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

TL;DR

This paper provides a Littlewood-Paley characterization of the $Q_{eta}( e^n)$ spaces, connecting them to fractional Laplacian-based Campanato spaces, extending and supplementing previous work by Baraka and Xiao.

Contribution

It establishes a Littlewood-Paley characterization for $Q_{eta}( e^n)$ spaces, linking them to fractional Laplacian operators, and extends prior results with a new proof approach.

Findings

$Q_{eta}( e^n)$ spaces characterized via Littlewood-Paley theory.

Connection established between $Q_{eta}( e^n)$ and fractional Laplacian-based Campanato spaces.

Provides a new proof method supplementing previous literature.

Abstract

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 $n\geq2$, $0<\alpha<1$. $(-\triangle)^{-\frac{\alpha}{2}}L^{2,n-2\alpha}$ 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].