Essential norm of composition operators on the Hardy space H 1 and the weighted Bergman spaces {A_{\alpha}^{p}} on the ball

We estimate the essential norm of a composition operator acting on the Hardy space H ¹ and the weighted Bergman spaces ${A_{\alpha}^{p}}$ on the unit ball. In passing, we recover (and somehow simplify the proof of) parts of the recent article by Demazeux, dealing with the same question for H ¹ of the unit disc. We also estimate the essential norm of a composition operator acting on ${A_{\alpha}^{p}}$ in terms of the angular derivatives of ${\phi}$ , under a mild condition on ${\phi}$ .