Composition operators on the polydisc induced by smooth symbols ✩

We study composition operators CΦ on the Hardy spaces Hp and weighted Bergman spaces A pα of the polydisc Dn in Cn. When Φ is of class C2 on Dn, we show that CΦ is bounded on Hp or A pα if and only if the Jacobian of Φ does not vanish on those points ζ on the distinguished boundary Tn such that Φ(ζ ) ∈ Tn. Moreover, we show that if ε >0 and if CΦ :A pα →A p α+ 1 2n−ε , then CΦ is bounded on A pα . © 2008 Elsevier Inc. All rights reserved