On inner functions with HpHp and BpBp derivatives in the unit ball of CnCn

Let BnBn be the unit ball in CnCn. If f is a bounded holomorphic function, we say that f is inner provided thatlimr→1−|f(rζ)|=1σ-a.e.,ζ∈Sn where SnSn is the unit sphere and σ   is normalized surface measure on SnSn. If β>−1β>−1 and p>0p>0 then Aβp denotes the weighted Bergman space of all holomorphic functions weighted by β(1−|z|2)(1−|z|2)β. For 00p>0 let HpHp denote the usual Hardy space of holomorphic functions on the ball. In this paper, we consider derivatives of inner functions in several spaces of holomorphic functions. If f   is an inner function, membership of the radial derivative, Rf=∑j=1nzj∂f∂zj, will be considered in the BpBp spaces for p>nn+1 and will be related to membership in weighted Dirichlet spaces, weighted Bergman spaces Aα2 for 0<α<10<α<1, and to the ApAp spaces for 11n>1, and either Rf∈B2n2n+1, Rf∈A3/2Rf∈A3/2, or Rf∈H1/2Rf∈H1/2 then f must be constant.