# Spectral decompositions and harmonic analysis on UMD spaces

Studia Mathematica (1994)

- Volume: 112, Issue: 1, page 13-49
- ISSN: 0039-3223

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topBerkson, Earl, and Gillespie, T.. "Spectral decompositions and harmonic analysis on UMD spaces." Studia Mathematica 112.1 (1994): 13-49. <http://eudml.org/doc/216134>.

@article{Berkson1994,

abstract = {We develop a spectral-theoretic harmonic analysis for an arbitrary UMD space X. Our approach utilizes the spectral decomposability of X and the multiplier theory for $L_X^p$ to provide on the space X itself analogues of the classical themes embodied in the Littlewood-Paley Theorem, the Strong Marcinkiewicz Multiplier Theorem, and the M. Riesz Property. In particular, it is shown by spectral integration that classical Marcinkiewicz multipliers have associated transforms acting on X.},

author = {Berkson, Earl, Gillespie, T.},

journal = {Studia Mathematica},

keywords = {spectral decomposition; multipliers; Fourier transforms; UMD space; Littlewood-Paley theorem; strong Marcinkiewicz multiplier theorem; Riesz property},

language = {eng},

number = {1},

pages = {13-49},

title = {Spectral decompositions and harmonic analysis on UMD spaces},

url = {http://eudml.org/doc/216134},

volume = {112},

year = {1994},

}

TY - JOUR

AU - Berkson, Earl

AU - Gillespie, T.

TI - Spectral decompositions and harmonic analysis on UMD spaces

JO - Studia Mathematica

PY - 1994

VL - 112

IS - 1

SP - 13

EP - 49

AB - We develop a spectral-theoretic harmonic analysis for an arbitrary UMD space X. Our approach utilizes the spectral decomposability of X and the multiplier theory for $L_X^p$ to provide on the space X itself analogues of the classical themes embodied in the Littlewood-Paley Theorem, the Strong Marcinkiewicz Multiplier Theorem, and the M. Riesz Property. In particular, it is shown by spectral integration that classical Marcinkiewicz multipliers have associated transforms acting on X.

LA - eng

KW - spectral decomposition; multipliers; Fourier transforms; UMD space; Littlewood-Paley theorem; strong Marcinkiewicz multiplier theorem; Riesz property

UR - http://eudml.org/doc/216134

ER -

## References

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- [2] N. Asmar, E. Berkson and T. A. Gillespie, Spectral integration of Marcinkiewicz multipliers, Canad. J. Math. 45 (1993), 470-482. Zbl0817.42005
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- [4] E. Berkson and T. A. Gillespie, Fourier series criteria for operator decomposability, ibid. 9 (1986), 767-789. Zbl0607.47026
- [5] E. Berkson and T. A. Gillespie, Stechkin's theorem, transference, and spectral decompositions, J. Funct. Anal. 70 (1987), 140-170. Zbl0607.47027
- [6] E. Berkson and T. A. Gillespie, Spectral decompositions and vector-valued transference, in: Analysis at Urbana II (Proceedings of Special Year in Modern Analysis at the Univ. of Ill., 1986-87), London Math. Soc. Lecture Note Ser. 138, Cambridge Univ. Press, Cambridge, 1989, 22-51.
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- [8] E. Berkson, T. A. Gillespie and P. S. Muhly, Abstract spectral decompositions guaranteed by the Hilbert transform, Proc. London Math. Soc. (3) 53 (1986), 489-517. Zbl0609.47042
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- [10] J. Bourgain, Some remarks on Banach spaces in which martingale difference sequences are unconditional, ibid. 21 (1983), 163-168. Zbl0533.46008
- [11] J. Bourgain, Vector-valued singular integrals and the ${H}^{1}$-BMO duality, in: Probability Theory and Harmonic Analysis (Mini-Conference on Probability and Harmonic Analysis, Cleveland, 1983), J.-A. Chao and W. A. Woyczyński (eds.), Monographs and Textbooks in Pure and Appl. Math. 98, Marcel Dekker, New York, 1986, 1-19.
- [12] D. L. Burkholder, A geometric condition that implies the existence of certain singular integrals of Banach-space-valued functions, in: Proc. Conf. on Harmonic Analysis in Honor of Antoni Zygmund (Chicago, 1981), W. Beckner et al. (eds.), Wadsworth, Belmont, Calif., 1983, 270-286.
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- [15] T. A. Gillespie, Commuting well-bounded operators on Hilbert spaces, Proc. Edinburgh Math. Soc. (2) 20 (1976), 167-172. Zbl0334.47025
- [16] J.-P. Kahane, Sur les sommes vectorielles $\sum \pm {u}_{n}$, C. R. Acad. Sci. Paris 259 (1964), 2577-2580.
- [17] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I (Sequence Spaces), Ergeb. Math. Grenzgeb. 92, Springer, Berlin, 1977. Zbl0362.46013
- [18] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces II (Function Spaces), Ergeb. Math. Grenzgeb. 97, Springer, Berlin, 1979. Zbl0403.46022
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- [20] D. J. Ralph, Semigroups of well-bounded operators and multipliers, Thesis, Univ. of Edinburgh, 1977.

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