Weighted Cesàro type operators between weighted Bergman spaces

Let $D$ be the open unit disk in the complex plane $C$. Let μ be a positive Borel measure on $[0,1)$. If $f\left(z\right)={\sum }_{n=0}^{\infty }{a}_n{z}^n$ is an analytic function in $D$, we consider for $\beta >0$ the following weighted Cesàro type operator$C_{\mu ,\beta }\left(f\right)\left(z\right)=\sum _{n=0}^{\infty }{\mu }_n\left(\sum _{k=0}^n\frac{\Gamma (n-k+\beta )}{(n-k)!\Gamma \left(\beta \right)}{a}_k\right)z^n,z\in D,$ where ${\mu }_n$ is the n-th moment of μ given by ${\mu }_n=\underset{[0,1)}{\int }{t}^{n}d\mu \left(t\right)$. We characterize boundedness of the weighted Cesàro type operator $C_{\mu ,\beta }$ from the weighted Bergman space $A_{\alpha }^p$ to $A_{\alpha }^q$ for $1\le p\le q<\infty$. Our method relies on a representation of $C_{\mu ,\beta }$ as an integral operator with a kernel and a generalized Schur's test.