Moscow Mathematical Journal

Volume 25, Issue 4, October–December 2025 pp. 557–586.

Wavelet Characterization of Besov Spaces Based on the Definition of Mixed-Norm Modulus of Smoothness

Authors: Junjian Zhao (1) and Zhitao Zhuang (2)
Author institution:(1) School of Mathematical Sciences, TianGong University, Tianjin 300387, People's Republic of China
(2) College of Mathematics and Information Science, North China University of Water Resources and Electric Power, Zhengzhou 450011, People's Republic of China

Summary:

Mixed-norm space is a generalization of classical Lebesgue space $L_p(\mathbb{R}^d)$. It considers functions with independent variables under possibly different meanings. In this paper, we give a research on mixed-norm Besov space $B_{\vec{p},q}^s(\mathbb{R}^d)$ defined by modulus of smoothness, which can be decomposed by another method based on the well-known Littlewood–Paley decomposition theory. Then we get two new results of mixed-norm Besov spaces' characterization. The first one is the Littlewood–Paley type characterization, which is received by studying the mutual control between the modulus of smoothness $\omega_{\vec{p}}^2$ and the Littlewood–Paley decomposition operator $\mathcal{L}_jf$ under mixed-norm. Based on this Littlewood–Paley type characterization, we get a characterization theorem for $B_{\vec{p},q}^s(\mathbb{R}^d)$ by wavelets.

2020 Math. Subj. Class. 41A35, 41A17, 42B08, 42C15, 42C20.

Keywords: Characterization, mixed-norm, Besov spaces, Littlewood–Paley decomposition, wavelet.

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