Some remarks on boundary values of vector-valued harmonic and analytic functions
Let X be a complex Banach space and D the open unit disc in \({\mathbb{C}}\), \[ h^ 1(X):=\{u: D\to X\quad harmonic:\quad \sup_{r<1}\int^{2\pi}_{0}\| u(re^{i\delta})\|_ X(d\delta /2\pi)<\infty \}, \] \[ N(X):=\{f: D\to X\quad analytic:\quad \sup_{r<1}\int^{2\pi}_{0}\ell n^+\| f(re^{i\delta})\|_ X(d\delta /2\pi)<\infty \}. \] The paper consists of amendments to \textit{A. V. Bukhvalov}'s and \textit{A. A. Danilevich}'s ``Boundary properties of analytic and harmonic functions with values in Banach space'' [Mat. Zametki 31, 203-214 (1982; Zbl 0496.30029)]. Specifically it is proved that if X has RNP, then every \(u\in h^ 1(X)\) possesses angular limits a.e. on \(\partial D\). If \(u\in h^ 1(X)\) has a.e. weak radial limits, then u has a.e. strong angular limits. If X is separable and Y has RNP, then very \(u\in h^ 1(L(X,Y))\) admits a.e. angular limits in the strong operator topology. In the analytic case, the same results hold, where \(h^ 1\) can be replaced by N and RNP by ARNP.
scientific article; zbMATH DE number 4140540
scientific article