On the theory and applications of Hardy and Bergman spaces

We study composition operators between higher orders weighted Bergman spaces. Certain growth conditions for generalized Nevanlinna counting functions of the inducing map are shown to be necessary and sufficient for such operators to be bounded or compact. Under a mind condition we show that a composition operators C is compact on the higher order weighted Bergman spaces and Hardy spaces of the open unit ball in nC if and only if 0 as |z | 1-