Molecular decomposition and Fourier multipliers for holomorphic Besov and Triebel–Lizorkin spaces

  • G. Cleanthous
  • A. G. Georgiadis
  • M. NielsenEmail author


Smooth molecular decompositions for holomorphic Besov and Triebel–Lizorkin spaces on the unit disk of the complex plane are constructed. The decompositions are used to obtain a boundedness result for Fourier multipliers. As further applications, we provide equivalent norms for the spaces under consideration, we consider the implications of the results on Hardy and Hardy–Sobolev spaces, and we study boundedness of coefficient multipliers.


Besov spaces Distributions Fourier multipliers Hardy spaces Hardy–Sobolev spaces Holomorphic functions Molecular decomposition Triebel–Lizorkin spaces 

Mathematics Subject Classification

30H25 42A16 (primary) and 30B30 30B40 30H10 42A05 42A45 (secondary) 



  1. 1.
    Bownik, M.: Atomic and molecular decompositions of anisotropic Besov spaces. Math. Z. 250(3), 539–571 (2005)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bownik, M., Ho, K.P.: Atomic and molecular decompositions of anisotropic Triebel–Lizorkin spaces. Trans. Am. Math. Soc. 358(4), 1469–1510 (2006)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Buckley, S.M., Ramanujan, M.S., Vukotić, D.: Bounded and compact multipliers between Bergman and Hardy spaces. Integral Equ. Oper. Theory 35(1), 1–19 (1999)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Choe, B.R., Koo, H., Smith, W.S.: Composition operators acting on holomorphic Sobolev spaces. Trans. Am. Math. Soc. 355(7), 2829–2855 (2003)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Cleanthous, G., Georgiadis, A.G., Nielsen, M.: Molecular decomposition of anisotropic homogeneous mixed-norm spaces with applications to the boundedness of operators. Appl. Comput. Harmonic Anal. (2017).
  6. 6.
    Dai, F., Gogatishvili, A., Yang, D., Yuan, W.: Characterizations of Besov and Triebel–Lizorkin spaces via averages on balls. J. Math. Anal. Appl. 433(2), 1350–1368 (2016)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Duren, P.L., Shields, A.L.: Coefficient multipliers of \(H^p\) and \(B^p\) spaces. Pac. J. Math. 32, 69–78 (1970)CrossRefGoogle Scholar
  8. 8.
    Flett, T.M.: Lipschitz spaces of functions on the circle and the disk. J. Math. Anal. Appl. 39, 125–158 (1972)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Frazier, M., Jawerth, B.: Decomposition of Besov spaces. Indiana Univ. Math. J. 34, 777–799 (1985)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Frazier, M., Jawerth, B.: A discrete transform and decompositions of distribution. J. Funct. Anal. 93, 34–170 (1990)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Grafakos, L.: Classical Fourier Analysis. Graduate Texts in Mathematics, Third edn, p. 249. Springer, New York (2014)zbMATHGoogle Scholar
  12. 12.
    Grafakos, L., Torres, R.: Pseudodifferential operators with homogeneous symbols. Mich. Math. J. 46, 261–269 (1999)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Guliev, V.S., Lizorkin, P.I.: \({\cal{B}}\)- and \({\cal{L}}\)-classes of harmonic and holomorphic functions in the disk, and classes of boundary values. Soviet Math. Dokl. 44, 215–219 (1992)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Ivanov, K., Petrushev, P.: Harmonic Besov and Triebel–Lizorkin spaces on the ball. J. Fourier Anal. Appl. 23(5), 1062–1096 (2017)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Kyriazis, G., Petrushev, P.: Rational bases for spaces of holomorphic functions in the disc. J. Lond. Math. Soc. (2) 89(2), 434–460 (2014)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Kyriazis, G., Petrushev, P., Xu, Y.: Decomposition of weighted Triebel–Lizorkin and Besov spaces on the ball. Proc. Lond. Math. Soc. (3) 97(2), 477–513 (2008)MathSciNetCrossRefGoogle Scholar
  17. 17.
    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(5), 1067–1111 (2012)MathSciNetCrossRefGoogle Scholar
  18. 18.
    MacGregor, T., Zhu, K.: Coefficient multipliers between Bergman and Hardy spaces. Mathematika 42(2), 413–426 (1995)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Narcowich, F., Petrushev, P., Ward, J.: Decomposition of Besov and Triebel–Lizorkin spaces on the sphere. J. Funct. Anal. 238(2), 530–564 (2006)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Ortega, J.M., Fábrega, J.: Holomorphic Triebel–Lizorkin spaces. J. Funct. Anal. 151, 177–212 (1997)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Ortega, J.M., Fábrega, J.: Hardy’s inequality and embeddings in holomorphic Triebel–Lizorkin spaces. Ill. J. Math. 43(4), 733–751 (1999)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Ortega, J.M., Fábrega, J.: Multipliers in Hardy–Sobolev spaces. Integral Equ. Oper. Theory 55(4), 535–560 (2006)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Oswald, P.: On Besov–Hardy–Sobolev spaces of analytic functions in the unit disc. Chech. Math. J. 33(108), 408–426 (1983)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Peetre, J.: New Thoughts on Besov Spaces. Duke University Mathematics Series 1. Department of Mathematics, Duke University, Durham (1976)zbMATHGoogle Scholar
  25. 25.
    Pekarskii, A.A.: Classes of analytic functions defined by best rational approximations in \(H_p\). Mat. Sb. (N.S.) 127(169)(1), 3–20 (1985)zbMATHGoogle Scholar
  26. 26.
    Torres, R.H.: Boundedness Results for Operators with Singular Kernels on Distribution Spaces, vol. 90, 442nd edn. Memoirs of the American Mathematical Society, Providence (1991)zbMATHGoogle Scholar
  27. 27.
    Triebel, H.: Theory of Function Spaces. Monographs in Mathematics, vol. 78. Birkhäuser Verlag, Basel (1983)CrossRefGoogle Scholar
  28. 28.
    Triebel, H.: Periodic spaces of Besov–Hardy–Sobolev type and related maximal inequalities for trigonometrical polynomials. Colloquia Mathematica Societatis János Bolyai, vol. 35, pp. 1201–1209. North-Holland, Amsterdam (1983)Google Scholar
  29. 29.
    Yuan, W., Sickel, W., Yang, D.: Morrey and Campanato Meet Besov, Lizorkin and Triebel. Lecture Notes in Mathematics, vol. 2005. Springer, Berlin (2010)CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of CyprusNicosiaCyprus
  2. 2.Department of Mathematical SciencesAalborg UniversityAalborg EastDenmark

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