Potential Analysis

, Volume 29, Issue 1, pp 1–16 | Cite as

Univalent Interpolation in Besov Spaces and Superposition into Bergman Spaces

  • Stephen M. Buckley
  • Dragan VukotićEmail author


We characterize the superposition operators from an analytic Besov space or the little Bloch space into a Bergman space in terms of the order and type of the symbol. We also determine when these operators are continuous or bounded and discuss their Montel compactness. Along the way, we prove new non-centered Trudinger–Moser inequalities and solve the problem of interpolation by univalent functions in analytic Besov spaces.


Superposition operator Trudinger–Moser inequalities Analytic Besov spaces Bergman spaces Little Bloch space Montel compactness Univalent interpolation Entire functions 

Mathematics Subject Classifications (2000)

47H30 31C25 30H05 


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  1. 1.
    Álvarez, V., Márquez, M.A., Vukotić, D.: Superposition operators between the Bloch space and Bergman spaces. Ark. Mat. 42(2), 205–216 (2004)CrossRefMathSciNetzbMATHGoogle Scholar
  2. 2.
    Anderson, J.M., Clunie, J., Pommerenke, Ch.: On Bloch functions and normal functions. J. Reine Angew. Math. 270, 12–37 (1974)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Appell, J., Zabrejko, P.P.: Nonlinear Superposition Operators. Cambridge University Press, Cambridge (1990)zbMATHGoogle Scholar
  4. 4.
    Arazy, J., Fisher, S.D., Peetre, J.: Möbius invariant function spaces. J. Reine Angew. Math. 363, 110–145 (1985)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Axler, S.J. Zero multipliers of Bergman spaces. Canad. Math. Bull. 28 (1985), 237–242.MathSciNetzbMATHGoogle Scholar
  6. 6.
    Beurling, A.: Études Sur un Problème de Majoration. Thèse pour le doctorat, Almquist & Wieksell, Upsalla (1933)zbMATHGoogle Scholar
  7. 7.
    Boas, R.P., Jr.: Entire Functions. Academic Press Inc., New York (1954)zbMATHGoogle Scholar
  8. 8.
    Böe, B.: Interpolating sequences for Besov spaces. J. Funct. Anal. 192, 319–341 (2002)CrossRefMathSciNetzbMATHGoogle Scholar
  9. 9.
    Buckley, S.M., Fernández, J.L., Vukotić, D.: Superposition operators on Dirichlet type spaces. Report Univ. Jyväskylä 83, 41–61 (2001) (papers on analysis: dedicated to Olli Martio on the occasion of his 60th Birthday, editors: Heinonen, J., Kilpeläinen, T., and Koskela, P.). Available from Google Scholar
  10. 10.
    Buckley, S.M., O’Shea, J.: Weighted Trudinger-type inequalities. Indiana Univ. Math. J. 48, 85–114 (1999)CrossRefMathSciNetzbMATHGoogle Scholar
  11. 11.
    Buckley, S.M., Vukotić, D.: Superposition operators and the order and type of entire functions. In: Recent Advances in Operator-related Function Theory. Contemporary Mathematics, vol. 393, pp. 51–57. American Mathematical Society, Providence, RI (2006)Google Scholar
  12. 12.
    Cámera, G.: Nonlinear superposition on spaces of analytic functions. In: Harmonic analysis and operator theory (Caracas, 1994). Contemporary Mathematics, vol. 189, pp.103–116. American Mathematical Society, Providence, RI (1995)Google Scholar
  13. 13.
    Cámera, G., Giménez, J.: Nonlinear superposition operators acting on Bergman spaces. Compositio Math. 93, 23–35 (1994)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Chang, S.-Y.A.: The Moser–Trudinger inequality and applications to some problems in conformal geometry. In: Hardt, R., Wolf, M. (eds.) Nonlinear Partial Differential Equations in Differential Geometry. Park City, Utah 1992. IAS/Park City Math. Ser., vol. 2, pp. 65–125. American Mathematical Society, Providence, RI (1996)Google Scholar
  15. 15.
    Chang, S.-Y.A., Marshall, D.E.: On a sharp inequality concerning the Dirichlet integral. Amer. J. Math. 107, 1015–1033 (1985)CrossRefMathSciNetzbMATHGoogle Scholar
  16. 16.
    Donaire, J.J., Girela, D., Vukotić, D.: On univalent functions in some Möbius invariant spaces. J. Reine Angew. Math. 553, 43–72 (2002)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Duren, P.L.: Theory of H  p Spaces. Academic Press, New York, 1970. Reprint: Dover, Mineola, NY (2000)Google Scholar
  18. 18.
    Duren, P.L., Schuster, A.P.: Bergman spaces. In: Math. Surveys and Monographs, vol. 100. American Mathematical Society, Providence, RI (2004)Google Scholar
  19. 19.
    Hardy, G.H., Littlewood, J.E.: Some properties of fractional integrals. II. Math. Z. 34, 403–439 (1932)CrossRefMathSciNetGoogle Scholar
  20. 20.
    Hedenmalm, H., Korenblum, B., Zhu, K.: Theory of Bergman spaces. In: Graduate Texts in Mathematics, vol. 199. Springer, New York (2000)Google Scholar
  21. 21.
    Holland, F., Walsh, D.: Growth estimates for functions in the Besov spaces A p. Proc. Roy. Irish Acad. Sect. A 88, 1–18 (1988)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Marhsall, D.E.: A new proof of a sharp inequality concerning the Dirichlet integral. Ark. Mat. 27(1), 131–137 (1989)CrossRefMathSciNetGoogle Scholar
  23. 23.
    Pommerenke, Ch.: Boundary Behaviour of Conformal Maps. Springer-Verlag, Berlin (1992)zbMATHGoogle Scholar
  24. 24.
    Titchmarsh, E.C.: The Theory of Functions, 2nd edn. Oxford University Press, Oxford (1988)Google Scholar
  25. 25.
    Walsh, D.: A property of univalent functions in A p. Glasgow Math. J. 42, 121–124 (2000)CrossRefzbMATHGoogle Scholar
  26. 26.
    Xiong, C.: Superposition operators between Q p spaces and Bloch-type spaces. Complex Var. Theory Appl. 50(12), 935–938 (2005)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Xu, W.: Superposition operators on Bloch-type spaces. Comput. Methods Funct. Theory 7(2), 501–507 (2007)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Zhu, K.: Analytic Besov spaces. J. Math. Anal. Appl. 157, 318–336 (1991)CrossRefMathSciNetzbMATHGoogle Scholar

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© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Department of MathematicsNational University of IrelandMaynoothIreland
  2. 2.Departamento de Matemáticas, C-XVUniversidad Autónoma de MadridMadridSpain

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