Weighted Differentiation Composition Operators from Mixed-Norm to Zygmund Spaces

Abstract

Abstract Let ϕ be an analytic self-map of the unit disk 𝔻, H(𝔻) the space of analytic functions on 𝔻 and u ∈ H(𝔻). The boundedness and compactness of the weighted differentiation composition operator , where n ∈ ℕ0, from mixed-norm space to the Zygmund spaces, and the little Zygmund spaces are investigated in this article. Keywords: BoundednessCompactnessMixed-norm spaceWeighted differentiation composition operatorsZygmund spaceAMS Subject Classification: 47B3847B3330H0530D4546E15 ACKNOWLEDGMENTS Y. L. Both thanks Professors Shaozong Yan and Xiaoman Chen for their encouragement and help while visiting Fudan University. Both authors gratefully acknowledge the support in part by the National Natural Science Foundation of China (No. 10471039) and the Grant of Higher Schools' Natural Science Basic Research of Jiangsu Province of China (Nos. 06KJD110175; 07KJB110115). The authors also thank the referees for their thoughtful comments and helpful suggestions.