A note on the spaces of variable integrability and summability of Almeida and Hästö
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- by Henning Kempka and Jan Vybíral PDF
- Proc. Amer. Math. Soc. 141 (2013), 3207-3212 Request permission
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 $\|\cdot |{\ell _{q(\cdot )}(L_{p(\cdot )})}\|$ 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 $\|\cdot |{\ell _{q(\cdot )}(L_{p(\cdot )})}\|$ 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 $\|\cdot |{\ell _{q(\cdot )}(L_{p(\cdot )})}\|$.References
- Alexandre Almeida and Peter Hästö, Besov spaces with variable smoothness and integrability, J. Funct. Anal. 258 (2010), no. 5, 1628–1655. MR 2566313, DOI 10.1016/j.jfa.2009.09.012
- L. Diening, P. Hästö, and S. Roudenko, Function spaces of variable smoothness and integrability, J. Funct. Anal. 256 (2009), no. 6, 1731–1768. MR 2498558, DOI 10.1016/j.jfa.2009.01.017
- Lars Diening, Petteri Harjulehto, Peter Hästö, and Michael Růžička, Lebesgue and Sobolev spaces with variable exponents, Lecture Notes in Mathematics, vol. 2017, Springer, Heidelberg, 2011. MR 2790542, DOI 10.1007/978-3-642-18363-8
- Ondrej Kováčik and Jiří Rákosník, On spaces $L^{p(x)}$ and $W^{k,p(x)}$, Czechoslovak Math. J. 41(116) (1991), no. 4, 592–618. MR 1134951
- W. Orlicz: Über konjugierte Exponentenfolgen, Studia Math. 3 (1931), 200–212.
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
- 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
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 141 (2013), 3207-3212
- MSC (2010): Primary 46E35
- DOI: https://doi.org/10.1090/S0002-9939-2013-11605-9
- MathSciNet review: 3068973