  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.
