Illinois Journal of Mathematics

Embeddings between operator-valued dyadic BMO spaces

Oscar Blasco and Sandra Pott

Full-text: Open access


We investigate a scale of dyadic operator-valued BMO spaces, corresponding to the different yet equivalent characterizations of dyadic BMO in the scalar case. In the language of operator spaces, we investigate different operator space structures on the scalar dyadic BMO space which arise naturally from the different characterizations of scalar BMO. We also give sharp dimensional growth estimates for the sweep of functions and its bilinear extension in some of those different dyadic BMO spaces.

Article information

Illinois J. Math., Volume 52, Number 3 (2008), 799-814.

First available in Project Euclid: 1 October 2009

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 42B30: $H^p$-spaces 42B35: Function spaces arising in harmonic analysis
Secondary: 47B35: Toeplitz operators, Hankel operators, Wiener-Hopf operators [See also 45P05, 47G10 for other integral operators; see also 32A25, 32M15]


Blasco, Oscar; Pott, Sandra. Embeddings between operator-valued dyadic BMO spaces. Illinois J. Math. 52 (2008), no. 3, 799--814. doi:10.1215/ijm/1254403715.

Export citation


  • O. Blasco, Hardy spaces of vector-valued functions: Duality, Trans. Amer. Math. Soc. 308 (1988), 495–507.
  • O. Blasco, Boundary values of functions in vector-valued Hardy spaces and geometry of Banach spaces, J. Funct. Anal. 78 (1988), 346–364.
  • O. Blasco, Dyadic BMO, paraproduct and Haar multipliers, Contemp. Math., vol. 445, Amer. Math. Soc., Providence, RI, pp. 11–18.
  • O. Blasco, Remarks on operator-valued BMO spaces, Rev. Un. Mat. Argentina 45 (2005), 63–78.
  • O. Blasco and S. Pott, Dyadic BMO on the bidisk, Rev. Mat. Iberoamericana 21 (2005), 483–510.
  • O. Blasco and S. Pott, Operator-valued dyadic BMO spaces, J. Oper. Theory, to appear.
  • J. Bourgain, Vector-valued singular integrals and the $H^1$-BMO duality, Probability theory and harmonic analysis, Cleveland, Ohio 1983, Monographs and Textbooks in Pure and Applied Mathematics, vol. 98, Dekker, New York, 1986.
  • E. G. Effros and Z. J. Ruan, Operator spaces, London Mathematical Society Monographs, vol. 23, Oxford Univ. Press, New York, 2000.
  • A. M. Garsia, Martingale inequalities: Seminar Notes on recent progress, Benjamin, Reading, 1973.
  • T. A. Gillespie, S. Pott, S. Treil and A. Volberg, Logarithmic growth for matrix martingale transforms, J. Lond. Math. Soc. (2) 64 (2001), 624–636.
  • B. Jacob and J. R. Partington, The Weiss conjecture on admissibility of observation operators for contraction semigroups, Integral Equations Operator Theory 40 (2001), 231–243.
  • B. Jacob, J. R. Partington and S. Pott, Admissible and weakly admissible observation operators for the right shift semigroup, Proc. Edinb. Math. Soc. (2) 45 (2002), 353–362.
  • N. H. Katz, Matrix valued paraproducts, J. Fourier Anal. Appl. 300 (1997), 913–921.
  • Y. Meyer, Wavelets and operators, Cambridge Univ. Press, Cambridge, 1992.
  • F. Nazarov, S. Treil and A. Volberg, Counterexample to the infinite dimensional Carleson embedding theorem, C. R. Acad. Sci. Paris 325 (1997), 383–389.
  • F. Nazarov, G. Pisier, S. Treil and A. Volberg, Sharp estimates in vector Carleson imbedding theorem and for vector paraproducts, J. Reine Angew. Math. 542 (2002), 147–171.
  • T. Mei, Operator valued Hardy spaces, Mem. Amer. Math. Soc., vol. 188.
  • T. Mei, Notes on matrix valued paraproducts, Indiana Univ. Math. J. 55 (2006), 747–760.
  • M. C. Pereyra, Lecture notes on dyadic harmonic analysis, Second summer school in analysis and mathematical physics (Cuernavaca, 2000), Contemp. Math., vol. 289, Amer. Math. Soc., Providence, RI, 2001, pp. 1–60.
  • S. Petermichl, Dyadic shifts and a logarithmic estimate for Hankel operators with matrix symbol, C. R. Acad. Sci. Paris Ser. I Math. 330 (2000), 455–460.
  • G. Pisier, Notes on Banach spaces valued Hp-spaces, non-commutative martingale inequalities and related questions, Preliminary Notes, 2000.
  • G. Pisier and Q. Xu, Non-commutative martingale inequalities, Comm. Math. Phys. 189 (1997), 667–698.
  • S. Pott and C. Sadosky, Bounded mean oscillation on the bidisk and operator BMO, J. Funct. Anal. 189 (2002), 475–495.
  • S. Pott and M. Smith, Vector paraproducts and Hankel operators of Schatten class via $p$-John-Nirenberg theorem, J. Funct. Anal. 217 (2004), 38–78.