Results in Mathematics

, Volume 72, Issue 1–2, pp 813–841 | Cite as

Equivalent Quasi-Norms of Besov–Triebel–Lizorkin-Type Spaces via Derivatives

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Abstract

In this paper, applying the Peetre maximal function characterizations and the boundedness of Fourier multipliers on Besov-type and Triebel–Lizorkin-type spaces, as well as Besov–Morrey and Triebel–Lizorkin–Morrey spaces, the authors present some equivalent quasi-norms of these spaces in terms of derivatives.

Keywords

Besov space Triebel–Lizorkin space Morrey space Peetre maximal function Fourier multiplier Lifting operator 

Mathematics Subject Classification

Primary 46E35 Secondary 42B15 

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References

  1. 1.
    Cho, Y.K.: Continuous characterization of the Triebel–Lizorkin spaces and Fourier multipliers. Bull. Korean Math. Soc. 47, 839–857 (2010)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Cho, Y.K., Kim, D.: A Fourier multiplier theorem on the Besov–Lipschitz spaces. Korean J. Math. 16, 85–90 (2008)Google Scholar
  3. 3.
    Fefferman, C., Stein, E.M.: Some maximal inequalities. Am. J. Math. 93, 107–115 (1971)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Frazier, M., Jawerth, B.: A discrete transform and decompositions of distribution spaces. J. Funct. Anal. 93, 34–170 (1990)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Kozono, H., Yamazaki, M.: Semilinear heat equations and the Navier–Stokes equation with distributions as initial data. C. R. Acad. Sci. Paris Sér. I Math. 317, 1127–1132 (1993)MathSciNetMATHGoogle Scholar
  6. 6.
    Liang, Y., Sawano, Y., Ullrich, T., Yang, D., Yuan, W.: New characterizations of Besov–Triebel–Lizorkin–Hausdorff spaces including coorbits and wavelets. J. Fourier Anal. Appl. 18, 1067–1111 (2012)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Sawano, Y., Tanaka, H.: Decompositions of Besov–Morrey spaces and Triebel–Lizorkin–Morrey spaces. Math. Z. 257, 871–905 (2007)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Sawano, Y., Yang, D., Yuan, W.: New applications of Besov-type and Triebel–Lizorkin-type spaces. J. Math. Anal. Appl. 363, 73–85 (2010)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Sickel, W.: Smoothness spaces related to Morrey spaces—a survey I. Eurasian Math. J. 3, 110–149 (2012)MathSciNetMATHGoogle Scholar
  10. 10.
    Stein, E.M.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, Princeton (1993)MATHGoogle Scholar
  11. 11.
    Tang, L., Xu, J.: Some properties of Morrey type Besov–Triebel spaces. Math. Nachr. 278, 904–917 (2005)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Triebel, H.: Fourier Analysis and Function Spaces, Teubner-Texte Math, vol. 7. Teubner, Leipzig (1977)Google Scholar
  13. 13.
    Triebel, H.: Theory of Function Spaces, Monographs in Math., vol. 78. Birkhäuser Verlag, Basel (1983)CrossRefGoogle Scholar
  14. 14.
    Triebel, H.: Theory of Function Spaces. II, Monographs in Math, vol. 84. Birkhäuser Verlag, Basel (1992)CrossRefGoogle Scholar
  15. 15.
    Triebel, H.: Tempered Homogeneous Function Spaces. European Mathematical Society (EMS), Zürich, EMS Series of Lectures in Mathematics (2015)Google Scholar
  16. 16.
    Ullrich, T.: Continuous characterizations of Besov–Lizorkin–Triebel spaces and new interpretations as coorbits. J. Funct. Spaces Appl. Art. ID 163213 (2012)Google Scholar
  17. 17.
    Yang, D., Yuan, W.: A new class of function spaces connecting Triebel–Lizorkin spaces and \(Q\) spaces. J. Funct. Anal. 255, 2760–2809 (2008)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Yang, D., Yuan, W.: New Besov-type spaces and Triebel–Lizorkin-type spaces including \(Q\) spaces. Math. Z. 265, 451–480 (2010)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Yang, D., Yuan, W.: Characterizations of Besov-type and Triebel–Lizorkin-type spaces via maximal functions and local means. Nonlinear Anal. 73, 3805–3820 (2010)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Yang, D., Yuan, W.: Relations among Besov-type spaces, Triebel–Lizorkin-type spaces and generalized Carleson measure spaces. Appl. Anal. 92, 549–561 (2013)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Yang, D., Yuan, W., Zhuo, C.: Fourier multipliers on Triebel–Lizorkin-type. J. Funct. Spaces Appl. Art. ID 431016 (2012)Google Scholar
  22. 22.
    Yuan, W., Sickel, W., Yang, D.: Morrey and Campanato Meet Besov, Lizorkin and Triebel. Lecture Notes in Math, vol. 2005. Springer-Verlag, Berlin (2010)Google Scholar

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© Springer International Publishing 2017

Authors and Affiliations

  1. 1.School of Mathematical SciencesBeijing Normal UniversityBeijingPeople’s Republic of China
  2. 2.Laboratory of Mathematics and Complex SystemsMinistry of EducationBeijingPeople’s Republic of China

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