Operator-valued martingale transforms in rearrangement invariant spaces and applications

Abstract

In this paper the operator-valued martingale transform inequalities in rearrangement invariant function spaces are proved. Some well-known results are generalized and unified. Applications are given to classical operators such as the maximal operator and the p-variation operator of vector-valued martingales, then we can very easily obtain some new vector-valued martingale inequalities in rearrangement invariant function spaces. These inequalities are closely related to both the geometrical properties of the underlying Banach spaces and the Boyd indices of the rearrangement invariant function spaces. Finally we give an equivalent characterization of UMD Banach lattices, and also prove the Fefferman-Stein theorem in the rearrangement invariant function spaces setting.