For pairs of rational inner functions on the bidisc with one boundary singularity each, the induced composition operator is bounded from A^2_{β/(2q)-2} to A^2_β whenever β > 2q, where q is a Lojasiewicz exponent.
Beslikas, Composition Operators and Rational Inner Functions on the bidisc, Proceedings of the American Mathematical Society, Vol
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Composition Operators and Rational Inner Functions II: Boundedness between two different Bergman Spaces
For pairs of rational inner functions on the bidisc with one boundary singularity each, the induced composition operator is bounded from A^2_{β/(2q)-2} to A^2_β whenever β > 2q, where q is a Lojasiewicz exponent.