Weighted Differentiation Composition Operators Between Normal Weight Zygmund Spaces and Bloch Spaces in the Unit Ball of $\hbox {C}^{\mathrm{n}}$ for $\hbox {n}>1$

Abstract

Let $\mu$ be a normal functions on [0, 1), and H(B) be the space of all holomorphic functions on the unit ball B of $\mathbf C^{n}$. Let $\varphi$ be a nonconstant holomorphic self-map on B, and $\psi$ be a holomorphic function on B. The weighted differentiation composition operator $\psi D_{\varphi }$ is defined on the space H(B) by $\psi D_{\varphi }(f)=\psi (Rf)\circ \varphi$, for all $f\in H(B)$. In this paper, the authors characterize the boundedness and compactness of the weighted differentiation composition operator $\psi D_{\varphi }$ from the normal weight Zygmund space $Z_{\mu }(B)$ to the normal weight Bloch space $\beta _{\mu }(B)$ for $n>1$. As a consequence of the main results, the authors give the briefly sufficient and necessary conditions that the differentiation composition operator $D_{\varphi }$ is compact from $Z_{\mu }(B)$ to $\beta _{\mu }(B)$ for $\displaystyle {\mu (r)=(1-r)^{s}\log ^{t}\frac{e}{1-r}}$.