Spectral decompositions and harmonic analysis on UMD spaces

• Autorzy: Earl Berkson , T. A. Gillespie
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 42A45: Multipliers
• 47B40: Spectral operators, decomposable operators, well-bounded operators, etc.

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 1994-1995
• Tom: 112
• Numer: 1
• Strony: 13-49

Daty

wydano

1994

otrzymano

1993-05-31

Twórcy

autor

Earl Berkson

• Department of Mathematics, University of Illinois, 1409 West Green St., Urbana, Illinois 61801, U.S.A.

autor

T. A. Gillespie

• Department of Mathematics, University of Edinburgh, James Clerk Maxwell Building, Edinburgh EH9 3JZ, Scotland

Bibliografia

• [1] N. Asmar, E. Berkson and T. A. Gillespie, Transferred bounds for square functions, Houston J. Math. 17 (1991), 525-550.
• [2] N. Asmar, E. Berkson and T. A. Gillespie, Spectral integration of Marcinkiewicz multipliers, Canad. J. Math. 45 (1993), 470-482.
• [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.
• [4] E. Berkson and T. A. Gillespie, Fourier series criteria for operator decomposability, ibid. 9 (1986), 767-789.
• [5] E. Berkson and T. A. Gillespie, Stechkin's theorem, transference, and spectral decompositions, J. Funct. Anal. 70 (1987), 140-170.
• [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] E. Berkson and T. A. Gillespie, Transference and extension of Fourier multipliers for $L^p(𝕋)$, J. London Math. Soc. (2) 41 (1990), 472-488.
• [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.
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