Composition operators on holomorphic Sobolev spaces in

We study the composition operator C Φ on holomorphic Sobolev spaces induced by an analytic self-map Φ of B n in C n that extends to be smooth on B n ¯ . We characterize the boundedness and the compactness of C Φ on A α , s p , and prove the jump phenomenon of C Φ on A α , s p . Moreover, we show an interesting result that the boundedness of C Φ on A α , s p is equivalent to the compactness of C Φ : A α , s p → A β , t q for appropriate A β , t q , for example A β , t q = A α + 1 / 4 , s p . We provide examples to show that our results are sharp.