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. 
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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