Boundedness criterion for compact differences of composition operators

Let $0<p,q<\infty$, let $\omega\in\widehat{\mathcal{D}}$, and let $\nu$ be a positive Borel measure on $\mathbb{D}$. Let $\varphi$ and $\psi$ be analytic self-maps of $\mathbb{D}$. Write $C_\varphi$ and $C_\psi$ for the corresponding composition operators, and let $\delta C_\varphi$ and $\delta C_\psi$ denote the associated difference-control operators. Boundedness criterion. The operator

$$C_\varphi-C_\psi:A^p_\omega\rightarrow L^q_\nu$$

is bounded if and only if $\delta C_\varphi$ and $\delta C_\psi$ are bounded from $A^p_\omega$ to $L^q_\nu$. This criterion characterizes bounded differences of composition operators through the two individual difference-control operators; the supplied text does not establish whether the statement is proved or remains open.