Geodesic
Spectral decompositions and harmonic analysis on UMD spaces
Studia Mathematica, Tome 112 (1994) no. 1, pp. 13-49
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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. 
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     title = {Spectral decompositions and harmonic analysis on {UMD} spaces},
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Earl Berkson; T. A. . Spectral decompositions and harmonic analysis on UMD spaces. Studia Mathematica, Tome 112 (1994) no. 1, pp. 13-49. doi: 10.4064/sm-112-1-13-49

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