Two newBMO spaces and sharp operators of B-valued martingale

For Banach space\|valued martingale, two new BMO spaces, namely BMO σ(p) r(X) , resp. BMO S(p) r(X) , and two new sharp operators, namely, f σ(p) r resp. f S(p) r (0r≤1) generated by the conditional p \|mean\|square resp. p\| mean\|square operator are introduced, and then, the connections between BMO σ(p) r(X) and BMO + p, BMO S(p) r(X) and BMO p, f σ(p) p and # p, f S(p) p and f # p are investigated. The results obtained here yield a new charactrization of the convexity and smoothness of Banach space. \ \