GENERALISED WEIGHTED COMPOSITION OPERATORS ON BERGMAN SPACES INDUCED BY DOUBLING WEIGHTS

Abstract

AbstractWe characterise bounded and compact generalised 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 $. On the way, we establish a new embedding theorem on weighted Bergman spaces$A^p_\omega $which generalises the well-known characterisation of the boundedness of the differentiation operator$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.