Tohoku Mathematical Journal

The dyadic structure and atomic decomposition of {$Q$} spaces in several real variables

Galia Dafni and Jie Xiao

Full-text: Open access


This paper contains several results relating $Q$ spaces in several real variables with their dyadic counterparts, which are analogues of theorems for BMO and for $Q$ spaces on the circle. In addition, it gives an atomic (or quasi-orthogonal) decomposition for these $Q$ spaces in terms of the same type of atoms used to decompose BMO.

Article information

Tohoku Math. J. (2), Volume 57, Number 1 (2005), 119-145.

First available in Project Euclid: 11 April 2005

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 42B35: Function spaces arising in harmonic analysis
Secondary: 46E30: Spaces of measurable functions (Lp-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 47B38: Operators on function spaces (general)

Q spaces BMO dyadic structure martingales fractional Carleson measures atomic decomposition quasi-orthogonal decomposition


Dafni, Galia; Xiao, Jie. The dyadic structure and atomic decomposition of {$Q$} spaces in several real variables. Tohoku Math. J. (2) 57 (2005), no. 1, 119--145. doi:10.2748/tmj/1113234836.

Export citation


  • R. Aulaskari, J. Xiao and R. H. Zhao, On subspaces and subsets of BMOA and UBC, Analysis 15 (1995) 101--121.
  • S.-Y. A. Chang and R. Fefferman, A continuous version of duality of $H^1$ and BMO, Ann. of Math. 112 (1980), 179--201.
  • G. Dafni and J. Xiao, Some new tent spaces and duality theorems for fractional Carleson measures and $Q_\alpha (\Rn)$, J. Funct. Anal. 208 (2004), 377--422.
  • M. Essén, S. Janson, L. Peng and J. Xiao, Q spaces of several real variables, Indiana Univ. Math. J. 49 (2000), 575--615.
  • M. Essén and J. Xiao, Some results on $Q_p$ spaces, $0 < p < 1$, J. Reine Angew. Math. 485 (1997), 173--195.
  • C. Fefferman, Characterizations of bounded mean oscillation, Bull. Amer. Math. Soc. 77 (1971), 587--588.
  • C. Fefferman and E. M. Stein, $H^p$ spaces of several variables, Acta Math. 129 (1972), 137--193.
  • M. Frazier and B. Jawerth, Decomposition of Besov spaces, Indiana Univ. Math. J. 34 (1985), 777--799.
  • M. Frazier, B. Jawerth and G. Weiss, Littlewood-Paley Theory and the Study of Function Spaces, CBMS Regional Conference Series in Math., 79, American Math. Society, 1991.
  • J. Garnett and P. W. Jones, BMO from dyadic BMO, Pacific J. Math. 99 (1982), 351--371.
  • S. Janson, On the space $Q_p$ and its dyadic counterpart, Complex analysis and differential equations (Uppsala, 1997), 194--205, Acta Univ. Upsaliensis Skr. Uppsala Univ. C Organ. Hist. 64, Uppsala Univ., Uppsala 1999.
  • F. John and L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14 (1961), 415--426.
  • Y. Meyer, Wavelets and operators; Translated from the 1990 French original by D. H. Salinger, Cambridge Studies in Advanced Mathematics, vol. 37, Cambridge University Press, Cambridge, 1992.
  • R. Rochberg and S. Semmes, A decomposition theorem for BMO and applications, J. Funct. Anal. 67 (1986), 228--263.
  • E. M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Univ. Press, Princeton, New Jersey, 1993.
  • A. Uchiyama, A constructive proof of the Fefferman-Stein decomposition of $\b$, Acta Math. 148 (1982), 215--241.
  • Z. Wu and C. Xie, Decomposition theorems for $Q_p$ spaces, Ark. Mat. 40 (2002), 383--401.