Abstract
In this paper, the objects of our investigation are some dyadic operators, including dyadic shifts, multilinear paraproducts and multilinear Haar multipliers. We mainly focus on the continuity and compactness of these operators. First, we consider the continuity properties of these operators. Then, by the Fréchet–Kolmogorov–Riesz–Tsuji theorem, the non-compactness properties of these dyadic operators will be studied. Moreover, we show that their commutators are compact with $\operatorname{CMO}$ functions, which is quite different from the non-compactness properties of these dyadic operators. These results are similar to those for Calderón–Zygmund singular integral operators.
Funding Statement
The authors were supported partly by NSFC (No. 11871101), 111 Project and the National Key Research and Development Program of China (Grant No. 2020YFA0712900). The third author was supported partly by Grant-in-Aid for Scientific Research (C) Nr. 15K04942, Japan Society for the Promotion of Science.
Acknowledgments
The authors want to express their sincere thanks to the referee for his or her valuable remarks and suggestions, which made this paper more readable.
Citation
Heng Gu. Qingying Xue. Kôzô Yabuta. "On Some Properties of Dyadic Operators." Taiwanese J. Math. 26 (3) 521 - 544, June, 2022. https://doi.org/10.11650/tjm/211101
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