Compact Differences of Weighted Composition Operators

Compact differences of two weighted composition operators acting from the weighted Bergman space $A^p_\omega$ to another weighted Bergman space $A^q_\nu$, where $0<p\le q<\infty$ and $\omega,\nu$ belong to the class $\mathcal{D}$ of radial weights satisfying two-sided doubling conditions, are characterized. On the way to the proof a new description of $q$-Carleson measures for $A^p_\omega$, with $\omega\in\mathcal{D}$, in terms of pseudohyperbolic discs is established. This last-mentioned result generalizes the well-known characterization of $q$-Carleson measures for the classical weighted Bergman space $A^p_\alpha$ with $-1<\alpha<\infty$ to the setting of doubling weights.