Spectral decompositions and harmonic analysis on UMD spaces
Volume 112 / 1994
Studia Mathematica 112 (1994), 13-49
DOI: 10.4064/sm-112-1-13-49
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.
