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Some essential properties ofQ p (∂Δ)-spaces

  • Jie Xiao
  • Jie Xiao
Article

Abstract

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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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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