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 $A^p_\alpha$, where $p>1$ and $\alpha\geq 0$. To be more precise, we give a sufficient condition for $ \|\int u_tC_{\phi_t}\, dt\|_e = \int \| u_tC_{\phi_t}\|_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.