The failure of Fatou's theorem on Poisson integrals of Pettis integrable functions
The problem of extending classical theorems in harmonic analysis to the setting of Banach space valued functions or measures has been studied by many authors. For instance, it is known as a Fatou's type theorem that for every infinite dimensional Banach space \(X\), every \(1\leq p<+\infty\) and every \(f\in L_p(T,X)\) (the Banach space of all \(X\)-valued \(p\)-th power Bochner integrable functions on the unit circle \(T)\), the \(X\)-valued harmonic function \(P_r * f\) defined (in the Bochner sense) as the convolution with the Poisson kernel \(P_r\) converges to \(f\) in the norm of \(L_p(T,X)\) as \(r\to 1\) and \(P_r * ft\to f(t)\) for almost all \(t\in T\). In this paper, authors investigate problems of such type in the case of Pettis integrable functions on \(T\). Inspired by \textit{S. J. Dilworth} and \textit{M. Girardi} [Quaest. Math. 18, No. 4, 365-380 (1995; Zbl 0856.28006)], they construct a strongly measurable \(X\)-valued \(p\)-Pettis integrable function \(F\) on \(T\) such that \(\| P_r * F(t)\|\to +\infty\) uniformly in \(t\in T\), as \(r\to 1\), and they show that this does not admit a conjugate function. They also note that \(P_r * F(t)\) is harmonic but not analytic on the unit disc. Concerning the problem of finding an analytic function failing Fatou's theorem, they give an analytic \(p\)-Pettis integrable function \(G\) on \(T\) such that \(\lim\sup_{r\to 1}\| P_r * G(t) \|=+\infty\) at any \(t\in T\), in the case that \(X\) does not have finite cotype. In the general case, they also give an \(X\)-valued analytic countably additive vector measure with the similar property.
analytic function failing Fatou's theorem
scientific article; zbMATH DE number 1246376