickiewicz University Poznań, Poznań (2012).

    Google Scholar 

  11. Triebel H., Theory of Function Spaces. II, Birkhäuser, Basel (1992).

    Book  MATH  Google Scholar 

  12. Schott T., “Function spaces with exponential weights. I,” Math. Nahr., vol. 189, 221–242 (1998).

    MathSciNet  MATH  Google Scholar 

  13. Schott T., “Function spaces with exponential weights. II,” Math. Nahr., vol. 196, 231–250 (1998).

    MathSciNet  MATH  Google Scholar 

  14. Wojciechowska A., “Local means and wavelets in function spaces with local Muckenhoupt weights,” in: Function Spaces IX, vol. 92, Banach Center, Warszawa (2011), 399–412.

  15. Chui C.K., An Introduction to Wavelets, Academic, New York (1992).

    MATH  Google Scholar 

  16. Wojtaszczyk P., A Mathematical Introduction to Wavelets, Cambridge University, Cambridge (1997).

    Book  MATH  Google Scholar 

  17. Ushakova E.P. and Ushakova K.E., “Localisation property of Battle–Lemarié wavelet’s sums,” J. Math. Anal. Appl., vol. 461, no. 1, 176–197 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  18. Battle G., “A block spin construction of ondelettes. Part I: Lemarié functions,” Comm. Math. Phys., vol. 110, 601–615 (1987).

    Article  MathSciNet  Google Scholar 

  19. Battle G., “A block spin construction of ondelettes. Part II: QFT connection,” Comm. Math. Phys., vol. 114, 93–102 (1988).

    Article  MathSciNet  Google Scholar 

  20. Lemarié P.G., “Une nouvelle base d’ondelettes de \( L^{2}({𝕉}^{N}) \),” J. Math. Pures Appl., vol. 67, 227–236 (1988).

    MathSciNet  Google Scholar 

  21. Ushakova E.P., “Spline wavelet bases in function spaces with Muckenhoupt weights,” Rev. Mat. Complut., vol. 33, 125–160 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  22. Oinarov O., “Boundedness of integral operators from weighted Sobolev space to weighted Lebesgue space,” Complex Var. Elliptic Equ., vol. 56, no. 10, 1021–1038 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  23. Kantorovich L.V. and Akilov G.P., Functional Analysis, Pergamon, Oxford and New York (1982).

    MATH  Google Scholar 

  24. Carl B. and Stephani I., Entropy, Compactness and the Approximation of Operators, Cambridge University, Cambridge (1990).

    Book  MATH  Google Scholar 

  25. Pietsch A., Eigenvalues and \( s \)-Numbers, Akademie, Leipzig (1987).

    MATH  Google Scholar 

  26. Haroske D. and Triebel H., “Wavelet bases and entropy numbers in weighted function spaces,” Math. Nachr., vol. 278, 108–132 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  27. Haroske D. and Skrzypczak L., “Entropy and approximation numbers of embeddings in function spaces with Muckenhoupt weights. I,” Rev. Mat.Complut., vol. 21, no. 1, 135–177 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  28. Nasyrova M.G. and Ushakova E.P., “Wavelet basis and entropy numbers of Hardy operator,” Anal. Math., vol. 44, no. 4, 543–576 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  29. Edmunds D.E. and Lang J., “Approximation numbers and Kolmogorov widths of Hardy-type operators in a non-homogeneous case,” Math. Nachr., vol. 279, no. 7, 727–742 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  30. Edmunds D.E. and Lang J., “Operators of Hardy type,” J. Comput. Appl. Math., vol. 208, 20–28 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  31. Lang J., “Improved estimates for the approximation numbers of Hardy-type operator,” J. Approx. Theory, vol. 121, no. 1, 61–70 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  32. Lang J., “Estimates for \( n \)-widths of the Hardy-type operators,” J. Approx. Theory, vol. 140, 141–146 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  33. Lang J., Nekvinda A., and Mendez O., “Asymptotic behavior of the approximation numbers of the Hardy-type operator from \( L^{p} \) into \( L^{q} \) (case \( 1<p\leq q\leq 2 \) or \( 2\leq p\leq q<\infty \)),” J. Inequal. Pure Appl. Math., vol. 5, no. 1, Article 18 (2004).

    MATH  Google Scholar 

  34. Lifshits M.A. and Linde W., “Approximation and entropy numbers of Volterra operators with application to Brownian motion,” Mem. Amer. Math. Soc., vol. 157, no. 745, 1–87 (2002).

    MathSciNet  MATH  Google Scholar 

  35. Lomakina E.N. and Stepanov V.D., “Asymptotic estimates for the approximation and entropy numbers of a one-weight Riemann–Liouville operator,” Siberian Adv. Math., vol. 17, no. 1, 1–36 (2007).

    Article  MathSciNet  Google Scholar 

Download references

Funding

The research in Section 5 was supported by the Russian Science Foundation (Grant no. 22–21–00579; https:// rscf.ru/project/22–21–00579/). The results of Sections 4 and 6 were obtained at the Steklov Institute of Mathematics and supported by Grant no. 19–11–00087 (https://rscf.ru/project/19–11–00087/). The rest of the article was carried out in the framework of the State Task with the Ministry of Science and Higher Education of the Russian Federation to the Trapeznikov Institute of Control Sciences and the Computing Center of the Far Eastern Branch of the Russian Academy of Sciences.

Author information

Authors and Affiliations

  1. Trapeznikov Institute of Control Sciences of the Russian Academy of Sciences, Moscow, Russia

    E. P. Ushakova

  2. Steklov Institute of Mathematics of the Russian Academy of Sciences, Moscow, Russia

    E. P. Ushakova

  3. Computing Center of the Far Eastern Branch of the Russian Academy of Sciences, Khabarovsk, Russia

    E. P. Ushakova

Authors
  1. E. P. Ushakova
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to E. P. Ushakova.

Additional information

Translated from Sibirskii Matematicheskii Zhurnal, 2022, Vol. 63, No. 6, pp. 1382–1410. https://doi.org/10.33048/smzh.2022.63.616

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ushakova, E.P. The Images of Integration Operators in Weighted Function Spaces. Sib Math J 63, 1181–1207 (2022). https://doi.org/10.1134/S0037446622060167

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0037446622060167

Keywords

UDC

Search

Navigation