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 $${\int }_r^1\omega (t)dt<C{\int }_{\frac{1+r}{2}}^1\omega (t)dt,0\le r<1.$$ 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 $T_g:A_{\omega }^p\to A_{\omega }^q$ are also investigated in this paper with $0<p\le q<\infty$, where $$T_{g}f(z)={\int }_0^{1}f(tz)\Re g(tz)\frac{dt}{t},f\in H(\mathbb{B}),z\in \mathbb{B}.$$

35 pages