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 $C_{\mathrm{\Phi }}:A_a^2({\mathbb{D}}^2)\to A_{\beta }^2({\mathbb{D}}^2)$ is bounded, whenever $a\ge -1$ and $\beta$ is positive, with the assumption that $\mathrm{\Phi }$ is induced by non-smooth Rational Inner Functions.