Estimates of essential norms of weighted composition operator from Bloch type spaces to Zygmund type spaces

Let $u$ be a holomorphic function and $\varphi$ a holomorphic self-map of the open unit disk $\mathbb{D}$ in the complex plane. We give some new characterizations for the boundedness of the weighted composition operators $uC_{\varphi}$ from Bloch type spaces to Zygmund type spaces in $\mathbb{D}$ in terms of $u, \varphi$, their derivatives and the $n$-th power $\varphi^n$ of $\varphi$. Moreover, we obtain some similar estimates for their essential norms. From which the sufficient and necessary conditions of compactness of the operators $uC_{\varphi}$ follows immediately.