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arXiv · 0908.4380

Littlewood-Paley characterization for $Q_α(R^n)$ spaces

Qifan Li
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Abstract

In Baraka's paper [2], he obtained the Littlewood-Paley characterization of Campanato spaces $L^{2,λ}$ and introduced $\mathcal {L}^{p,λ,s}$ spaces. He showed that $\mathcal {L}^{2,λ,s}=(-\triangle)^{-\frac{s}{2}}L^{2,λ}$ for $0\leqλ<n+2$. In [7], by using the properties of fractional Carleson measures, J Xiao proved that for $n\geq2$, $0<α<1$. $(-\triangle)^{-\fracα{2}}L^{2,n-2α}$ is essential the $Q_α(\mathbb{R}^n)$ spaces which were introduced in [4]. Then we could conclude that $Q_α(\mathbb{R}^n)=\mathcal {L}^{2,n-2α,α}$ 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].

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Qifan Li. 2009-08-31. Littlewood-Paley characterization for $Q_α(R^n)$ spaces. https://arxiv.org/abs/0908.4380

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