The existence of radial limits of analytic functions in Banach spaces (Q5955021)

A complex Banach space \(X\) has the analytic Radon-Nikodym property (ARNP) if every \(f\in H^\infty({\mathbf D}, X)\), the space of bounded analytic functions from the open unit disk \({\mathbf D}\) of \(\mathbb{C}\) into \(X\) has radial limits almost everywhere on \({\mathbf T}=\partial {\mathbf D}\). The author studies what happens when \(X\) does not have the ARNP. In previous papers, he showed that, in this case: (1) there exists \(F\in H^\infty({\mathbf D},X)\) such that \(\|F\|_\infty\leq 1\) and, for almost all \(\theta\in [0,2\pi]\), \[ \limsup_{r,s\uparrow 1} \|F(r e^{i\theta}) -F(s e^{i\theta})\|\geq 3/4 \] [\textit{S. Bu}, Ann. Fac. Sci. Toulouse, V. Sér., Math. 11, No. 2, 79-89 (1990; Zbl 0731.46006)]; (2) there exists \(F\in H^\infty({\mathbf D},X)\) and \(0< r_n\uparrow 1\) such that, for all \(\alpha, \beta\in [0,2\pi]\), and all \(m\not= n\), \[ \|F(r_n e^{i\alpha}) - F(r_m e^{i\beta})\|\geq 1 \] [\textit{S. Bu} and \textit{B. Khaoulani}, Math. Ann. 288, No. 2, 345-360 (1990; Zbl 0699.46012)]. In the present paper, the author investigates how big the set of such ``bad'' functions is, and the nature of the set of boundary limit points for such a function. Precisely, he shows that, if \(X\) does not have the ARNP: (a) the set of \(f\in H^\infty({\mathbf D}, X)\) for which there exists an \(\varepsilon >0\) such that \(\limsup_{r,s\uparrow 1} \|f(r e^{i\theta}) -f(s e^{i\theta})\|\geq \varepsilon\) for almost all \(\theta\in [0,2\pi]\) is a dense open set of \(H^\infty({\mathbf D}, X)\) (Theorem 1); (b) for each (non empty) open subset \(A\subseteq {\mathbf T}\), there exists an \(F\in H^\infty({\mathbf D},X)\) such that \(F\) has boundary values everywhere on \(A^c\), but radial limits nowhere on \(A\) (Theorem 4); (c) for every subset \(A\subseteq {\mathbf T}\) of positive measure, there exists \(F\in H^\infty({\mathbf D},X)\) which has nontangential limits almost everywhere on \(A^c\), but radial limits almost nowhere on \(A\) (Theorem 5).