SPECTRAL DECOMPOSITIONS AND HARMONIC-ANALYSIS ON UMD SPACES

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 ana logues of the classical themes embodied in the Littlewood-Paley Theore m, the Strong Marcinkiewicz Multiplier Theorem, and the M. Riesz Prope rty. In particular, it is shown by spectral integration that classical Marcinkiewicz multipliers have associated transforms acting on X.