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A note on the spaces of variable integrability and summability of Almeida and Hästö


Authors: Henning Kempka and Jan Vybíral
Journal: Proc. Amer. Math. Soc. 141 (2013), 3207-3212
MSC (2010): Primary 46E35
DOI: https://doi.org/10.1090/S0002-9939-2013-11605-9
Published electronically: May 31, 2013
MathSciNet review: 3068973
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Abstract | References | Similar Articles | Additional Information

Abstract: We address an open problem posed recently by Almeida and Hästö. They defined the spaces $ {\ell _{q(\cdot )}(L_{p(\cdot )})}$ of variable integrability and summability and showed that $ \Vert\cdot \vert{\ell _{q(\cdot )}(L_{p(\cdot )})}\Vert$ is a norm if $ q\ge 1$ is constant almost everywhere or if $ 1/p(x)+1/q(x)\le 1$ for almost every $ x\in \mathbb{R}^n$. Nevertheless, the natural conjecture (expressed also by Almeida and Hästö) is that the expression is a norm if $ p(x),q(x)\ge 1$ almost everywhere. We show that $ \Vert\cdot \vert{\ell _{q(\cdot )}(L_{p(\cdot )})}\Vert$ is a norm if $ 1\le q(x)\le p(x)$ for almost every $ x\in \mathbb{R}^n$. Furthermore, we construct an example of $ p(x)$ and $ q(x)$ with $ \min (p(x),q(x))\ge 1$ for every $ x\in \mathbb{R}^n$ such that the triangle inequality does not hold for $ \Vert\cdot \vert{\ell _{q(\cdot )}(L_{p(\cdot )})}\Vert$.


References [Enhancements On Off] (What's this?)

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Additional Information

Henning Kempka
Affiliation: Mathematical Institute, Friedrich-Schiller-University, D-07737 Jena, Germany
Email: henning.kempka@uni-jena.de

Jan Vybíral
Affiliation: RICAM, Austrian Academy of Sciences, Altenbergstrasse 69, A-4040 Linz, Austria
Email: jan.vybiral@oeaw.ac.at

DOI: https://doi.org/10.1090/S0002-9939-2013-11605-9
Keywords: Triangle inequality, Lebesgue spaces with variable exponent, iterated Lebesgue spaces
Received by editor(s): February 1, 2011
Received by editor(s) in revised form: November 29, 2011
Published electronically: May 31, 2013
Additional Notes: The first author acknowledges the financial support provided by DFG project HA 2794/5-1, “Wavelets and Function Spaces on Domains”.
The second author acknowledges the financial support provided by FWF project Y 432-N15 START-Preis, “Sparse Approximation and Optimization in High Dimensions”.
The authors would like to thank the referee for useful hints, which helped to improve the paper.
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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