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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Sharp constants related to the triangle inequality in Lorentz spaces


Authors: Sorina Barza, Viktor Kolyada and Javier Soria
Journal: Trans. Amer. Math. Soc. 361 (2009), 5555-5574
MSC (2000): Primary 46E30, 46B25
Published electronically: May 20, 2009
MathSciNet review: 2515823
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Abstract: We study the Lorentz spaces $ L^{p,s}(R,\mu)$ in the range $ 1<p<s\le \infty$, for which the standard functional

$\displaystyle \vert\vert f\vert\vert _{p,s}=\left(\int_0^\infty (t^{1/p}f^*(t))^s\frac{dt}{t}\right)^{1/s} $

is only a quasi-norm. We find the optimal constant in the triangle inequality for this quasi-norm, which leads us to consider the following decomposition norm:

$\displaystyle \vert\vert f\vert\vert _{(p,s)}=\inf\bigg\{\sum_{k}\vert\vert f_k\vert\vert _{p,s}\bigg\}, $

where the infimum is taken over all finite representations $ f=\sum_{k}f_k. $ We also prove that the decomposition norm and the dual norm

$\displaystyle \vert\vert f\vert\vert _{p,s}'= \sup\left\{ \int_R fg d\mu: \vert\vert g\vert\vert _{p',s'}=1\right\}$

agree for all values of $ p,s>1$.


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Additional Information

Sorina Barza
Affiliation: Department of Mathematics, Karlstad University, SE-65188 Karlstad, Sweden
Email: sorina.barza@kau.se

Viktor Kolyada
Affiliation: Department of Mathematics, Karlstad University, SE-65188 Karlstad, Sweden
Email: viktor.kolyada@kau.se

Javier Soria
Affiliation: Department of Applied Mathematics and Analysis, University of Barcelona, E-08007 Barcelona, Spain
Email: soria@ub.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-09-04739-4
PII: S 0002-9947(09)04739-4
Keywords: Equivalent norms, level function, Lorentz spaces, sharp constants
Received by editor(s): June 25, 2007
Received by editor(s) in revised form: January 8, 2008
Published electronically: May 20, 2009
Additional Notes: An essential part of this work was performed while the first and second authors stayed at the University of Barcelona as invited researchers. We express our gratitude to the Department of Mathematics of the University of Barcelona for the hospitality and excellent conditions.
The third author was partially supported by grants MTM2007-60500 and 2005SGR00556.
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.