Littlewood-Paley characterization for $Q_{\alpha }(R^n)$ spaces

In Baraka’s paper [2], he obtained the Littlewood-Paley characterization of Campanato spaces $L^{2,\lambda }$ and introduced ${\mathcal{L}}^{p,\lambda ,s}$ spaces. He showed that $\mathcal {L}^{2,\lambda,s}=(-\triangle)^{-\frac{s}{2}}L^{2,\lambda}$for$0\leq\lambda<n+2$. In [7], by using the properties of fractional Carleson measures, J Xiao proved that for $n\ge 2$, $0<\alpha <1$. $(-\mathrm{△}{)}^{-\frac{\alpha }{2}}{L}^{2,n-2\alpha }$ is essential the $Q_{\alpha }({\mathbb{R}}^n)$ spaces which were introduced in [4]. Then we could conclude that $Q_{\alpha }({\mathbb{R}}^n)={\mathcal{L}}^{2,n-2\alpha ,\alpha }$ for $0<\alpha <1$. In fact, this result could be also obtained directly by using the method in [2]. In this paper, We proved this result in the spirit of [2]. This paper could be considered as the supplement of Baraka’s work [2].