Continuous Frames Under the Weighted Composition Operators on the Weighted Bergman Space

We investigate the behavior of continuous frames in the weighted Bergman space Aα2 over the unit disc under the action of weighted composition operators. Motivated by developments in the discrete frame setting, we provide a comprehensive characterization of those weighted composition operators that preserve continuous frames, including tight and Parseval frames. Additionally, we examine the structure of dual frames in this context and establish necessary and sufficient conditions under which dual frame pairs are preserved by such operators. Explicit constructions of dual pairs induced by weighted composition operators are also presented. The study concludes with an analysis of the scalability of continuous frames and explores its invariance under the action of weighted composition operators.