Maximal functions and Calderon-Zygmund theory for vector-valued functions with operator weights

Abstract

We generalize the Hardy-Littlewood maximal functions to vector-valued functions taking values in a Banach space with a varying weight, t →ρ t , which satisfies a reverse Holder condition, and prove estimates on the bounds of the maximal function. We also prove the existence of a Calderon-Zygmund decomposition for functions in L P (t → ρ t ) when the weight satisfies a reverse Holder inequality. This decomposition is used to prove an extrapolation theorem for the martingale transform on weighted spaces.