Essential norm and integration of a family of weighted composition operators

We study the interchange of essential norm and integration of certain families of weighted composition operators acting on the standard weighted Bergman spaces $A_{\alpha }^p$, where $p>1$ and $\alpha \ge 0$. To be more precise, we give a sufficient condition for $\mathrm{\parallel }\int u_t{C}_{{\phi }_t}dt{\mathrm{\parallel }}_e=\int \mathrm{\parallel }{u}_t{C}_{{\phi }_t}{\mathrm{\parallel }}_{e}dt$ to hold in terms of geometric properties of $u_t$ and ${\phi }_t$. We also provide some necessary conditions for the equality to hold and calculate the essential norm of some integral operators such as some Volterra operators.