Composition operators on holomorphic Sobolev spaces in B-n

Abstract

We study the composition operator C-Phi on holomorphic Sobolev spaces induced by an analytic self-map Phi of B-n in C-n that extends to be smooth on (B) over bar (n). We characterize the boundedness and the compactness of C-Phi on A(alpha,s)(p), and prove the jump phenomenon of C-Phi on A(alpha,s)(p). moreover, we show an interesting result that the boundedness of C-Phi on A(alpha,s)(p) is equivalent to the compactness of C-Phi : A(alpha,s)(p) -> A(beta,t)(q) for appropriate A(beta,t)(q), for example A(beta,t)(q) = A(alpha+1/4,s)(p). We provide examples to show that our results are sharp. (C) 2010 Elsevier Inc. All rights reserved.