Weighted Bergman spaces induced by doubling weights in the unit ball of $\mathbb {C}^n$

Abstract

This paper is devoted to the study of the weighted Bergman space $A_\omega ^p$ in the unit ball $\mathbb {B}$ of $\mathbb {C}^n$ with doubling weight $\omega$ satisfying

$$\begin{aligned} \int _r^1\omega (t)dt<C \int _{\frac{1+r}{2}}^1\omega (t)dt ,\quad 0\le r<1. \end{aligned}$$

The q-Carleson measures for $A_\omega ^p$ are characterized in terms of a neat geometric condition involving Carleson block. Some equivalent characterizations for $A_\omega ^p$ are obtained by using the radial derivative and admissible approach regions. The boundedness and compactness of Volterra integral operator $T_g:A_\omega ^p\rightarrow A_\omega ^q$ are also investigated in this paper with $0<p\le q<\infty$, where

$$\begin{aligned} T_gf(z)=\int _0^1 f(tz)\mathfrak {R}g(tz)\frac{dt}{t}, \quad f\in H(\mathbb {B}), \quad z\in \mathbb {B}. \end{aligned}$$