Littlewood-Paley characterization for Qα(R) spaces

In Baraka’s paper [2], he obtained the Littlewood-Paley characterization of Campanato spaces L and introduced Lp,λ,s spaces. He showed that L2,λ,s = (−△)− s 2L for 0 ≤ λ < n+ 2. In [7], by using the properties of fractional Carleson measures, J Xiao proved that for n ≥ 2, 0 < α < 1. (−△)−α2 L is essential the Qα(R) spaces which were introduced in [4]. Then we could conclude that Qα(R) = L2,n−2α,α for 0 < α < 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].