Composition operators on holomorphic Sobolev spaces in Bn

We study the composition operator CΦ on holomorphic Sobolev spaces induced by an analytic self-map Φ of Bn in Cn that extends to be smooth on . We characterize the boundedness and the compactness of CΦ on , and prove the jump phenomenon of CΦ on . Moreover, we show an interesting result that the boundedness of CΦ on is equivalent to the compactness of for appropriate , for example . We provide examples to show that our results are sharp.