Weighted Composition Operators from $A^{p}_{\alpha}$ to $A^{\infty}(\varphi)$

Abstract:

      Suppose $u\in H (D)$ and $\phi$ is an analytic map of the unit disk $D$ into itself. Define the weighted composition operator $uC_\phi: uC_\phi (f)= u f\circ \phi$, for all $f\in H(D)$. In this paper, we get necessary and sufficient conditions for the bounded and compact weighted composition operators from the weighted Bergman spaces $A_{\alpha}^{p}$ to $A^{\infty}(\varphi) \ (A_{0}^{\infty}(\varphi))$ spaces.