Identification of fully measurable grand Lebesgue spaces
type: Item
instance of: scholarly article
title: Identification of fully measurable grand Lebesgue spaces
zbMATH Open document ID: 1386.46029
DOI: 10.1155/2017/3129186
author: G. Anatriello
author: Ralph Chill
author: A. Fiorenza
published in: Journal of Function Spaces
publication date: 2017-11-01T00:00:00Z
review text: Summary: We consider the Banach function spaces, called fully measurable grand Lebesgue spaces, associated with the function norm \(\rho(f)=\operatorname{ess}\sup_{x\in X}\delta(x)\rho_{p(x)}(f)\), where \(\rho_{p(x)}\) denotes the norm of the Lebesgue space of exponent \(p(x)\), and \(p(\cdot)\) and \(\delta(\cdot)\) are measurable functions over a measure space \((X,\nu)\), \(p(x)\in [1,\infty]\), and \(\delta(x)\in (0,1]\) almost everywhere. We prove that every such space can be expressed equivalently replacing \(p(\cdot)\) and \(\delta(\cdot)\) with functions defined everywhere on the interval \((0,1)\), decreasing and increasing, respectively (hence the full measurability assumption in the definition does not give an effective generalization with respect to the pointwise monotone assumption and the essential supremum can be replaced with the simple supremum). In particular, we show that, in the case of bounded \(p(\cdot)\), the class of fully measurable Lebesgue spaces coincides with the class of generalized grand Lebesgue spaces introduced by Capone, Formica, and Giova [\textit{C. Capone} et al., Nonlinear Anal., Theory Methods Appl., Ser.~A, Theory Methods 85, 125--131 (2013; Zbl 1286.46030)].
Mathematics Subject Classification ID: 46E30
zbMATH DE Number: 6801994
zbMATH Keywords: grand Lebesgue spaces
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2017/3129186
OpenAlex ID: W2762369053
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description: scientific article; zbMATH DE number 6801994