Interpolation of vector measures
type: Item
instance of: scholarly article
title: Interpolation of vector measures
zbMATH Open document ID: 1236.46016
author: Fernando Mayoral
author: Francisco Naranjo
author: Antonio Fernández
author: E. A. Sánchez-Pérez
author: Ricardo del Campo
publication date: 2011-06-24T00:00:00Z
review text: Let \((\Omega,\Sigma)\) be a measurable space and \(m_{0}: \Sigma \rightarrow X_{0}\) and \(m_{1}: \Sigma \rightarrow X_{1}\) be positive vector measures with values in the Banach Köthe function spaces \(X _{0}\) and \(X _{1}\). If \(0 < \alpha < 1\), the authors define a new vector measure \([m_{0}, m_{1}]_{\alpha}\) with values in the Calderón lattice interpolation space \(X_{0}^{1-\alpha} X_{1}^{\alpha}\). They analyse the space of integrable functions with respect to the measure \([m_{0}, m_{1}]_{\alpha}\) in order to prove suitable extensions of the classical Stein-Weiss formulas that hold for the complex interpolation of \(L^{p}\)-spaces. More precisely, their aim is to study under which assumptions on the vector measures and the indices involved the formula \([L^{p_0}(m_0), L^{p_1}(m_1)]_\theta=L^p([m_0,m_1]_\alpha)\) holds isometrically, where \(1/p=(1-\theta)/p_0 + \theta/p_1\) and \(\alpha=p \theta/p_1\). For the case \(m_0=m_1\), this is essentially known to be true by a result of the authors (minus the first named author) [Collect.\ Math.\ 61, No.\ 3, 241--252 (2010; Zbl 1228.46017)]. However, in the more general case, matters turn out to be more delicate (the authors provide a counterexample). Eventually, the authors prove that the above formula (where the complex interpolation space may have to be replaced by the Calderón lattice interpolation space to avoid more restrictions) holds whenever the same formula holds for the case \(p_0=p_1=1\) and \(\theta=\alpha\) (the authors state some affirmative examples). As applications, they provide a Riesz-Thorin theorem for spaces of \(p\)-integrable functions with respect to vector measures and a formula for representing the interpolation of the injective tensor product of such spaces.
Mathematics Subject Classification ID: 46B70
Mathematics Subject Classification ID: 46G10
Mathematics Subject Classification ID: 46E30
zbMATH DE Number: 5912873
zbMATH Keywords: interpolation
zbMATH Keywords: Banach function space
zbMATH Keywords: vector measure
reviewed by: Carsten Michels
MaRDI profile type: Publication
full work available at URL: s10114-011-9542-8
OpenAlex ID: W2166530649
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DOI: 10.1007/S10114-011-9542-8
published in: Acta Mathematica Sinica, English Series
description: scientific article; zbMATH DE number 5912873