Bergman and Dirichlet spaces in the unit ball and symmetric lifting operator

Abstract

Let $\mathbb{B}_n$ be the open unit ball in $\mathbb{C}^n$ and $\mathbb{B}_n^2 = \mathbb{B}_n \times \mathbb{B}_n$. The symmetric lifting operator which lifts analytic functions from $H(\mathbb{B}_n)$ to $H(\mathbb{B}_n^2)$ is defined as follow\[L(f)(z,w) = \frac{f(z) - f(w)}{z-w}.\]In this paper we investigate the action of symmetric lifting operator on the Bergman space in the unit ball. Also, we state a characterization for Dirichlet space and consider symmetric lifting operator on the Dirichlet space in the unit ball.

Keywords

• symmetric lifting operator‎
• bergman space‎
• dirichlet space
• ‎ pseudo-hyperbolic metric