Compact differences of composition operators on Bergman spaces induced by doubling weights

Abstract

Bounded and compact differences of two composition operators acting from the weighted Bergman space $A_{\omega }^p$ to the Lebesgue space $L_{\nu }^q$, where $0<q<p<\infty$ and $\omega$ belongs to the class $\mathcal{D}$ of radial weights satisfying a two-sided doubling condition, are characterized. On the way to the proofs a new description of $q$-Carleson measures for $A_{\omega }^p$, with $p>q$ and $\omega \in \mathcal{D}$, involving pseudohyperbolic discs is established. This last-mentioned result generalizes the well-known characterization of $q$-Carleson measures for the classical weighted Bergman space $A_{\alpha }^p$ with $-1<\alpha <\infty$ to the setting of doubling weights. The case $\omega \in \hat{\mathcal{D}}$ is also briefly discussed and an open problem concerning this case is posed.

Cited by in corpus (1)

• Generalized weighted composition operators on Bergman spaces induced by doubling weights