Uniform convexity in terms of martingale \(H^1\) and BMO spaces (Q2744642)

It is proved that the dual of the Hardy space \(H_{\mathbf B}^1\) containing \({\mathbf B}\)-valued martingales is the space BMO\(_{2,{\mathbf B}^*}^-\) if and only if \({\mathbf B}^*\) has Radon-Nikodym property, where \({\mathbf B}^*\) denotes the dual of the Banach space \({\mathbf B}\). Moreover, a new characterization of martingale type and cotype properties of \({\mathbf B}\) is given in terms of the boundedness of \(S_qf=(\sum_{k=1}^ \infty \|d_kf\|_{\mathbf B}^q)^{1/q}\) in \(H^1\) and BMO spaces.