Fefferman-Stein decomposition for $Q$-spaces and micro-local quantities

In this paper, we consider the Fefferman-Stein decomposition of $Q_{\alpha}(\mathbb{R}^{n})$ and give an affirmative answer to an open problem posed by M. Essen, S. Janson, L. Peng and J. Xiao in 2000. One of our main methods is to study the structure of the predual space of $Q_{\alpha}(\mathbb{R}^{n})$ by the micro-local quantities. This result indicates that the norm of the predual space of $Q_{\alpha}(\mathbb{R}^{n})$ depends on the micro-local structure in a self-correlation way.