A direct approach to the duality of grand and small Lebesgue spaces
type: Item
instance of: scholarly article
title: A direct approach to the duality of grand and small Lebesgue spaces
zbMATH Open document ID: 1184.46032
author: Giovanni Di Fratta
author: A. Fiorenza
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2009-03-25T00:00:00Z
review text: Let \(\mathcal{M}_{0}\) be the set of all Lebesgue measurable function in the interval \((0,1)\), finite a.e.\ in it and let \(\mathcal{M}_{0}^{+}\) be the class of all nonnegative functions of \(\mathcal{M}_{0}\). For \(f \in \mathcal{M}_{0}\), the decreasing rearrangement \(f^{*}\) of \(f\) is defined by \[ f^{*} = \inf\{ \lambda > 0: |\{x \in (0,1): |f(x)| > \lambda\}| \leq t \}, \] where the symbol \(|E|\) denotes the Lebesgue measure of \(E\). Let \(1 < p < \infty\) and \(\Theta > 0\). For all \(f \in \mathcal{M}_{0}\), we set \[ |\lceil f \rceil|_{p,\Theta} := \sup_{0<t<p}(1 - \log t) ^{-\frac{\Theta}{p}} \left(\int_{0}^{1}f^{*}(s)^{p}\,ds \right)^{\frac{1}{p}} \] and \[ \|f\|_{p,\Theta} := \sup_{0<\varepsilon<p-1}\varepsilon^{\Theta} \left(\int_{0}^{1} |f^{*}(x)|^{p-\varepsilon}\,dx \right)^{\frac{1}{p-\varepsilon}}. \] The authors prove that \[ c_{1}\|f\|_{p,\Theta} \leq |\lceil f \rceil|_{p,\Theta} \leq \|f\|_{p,\Theta} \] for some \(c_{1}, c_{2} > 0\) and for every \(f \in \mathcal{M}_{0}^{+}\). A Hölder type inequality is also proved.
reviewed by: Andrzej Smajdor
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 26D15
zbMATH DE Number: 5532904
zbMATH Keywords: Banach function spaces
zbMATH Keywords: Hölder's inequality
zbMATH Keywords: associate spaces
zbMATH Keywords: grand Lebesgue spaces
zbMATH Keywords: small Lebesgue spaces
zbMATH Keywords: Euler's gamma function
MaRDI profile type: Publication
full work available at URL: j.na.2008.03.044
OpenAlex ID: W1975403860
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DOI: 10.1016/J.NA.2008.03.044
description: scientific article; zbMATH DE number 5532904