Czechoslovak Mathematical Journal, Vol. 76, No. 2, pp. 401-458, 2026
Some new estimates for Littlewood-Paley square operators on weighted Morrey-Campanato spaces
Hua WANG
Received May 2, 2025. Published online April 14, 2026.
Abstract: Let $1\leq p<\infty$ and $-n/p<\alpha\leq1$. A distinguished subset $\mathcal{C}^{\alpha,p}_{\ast}(\omega)$ of the weighted Morrey-Campanato space $\mathcal{C}^{\alpha,p}(\omega)$ on $\mathbb R^n$ is introduced and studied. This new class is a proper subset of $\mathcal{C}^{\alpha,p}(\omega)$. We establish John-Nirenberg-type inequalities suitable for the Morrey-Campanato spaces $\mathcal{C}^{\alpha,p}(\omega)$ and $\mathcal{C}^{\alpha,p}_{\ast}(\omega)$ with $\omega\in A_1$. Based on this result, some new equivalent characterizations of the Morrey-Campanato spaces $\mathcal{C}^{\alpha,p}(\omega)$ and $\mathcal{C}^{\alpha,p}_{\ast}(\omega)$ are also given. Let $T(f)$ denote the Littlewood-Paley square operators, including the Littlewood-Paley $g$-function $\mathcal{G}_{\psi}(f)$, Lusin's area integral $\mathcal{S}_{\psi}(f)$ and Stein's function $\mathcal{G}^{\ast}_{\lambda,\psi}(f)$ with $\lambda>2$. Here $\psi$ is a Littlewood-Paley function on $\mathbb R^n$. We establish the boundedness of Littlewood-Paley square operators on weighted Morrey-Campanato spaces. It is proved that if $T(f)(x_0)$ is finite for a single point $x_0\in\mathbb R^n$, then $T(f)(x)$ is finite almost everywhere in $\mathbb R^n$. Moreover, it is shown that $T(f)$ is bounded from $\mathcal{C}^{\alpha,p}(\omega)$ into $\mathcal{C}^{\alpha,p}_{\ast}(\omega)$ for $1\leq p<\infty$ and $0<\alpha\leq1$, provided that $\omega\in A_1$.
Affiliations: Hua Wang, School of Mathematics and Information Science, Xiangnan University, Suxian District, Chenzhou 423000, P. R. China, e-mail: wanghua@pku.edu.cn
