Weighted Composition Operators on Some Weighted Spaces in the Unit Ball

Abstract

Let ${\mathbb{B}}_{n}$ be the unit ball of ${\mathbb{C}}^{n}$, $H({\mathbb{B}}_{n})$ the space of all holomorphic functions in ${\mathbb{B}}_{n}$. Let $u\in H({\mathbb{B}}_{n})$ and $\alpha $ be a holomorphic self-map of ${\mathbb{B}}_{n}$. For $f\in H({\mathbb{B}}_{n})$, the weigthed composition operator $u{C}_{\alpha }$ is defined by $(u{C}_{\alpha }f)(z)=u(z)f(\alpha (z)),z\in {\mathbb{B}}_{n}.$ The boundedness and compactness of the weighted composition operator on some weighted spaces on the unit ball are studied in this paper.