Fefferman-Stein decomposition for QQ-spaces and micro-local quantities

In this paper, we study the Fefferman-Stein decomposition of Qα(Rn)Qα(Rn) and give an affirmative answer to an open problem posed by Essén et al. (2000). One of our main methods is to characterize the structure of the predual of Qα(Rn)Qα(Rn) by the micro-local quantities. This result indicates that the norm of the predual space of Qα(Rn)Qα(Rn) depends on the micro-local structure in a self-correlation way.