Composition operators on the polydisc induced by smooth symbols

We study composition operators C_Φ on the Hardy spaces H^p and weighted Bergman spaces A_α^p of the polydisc Dⁿ in Cⁿ. When Φ is of class C² on over(Dⁿ, -), we show that C_Φ is bounded on H^p or A_α^p 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 C_Φ : A_α^p → A_{α + frac(1, 2 n) - ε}^p, then C_Φ is bounded on A_α^p.