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