Using recent characterizations of the compactness of composition operators on HardyOrlicz and Bergman-Orlicz spaces on the ball ([2, 3]), we first show that a composition operator which is compact on every Hardy-Orlicz (or Bergman-Orlicz) space has to be compact on H∞. Then, although it is well-known that a map whose range is contained in some nice Korányi approach region induces a compact comp...

We show that the approximation numbers of a compact composition operator on the weighted Bergman spaces Bα of the unit disk can tend to 0 arbitrarily slowly, but that they never tend quickly to 0: they grow at least exponentially, and this speed of convergence is only obtained for symbols which do not approach the unit circle. We also give an upper bounds and explicit an example. Mathematics Su...

We obtain sharp L p → q $L^p\rightarrow L^q$ hypercontractive inequalities for the weighted Bergman spaces on unit disk D $\mathbb {D}$ with usual weights α − 1 π ( | z 2 ) , > $\frac{\alpha -1}{\pi }{(1-|z|^2)}^{\alpha -2},\alpha >1$ ⩾ $q\geqslant 2$ thus solving an interesting case of a problem from Janson [Ark. Math. 21 (1983), no. 1, 97–110]. also give some estimates 0 < $0<q<2$ .