New characterizations for weighted composition operator from Zygmund type spaces to Bloch type spaces

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