Composition operators on strictly pseudoconvex domains with smooth symbol

It is well known that the composition operator C_Φ is unbounded on Hardy and Bergman spaces on the unit ball B_n in ℂⁿ when n > 1 for a linear holomorphic self-map Φ of B_n. We find a sufficient and necessary condition for a composition operator with smooth symbol to be bounded on Hardy or Bergman spaces over a bounded strictly pseudoconvex domain in ℂⁿ. Moreover, we show that this condition is equivalent to the compactness of the composition operator from a Hardy or Bergman space into the Bergman space whose weight is 1/4 bigger. We also prove that a certain jump phenomenon occurs when the composition operator is not bounded. Our results generalize known results on the unit ball to strictly pseudoconvex domains.