Spectral decompositions and harmonic analysis on UMD spaces

Earl Berkson; T. Gillespie

Studia Mathematica (1994)

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

Abstract

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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.

How to cite

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Berkson, 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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  1. [1] N. Asmar, E. Berkson and T. A. Gillespie, Transferred bounds for square functions, Houston J. Math. 17 (1991), 525-550. Zbl0784.43003
  2. [2] N. Asmar, E. Berkson and T. A. Gillespie, Spectral integration of Marcinkiewicz multipliers, Canad. J. Math. 45 (1993), 470-482. Zbl0817.42005
  3. [3] E. Berkson, J. Bourgain and T. A. Gillespie, On the almost everywhere convergence of ergodic averages for power-bounded operators on L p -subspaces, Integral Equations Operator Theory 14 (1991), 678-715. Zbl0786.47003
  4. [4] E. Berkson and T. A. Gillespie, Fourier series criteria for operator decomposability, ibid. 9 (1986), 767-789. Zbl0607.47026
  5. [5] E. Berkson and T. A. Gillespie, Stechkin's theorem, transference, and spectral decompositions, J. Funct. Anal. 70 (1987), 140-170. Zbl0607.47027
  6. [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. 
  7. [7] E. Berkson and T. A. Gillespie, Transference and extension of Fourier multipliers for L p ( ) , J. London Math. Soc. (2) 41 (1990), 472-488. Zbl0693.42009
  8. [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
  9. [9] E. Berkson, T. A. Gillespie and P. S. Muhly, Generalized analyticity in UMD spaces, Ark. Mat. 27 (1989), 1-14. 
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  11. [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. 
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  13. [13] H. R. Dowson, Spectral Theory of Linear Operators, London Math. Soc. Monographs 12, Academic Press, New York, 1978. Zbl0384.47001
  14. [14] R. E. Edwards and G. I. Gaudry, Littlewood-Paley and Multiplier Theory, Ergeb. Math. Grenzgeb. 90, Springer, Berlin, 1977. 
  15. [15] T. A. Gillespie, Commuting well-bounded operators on Hilbert spaces, Proc. Edinburgh Math. Soc. (2) 20 (1976), 167-172. Zbl0334.47025
  16. [16] J.-P. Kahane, Sur les sommes vectorielles ∑ ± u n , C. R. Acad. Sci. Paris 259 (1964), 2577-2580. 
  17. [17] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I (Sequence Spaces), Ergeb. Math. Grenzgeb. 92, Springer, Berlin, 1977. Zbl0362.46013
  18. [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. [20] D. J. Ralph, Semigroups of well-bounded operators and multipliers, Thesis, Univ. of Edinburgh, 1977. 

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