Composition Operators and Rational Inner Functions II: Boundedness between two different Bergman Spaces

In this note we provide a sufficient condition on when the composition operator $C_Φ:A^2_{a}(\mathbb{D}^2)\to A^2_β(\mathbb{D}^2)$ is bounded, whenever $a\ge-1$ and $β$ is positive, with the assumption that $Φ$ is induced by non-smooth Rational Inner Functions.