Some essential properties ofQ p (∂Δ)-spaces

  • Jie Xiao
  • Jie Xiao


For p∈(−∞, ∞) letQ p (∂Δ) be the space of all complex-valued functions f on the unit circle ∂Δ satisfying
$$\mathop {\sup }\limits_{I \subset \partial \Delta } \left| I \right|^{ - p} \int_I {\int_I {\frac{{\left| {f(z) - f(w)} \right|^2 }}{{\left| {z - w} \right|^{2 - p} }}\left| {dz} \right|\left| {dw} \right|< \infty } } $$
, where the supremum is taken over all subarcs I ⊃ ∂Δ with the arclength |I|. In this paper, we consider some essential properties ofQ p (∂Δ). We first show that if p>1, thenQ p (∂Δ)=BMO(∂Δ), the space of complex-valued functions with bounded mean oscillation on ∂Δ. Second, we prove that a function belongs toQ p (∂Δ) if and only if it is Möbius bounded in the Sobolev spaceL p 2 (∂Δ). Finally, a characterization ofQ p (∂Δ) is given via wavelets.

Math Subject Classifications

42A45 46E15 

Keywords and Phrases

Qp(∂Δ)-space Sobolev space Möbius boundedness wavelet 


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  1. [1]
    Aulaskari, R., Xiao, J., and Zhao, R. (1995). On subspaces and subsets of BMOA and UBC,Analysis,15, 101–121.MATHMathSciNetGoogle Scholar
  2. [2]
    Carleson, L. (1980). An explicit unconditional basis inH 1,Bull. Sci. Math.,104, 405–416.MATHMathSciNetGoogle Scholar
  3. [3]
    Daubechies, I. (1988). Orthonormal bases of compactly supported wavelets,Comm. Pure Appl. Math.,41, 909–996.MATHMathSciNetGoogle Scholar
  4. [4]
    Essén, M. (1998). Private communication.Google Scholar
  5. [5]
    Essén, M. and Xiao, J. (1997). Some results on Qp spaces, 0<p<1,J. reine angew. Math.,485, 173–195.MATHMathSciNetGoogle Scholar
  6. [6]
    Garnett, J. and Jones, P. (1982). BMO from dyadic BMO,Pacific J. Math.,99, 351–371.MATHMathSciNetGoogle Scholar
  7. [7]
    Janson, S. (1999). On the space Qp and its dyadic counterpart,Acta Univ. Upsaliensis, C:64, Kiselman, C., Ed., Complex Analysis and Differential Equations, 194–205.Google Scholar
  8. [8]
    John, F. and Nirenberg, L. (1961). On functions of bounded mean oscillation.Comm. Pure Appl. Math.,14, 415–426.MATHMathSciNetGoogle Scholar
  9. [9]
    Lemarié, P. and Meyer, Y. (1986). Ondelettes et bases hilbertiennes.Rev. Mat. Iberoamer,2, 1–18.Google Scholar
  10. [10]
    Meyer, Y. (1992).Wavelets and Operators, Cambridge University Press.Google Scholar
  11. [11]
    Nicolau, A. and Xiao, J. (1997). Bounded functions in Möbius invariant Dirichlet spaces,J. Funct. Anal.,150, 383–425.MATHMathSciNetCrossRefGoogle Scholar
  12. [12]
    Stein, E.M. (1993).Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, NJ.MATHGoogle Scholar
  13. [13]
    Strömberg, J.O. (1983). A modified Franklin system and higher-order spline systems onR n as unconditional bases for Hardy spaces, Conference in Harmonic Analysis in Honor of A. Zygmund, Vol. II, Beckner, W., et al., Eds.,Wadsworth Math. Series, 475–493.Google Scholar
  14. [14]
    Twomey, J. B. (1997). Radial variation of functions in Dirichlet-type spaces,Mathematika,44, 267–277.MATHMathSciNetCrossRefGoogle Scholar
  15. [15]
    Wojtaszczyk, P. (1982). The Franklin system is an unconditional basis inH 1,Ark. Math.,20, 293–300.MATHMathSciNetCrossRefGoogle Scholar
  16. [16]
    Wojtaszczyk, P. (1997).A Mathematical Introduction to Wavelets, Cambridge University Press.Google Scholar

Copyright information

© Birkhäuser 2000

Authors and Affiliations

  • Jie Xiao
    • 1
  • Jie Xiao
    • 2
  1. 1.Department of MathematicsPeking UniversityBeijingChina
  2. 2.Institute of AnalysisTU Braunschweig, PK 14Germany

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