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

TL;DR

This paper characterizes the boundedness and essential norms of weighted composition operators from Bloch type spaces to Zygmund type spaces in the unit disk, providing new criteria based on the functions and their derivatives.

Contribution

It introduces new characterizations for the boundedness and essential norms of weighted composition operators between specific function spaces, including criteria for compactness.

Findings

Derived new boundedness criteria involving $u, \, \, \, \, \, \, \, \\varphi$ and derivatives.

Established estimates for the essential norms of the operators.

Provided necessary and sufficient conditions for the compactness of these operators.

Abstract

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.