Upper and Lower Bounds for Essential Norm of Weighted Composition Operators from Bergman Spaces with Békollé Weights

Abstract

Let $\sigma$ be a weight function such that $\sigma /{\left(1-{\left|z\right|}^2\right)}^{\alpha }$ is in the class $B_{p_0}\left(\alpha \right)$ of Békollé weights, $\mu$ a normal weight function, $\psi$ a holomorphic map on $\mathbb{D}$ , and $\phi$ a holomorphic self-map on $\mathbb{D}$ . In this paper, we give upper and lower bounds for essential norm of weighted composition operator $W_{\psi ,\phi }$ acting from weighted Bergman spaces ${\mathcal{A}}^p\left(\sigma \right)$ to Bloch-type spaces ${\mathcal{B}}_{\mu }.$