Let ψ∈H(Bn)\documentclass[12pt]{minimal}
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\begin{document}$$\psi \in H(\mathbb {B}_n)$$\end{document} be the space of all holomorphic functions on the unit ball Bn\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {B}_n$$\end{document} of Cn\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {C}^n$$\end{document} and φ=(φ1,…,φn)∈S(Bn)\documentclass[12pt]{minimal}
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\begin{document}$$\varphi = (\varphi _1, \ldots , \varphi _n) \in S(\mathbb {B}_n)$$\end{document} the set of holomorphic self-maps of Bn\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {B}_n$$\end{document}. Let Wψ,φ:Bν(orBν,0)→Bμ(orBμ,0)\documentclass[12pt]{minimal}
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\begin{document}$$W_{\psi , \varphi }:\mathscr {B}_\nu \, (\text {or }\mathscr {B}_{\nu ,0}) \rightarrow \mathscr {B}_\mu \, (\text {or } \mathscr {B}_{\mu ,0})$$\end{document}, f↦ψ·(f∘φ)\documentclass[12pt]{minimal}
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\begin{document}$$f\mapsto \psi \hspace{1.111pt}{\cdot }\hspace{1.111pt}(f \hspace{0.55542pt}{\circ }\hspace{1.111pt}\varphi )$$\end{document}, be weighted composition operators between (little) Bloch-type spaces where ν,μ\documentclass[12pt]{minimal}
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\begin{document}$$\nu , \mu $$\end{document} are normal weights on Bn\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {B}_n$$\end{document}. We characterize the boundedness, compactness and give the asymptotic estimates of the norms of Wψ,φ\documentclass[12pt]{minimal}
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\begin{document}$$ W_{\psi ,\varphi } $$\end{document} in terms of function theoretic properties of the symbol ψ\documentclass[12pt]{minimal}
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\begin{document}$$\psi $$\end{document} and of the modulus |φk|\documentclass[12pt]{minimal}
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\begin{document}$$|\varphi _k|$$\end{document} of a component function φk\documentclass[12pt]{minimal}
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\begin{document}$$\varphi _k$$\end{document} of the holomorphic symbol φ\documentclass[12pt]{minimal}
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\begin{document}$$\varphi $$\end{document}.
