Composition operators on holomorphic Sobolev spaces in B_n

We study the composition operator C_Φ on holomorphic Sobolev spaces induced by an analytic self-map Φ of B_n in Cⁿ 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.