VECTOR-VALUED HOLOMORPHIC FUNCTIONS ON THE COMPLEX BALL AND THE ANALYTIC RADON-NIKODYM PROPERTY

The complex Banach spaces X with values in which every bounded holomorphic function in the unit ball B of Cd(d > 1) has boundary limits almost surely are exactlythe spaces with the analytic Radon-Nikodym property.The proof is based on inner Hardymartingales introduced here. The inner Hardy martingales are constructed in terms ofinner functions in B and are reasonable discrete approximations for the image processesof the holomorphic Brownian motion under X-valued holomorpkic functions in B.