Weighted Bergman spaces induced by doubling weights in the unit ball of $\mathbb{C}^n$ (1903.03748v2)

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 $$\int_r^1\omega(t)dt <C \int_{\frac{1+r}{2}}^1\omega(t)dt ,\,\, 0\leq r<1.$$ 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 $0<p\leq q<\infty$, where $$T_gf(z)=\int_0^1 f(tz)\Re g(tz)\frac{dt}{t}, \qquad~~f\in H(\mathbb{B}), ~~z\in \mathbb{B}. $$