Generalized weighted composition operators on Bergman spaces induced by doubling weights

Bounded and compact generalized weighted composition operators acting from the weighted Bergman space $A^p_\omega$, where $0<p<\infty$ and $\omega$ belongs to the class $\mathcal{D}$ of radial weights satisfying a two-sided doubling condition, to a Lebesgue space $L^q_\nu$ are characterized. On the way to the proofs a new embedding theorem on weighted Bergman spaces $A^p_\omega$ is established. This last-mentioned result generalizes the well-known characterization of the boundedness of the differentiation operator $D^n(f)=f^{(n)}$ from the classical weighted Bergman space $A^p_\alpha$ to the Lebesgue space $L^q_\mu$, induced by a positive Borel measure $\mu$, to the setting of doubling weights.