Weighted Differentiation Composition Operators Between Normal Weight Zygmund Spaces and Bloch Spaces in the Unit Ball of $$\hbox {C}^{\mathrm{n}}$$ C n for $$\hbox {n}>1$$ 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 \(\displaystyle {\mu (r)=(1-r)^{s}\log ^{t}\frac{e}{1-r}}\).