Weighted Cesàro type operators on the weighted Bergman spaces in the unit ball

In this paper, we study weighted Cesàro type operators $${\mathcal{C}}_{\mu ,\beta }^{\xi }:A_{\alpha }^p({\mathbb{B}}_n)⟶A_{\alpha }^q({\mathbb{B}}_n)$$ induced by measures on $[0,1)$, and obtain boundedness characterizations throughout the range $0<p,q\le \mathrm{\infty }$. We first extend the corresponding results on the unit disk to the unit ball in the range $1\le p\le q<\mathrm{\infty }$. By means of atomic decomposition, we further treat the case $0<p<1$ and $p\le q<\mathrm{\infty }$, thereby obtaining a complete characterization for $0<p\le q<\mathrm{\infty }$. The latter result is new even in one dimension. For the range $0<q<p\le \mathrm{\infty }$, we characterize the boundedness in terms of an integral condition involving the tail function of the inducing measure. This result is also new even on the unit disk. Finally, we characterize the boundedness of the corresponding operators when the target space is $H^{\mathrm{\infty }}({\mathbb{B}}_n)$.