Boundedness of vector-valued martingale transforms on extreme points and applications

Let B-1, B-2 be a pair of Banach spaces and T be a vector valued martingale transform (with respect to general filtration) which maps B-1-valued martingales into B-2-valued martingales. Then, the following statements are equivalent: T is bounded from L-B1(p), into L-B2(p), for some p (or equivalently for every p) in the range 1 < p < infinity; T is bounded from L-B1(infinity) into BMOB2; T is bounded from BMOB1, into BMOB2; T is bounded from H-B1(1) into H-B2(1). Applications to UMD and martingale cotype properties are given. We also prove that the Hardy space H-B(1) defined in the case of a general filtration has nice dense sets and nice atomic decompositions if and only if B has the Radon-Nikodym property.