THE EXISTENCE OF RADIAL LIMITS OF ANALYTIC FUNCTIONS IN BANACH SPACES

Let X be a complex Banach space without the analytic Radon-Nikodym property. Theauthor shows that G = {f ∈ H∞(D,X): there exists e ＞ 0, such that for almost all θ ∈[0, 2π], limsup ‖f(rei) - f(sei)‖ ＞ ∈ } is a dense open subset of H (D, X). It is also shown r,s↑1that for every open subset B of T, there exists F ∈ H∞(D,X), such that F has boundary values everywhere on Bc and F has radial limits nowhere on B. When A is a measurable subset of T with positive measure, there exists f ∈ H∞(D, X), such that f has nontangential limits almost eyerywhere on Ac and f has radial limits almost nowhere on A.