Essential Norms of Weighted Composition Operators From the Bergman Space To Weighted-Type Spaces on The Unit Ball

Abstract

We estimate the essential norm of the weighted composition operator \(uC_{\varphi}\) from the weighted Bergman space \(A^{p}_{\alpha}(\mathbb{B})\) to the weighted space \(H^{\infty}_{\mu}(\mathbb{B})\) on the unit ball \(\mathbb{B}\), when \(p > 1\) and \(\alpha \geq -1\) (for \(\alpha = -1\), \(A^{p}_{\alpha}\) is the Hardy space \(H^{p}(\mathbb{B})\)). We also give a necessary and sufficient condition for the operator \(uC_{\varphi} : A^{p}_{\alpha}(\mathbb{B}) \to H^{\infty}_{\mu}(B)\) to be compact, and for the operator \(uC_{\varphi} : A^{p}_{\alpha}(\mathbb{B}) \to H^{\infty}_{\mu,0}(\mathbb{B})\) to be bounded or compact, when \(p > 0\), \(\alpha \geq -1\).