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 Aᵖ_α A α p , where $$p>1$$ p > 1 and α ≥ 0 α ≥ 0 . To be more precise, we give a sufficient condition for \| ∫ uₜCφ ₜ\, dt\| ₑ = ∫ \| uₜCφ ₜ\| ₑ \, dt ∫ u t C ϕ t d t e = ∫ u t C ϕ t e d t to hold in terms of geometric properties of uₜ u t and φ ₜ ϕ 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.