Compact composition operators on Hardy-Orlicz and weighted Bergman-Orlicz spaces on the ball

Using recent characterizations of the compactness of composition operators on Hardy-Orlicz and Bergman-Orlicz spaces on the ball, we first show that a composition operator which is compact on every Hardy-Orlicz (or Bergman-Orlicz) space has to be compact on H^{\infty}. Then, we prove that, for each Koranyi region \Gamma, there exists a map \phi taking the ball into \Gamma such that, C_{\phi} is not compact on H^{\psi}\left(\mathbb{B}_{N}\right), when \psi grows fast. Finally, we give another characterization of the compactness of composition operator on weighted Bergman-Orlicz spaces in terms of an Orlicz "Angular derivative"-type condition. This extends (and simplify the proof of) a result by K. Zhu for the classical Bergman case. Moreover, we deduce that the compactness of composition operators on weighted Bergman-Orlicz spaces does not depend on the weight anymore, when the Orlicz function grows fast.