The failure of Fatou's theorem on Poisson integrals of Pettis integrable functions
type: Item
instance of: scholarly article
title: The failure of Fatou's theorem on Poisson integrals of Pettis integrable functions
zbMATH Open document ID: 0922.28010
author: Francisco J. Freniche
author: Juan Carlos García-Vázquez
author: Luis Rodríguez-Piazza
published in: Journal of Functional Analysis
publication date: 1999-05-18T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 28B05
Mathematics Subject Classification ID: 31A05
zbMATH DE Number: 1246376
zbMATH Keywords: Pettis integrable functions
zbMATH Keywords: analytic function failing Fatou's theorem
zbMATH Keywords: countably additive vector measure
reviewed by: Minoru Matsuda
MaRDI profile type: Publication
OpenAlex ID: W2041498809
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DOI: 10.1006/JFAN.1998.3338
description: scientific article; zbMATH DE number 1246376