Skip to main content
SpringerLink
Log in

Pseudo-Differential Operators on Matrix Weighted Besov–Triebel–Lizorkin Spaces

  • Original Paper
  • Published:
Bulletin of the Iranian Mathematical Society Aims and scope Submit manuscript

Abstract

We characterize the matrix-weighted Triebel–Lizorkin spaces and Besov spaces by Peetre maximal function and approximation. Using these characterizations, we obtain the boundedness of pseudo-differential operators with symbol in Hörmander’s class on matrix weighted Besov and Triebel–Lizorkin 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

Data availability

Our manuscript has no associated data.

References

  1. Abels, H.: De Gruyter Graduate Lectures. De Gruyter, Berlin (2012)

    Google Scholar 

  2. Bu, F., Yang, D., Yuan, W.: Real-variable characterizations and their applications of matrix-weighted Besov spaces on spaces of homogeneous type, Math. Z. (to appear)

  3. Dintelmann, P.: On the boundedness of pseudo-differential operators on weighted Besov–Triebel spaces. Math. Nachr. 183, 43–53 (1997)

    Article  MathSciNet  Google Scholar 

  4. Fefferman, C., Stein, E.M.: Some maximal inequalities. Am. J. Math. 93, 107–115 (1971)

    Article  MathSciNet  Google Scholar 

  5. Frazier, M., Roudenko, S.: Matrix-weighted Besov spaces and conditions of \(A_{p}\) type for \(0<p\le 1\). Indiana Univ. Math. J. 53(5), 1225–1254 (2004)

    Article  MathSciNet  Google Scholar 

  6. Frazier, M., Roudenko, S.: Littlewood–Paley theory for matrix-weighted function spaces. Math. Ann. 380, 487–537 (2021)

    Article  MathSciNet  Google Scholar 

  7. Ghaemi, M.B., Birgani, M.J., Wong, M.W.: Characterizations of nuclear pseudo-differential operators on \({\mathbb{S} }^{1}\) with applications to adjoints and products. J. Pseudo-Differ. Oper. Appl. 8, 191–201 (2017)

    Article  MathSciNet  Google Scholar 

  8. Goldberg, M.: Matrix \(A_{p}\) weights via maximal functions. Pac. J. Math. 211(2), 201–220 (2003)

    Article  Google Scholar 

  9. Grubb, G.: Distributions and Operators, Graduate Texts in Mathematics. Springer, New York (2009)

    Google Scholar 

  10. Haroske, D.D., Skandera, P.: Embeddings of doubling weighted Besov spaces. In: Function Spaces X, Banach Center Publicatons, pp. 105–119. Polish Academy of Sciences Institute of Mathematics, Warsaw (2014)

  11. Hörmander, L.: Pseudo-differential operators. Commun. Pure Appl. Math. 18, 501–517 (1965)

    Article  MathSciNet  Google Scholar 

  12. Itkin, A.: Pricing Derivatives Under Lévy Models. Modern Finite-Difference and Pseudo-Differential Operators Approach. Birkhäuser, New York (2017)

    Book  Google Scholar 

  13. Kohn, J.J., Nirenberg, L.: An algebra of pseudo-differential operators. Commun. Pure Appl. Math. 18, 269–305 (1965)

    Article  MathSciNet  Google Scholar 

  14. Kumanp-Go, H.: Pseudo-Differential Operators. MIT Press, Cambridge (1974)

    Google Scholar 

  15. Moussai, M., Djeriou, A.: Boundedness of some pseudo-differential operators on generalized Triebel–Lizorkin spaces. Analysis (Munich) 31, 13–29 (2011)

    MathSciNet  Google Scholar 

  16. Park, B.J.: On the boundedness of pseudo-differential operators on Triebel–Lizorkin and Besov spaces. J. Math. Anal. Appl. 461, 544–576 (2018)

    Article  MathSciNet  Google Scholar 

  17. Park, B.J.: Boundedness of pseudo-differential operators of type (0,0) on Triebel–Lizorkin and Besov spaces. Bull. Lond. Math. Soc. 51(6), 1039–1060 (2019)

    Article  MathSciNet  Google Scholar 

  18. Park, B.J.: Sharp estimates for pseudo-differential operators of type (1,1) on Triebel–Lizorkin and Besov spaces. Stud. Math. 250(2), 129–162 (2020)

    Article  MathSciNet  Google Scholar 

  19. Roudenko, S.: Matrix-weighted Besov spaces. Trans. Am. Math. Soc. 355(1), 273–314 (2003)

    Article  MathSciNet  Google Scholar 

  20. Roudenko, S.: Duality of matrix-weighted Besov spaces. Stud. Math. 160(2), 129–156 (2004)

    Article  MathSciNet  Google Scholar 

  21. Runst, T.: Pseudodifferential operators of the exotic class \(L_{1,1}^{0}\) in spaces of Besov and Triebel–Lizorkin type. Ann. Glob. Anal. Geom. 3, 13–28 (1985)

    Article  Google Scholar 

  22. Sawano, Y.: Theory of Besov spaces. In: Developments in Mathematics, vol. 56. Springer, Singapore (2018)

  23. Srivastava, H.M., Upadhyay, S.K., Khatterwani, K.: A family of pseudo-differential operators on the Schwartz space associated with the fractional Fourier transform. Russ. J. Math. Phys. 24, 534–543 (2017)

    Article  MathSciNet  Google Scholar 

  24. Sugimoto, M.: Pseudo-differential operators on Besov spaces. Tsukuba J. Math. 12, 43–63 (1988)

    Article  MathSciNet  Google Scholar 

  25. Taylor, M. E. Pseudodifferential Operators and Nonlinear PDE. In: Progress in Mathematics, vol. 100. Birkhäuser, Boston (1991)

  26. Taylor, M. E. Tools for PDE: pseudodifferential operators, paradifferential operators, and layer potentials. In: Mathematical Surveys and Monographs, vol. 81. American Mathematical Society, Providence (2000)

  27. Toft, J.: The Bargmann transform on modulation and Gelfand–Shilov spaces, with applications to Toeplitz and pseudo-differential operators. J. Pseudo-Differ. Oper. Appl. 3, 145–227 (2012)

    Article  MathSciNet  Google Scholar 

  28. Treil, S., Volberg, A.: Wavelets and the angle between past and future. J. Funct. Anal. 143, 269–308 (1997)

    Article  MathSciNet  Google Scholar 

  29. Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. VEB Deutscher Verlag der Wissenschaften, Berlin/North Holland Publishing Company, Amsterdam (1978)

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

    Book  Google Scholar 

  31. Unterberger, A.: Pseudodifferential Methods in Number Theory. Pseudo-Differential Operators Theory and Applications, vol. 13. Birkhäuser, Cham (2018)

  32. Wang, Q., Yang, D., Zhang, Y.: Real-variable characterizations and their applications of matrix-weighted Triebel–Lizorkin spaces. https://arxiv.org/pdf/2207.08474.pdf

  33. Wong, M. W. An introduction to pseudo-differential operators, 3rd edn. In: Series on Analysis, Applications and Computation, vol. 6. World Scientific Publishing Co. Pte. Ltd., Hackensack (2014)

Download references

Acknowledgements

We are grateful to referees for their carefully reading and suggestions. The work is supported by the National Natural Science Foundation of China (Grant No. 12161022).

Author information

Authors and Affiliations

  1. College of Mathematics and Statistics, Hainan Normal University, Haikou, 571158, Hainan, China

    Tengfei Bai

  2. Center for Applied Mathematics of Guangxi, Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation, School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin, 541004, Guangxi, China

    Jingshi Xu

Authors
  1. Tengfei Bai
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jingshi Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jingshi Xu.

Ethics declarations

Conflict of interest

The authors declare that there is no conflict of interest.

Additional information

Communicated by Saeid Maghsoudi.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bai, T., Xu, J. Pseudo-Differential Operators on Matrix Weighted Besov–Triebel–Lizorkin Spaces. Bull. Iran. Math. Soc. 50, 31 (2024). https://doi.org/10.1007/s41980-024-00869-w

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s41980-024-00869-w

Keywords

Mathematics Subject Classification

Search

Navigation