Hostname: page-component-77c89778f8-5wvtr Total loading time: 0 Render date: 2024-07-20T20:47:57.191Z Has data issue: false hasContentIssue false
  • English
  • Français

A note on UMD spaces and transference in vector-valued function spaces*

Published online by Cambridge University Press:  20 January 2009

N. H. Asmar ,
B. P. Kelly  and
S. Montgomery-Smith
N. H. Asmar
Affiliation:
Department of MathematicsUniversity of Missouri-ColumbiaColumbia, Missouri 65211, U.S.A.
B. P. Kelly
Affiliation:
Department of MathematicsUniversity of Missouri-ColumbiaColumbia, Missouri 65211, U.S.A.
S. Montgomery-Smith
Affiliation:
Department of MathematicsUniversity of Missouri-ColumbiaColumbia, Missouri 65211, U.S.A.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A Banach space X is called an HT space if the Hilbert transform is bounded from Lp(X) into Lp(X), where 1 < p < ∞. We introduce the notion of an ACF Banach space, that is, a Banach space X for which we have an abstract M. Riesz Theorem for conjugate functions in Lp(X), 1 < p < ∞. Berkson, Gillespie and Muhly [5] showed that X ∈ HT ⇒ X ∈ ACF. In this note, we will show that X ∈ ACF ⇒ X ∈ UMD, thus providing a new proof of Bourgain's result X ∈ HT ⇒ X ∈ UMD.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 39 , Issue 3 , October 1996 , pp. 485 - 490
Copyright
Copyright © Edinburgh Mathematical Society 1996

Footnotes

*

The work of the first and third authors was partially funded by NSF grants. The second and third authors' work was partially funded by the University of Missouri Research Board.

References

REFERENCES

1. Asmar, N., Berkson, E. and Gillespie, T. A., Distributional control and generalized analyticity, Integral Equations and Operator Theory 14 (1991), 311–341.CrossRefGoogle Scholar
2. Asmar, N., Berkson, E. and Gillespie, T. A., Representations of groups with ordered duals and generalized analyticity, J. Funct. Anal. 90 (1990), 206–235.CrossRefGoogle Scholar
3. Asmar, N. and Hewitt, E., Marcel Riesz's theorem on conjugate Fourier series and its descendants, in Proceedings, Analysis Conference, Singapore, 1986 (Choy, S. T. L. et al. , eds., Elsevier Science, New York, 1988), 1–56.Google Scholar
4. Asmar, N. and Montgomery-Smith, S., Hahn's Embedding Theorem for orders and harmonic analysis on groups with ordered duals, Colloq. Math. 70 (1996), 235–252.CrossRefGoogle Scholar
5. Berkson, E., Gillespie, T. A. and Muhly, P. S., Generalized analyticity in UMD spaces, Ark. Math. 27 (1989), 1–14.CrossRefGoogle Scholar
6. Bourgain, J., Some remarks on Banach spaces in which martingale difference sequences are unconditional, Ark. Math. 21 (1983), 163–168.CrossRefGoogle Scholar
7. Burkholder, D. L., A geometrical characterization of Banach spaces in which martingale difference sequences are unconditional, Ann. Probab. 9 (1981), 997–1011.CrossRefGoogle Scholar
8. Burkholder, D. L., A geometric condition that implies the existence of certain singular integrals of Banach-space-valued functions, in Proceedings, Conference on Harmonic Analysis in Honor of A. Zygmund, Chicago, 1981 (Becker, W. et al. , eds., Wadsworth, Belmont, CA, 1983), 270–286.Google Scholar
9. Burkholder, D. L., Martingales and Fourier analysis in Banach spaces, in C.I.M.E. Lectures, Varenna, Italy, 1985 (Lecture Notes in Mathematics, 1206, 1986), 61–108.Google Scholar
10. Calderón, A. P., Ergodic theory and translation-invariant operators, Proc. Nat. Acad. Sci., U.S.A. 59 (1968), 349–353.CrossRefGoogle ScholarPubMed
11. Coifman, R. R. and Weiss, G., Transference methods in analysis (Regional Conference Series in Math., 31, Amer. Math. Soc., Providence, R. I., 1977).CrossRefGoogle Scholar
12. Fuchs, L., Partially ordered algebraic systems (Pergamon Press, Oxford, New York, 1960).Google Scholar
13. Kelly, B. P., Distributional controlled representations acting on vector-valued functions spaces, Doctoral Dissertation, University of Missouri, 1994.Google Scholar
14. Maurey, B., Système de Haar (Séminaire Maurey-Schwartz, 1974–1975, École Poly-technique, Paris, 1975).Google Scholar