Weighted composition operators on weighted Bergman spaces induced by double weights

In this paper, we investigate the boundedness, compactness, essential norm and the Schatten class of weighted composition operators $uC_\varphi$ on Bergman type spaces $A_\omega^p$ with double weight $\omega$. Let $X=\{u\in H(D): uC_\varphi:A_\omega^p\to A_\omega^p \mbox{ is bounded}\}.$ For some regular weights $\omega$, we obtain that $X=H^\infty$ if and only if $\varphi$ is a finite Blaschke product.