Paper detail

Generalized weighted composition operators on Bergman spaces induced by doubling weights

Bounded and compact generalized weighted composition operators acting from the weighted Bergman space $A^p_ω$, where $0<p<\infty$ and $ω$ belongs to the class $\mathcal{D}$ of radial weights satisfying a two-sided doubling condition, to a Lebesgue space $L^q_ν$ are characterized. On the way to the proofs a new embedding theorem on weighted Bergman spaces $A^p_ω$ 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 $A^p_α$ to the Lebesgue space $L^q_μ$, induced by a positive Borel measure $μ$, to the setting of doubling weights.