Littlewood Subordination Theorem and Composition Operators on Function Spaces with Variable Exponents

This study concerns a detailed analysis of composition opera-tors Cϕ on the classical Bergman spaces, as well as on the Hardy and Bergman spaces with variable exponents. Here, ϕ is an an-alytic self-map of the open unit disk in the complex plane. Ac-cordingly, conditions for the boundedness of these operators are obtained. It is worth mentioning that the Littlewood subordi-nation theorem plays a fundamental role in proving the stated theorems in which we use the Rubio de Francia extrapolation theorem.