Sharp constants related to the triangle inequality in Lorentz spaces
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- by Sorina Barza, Viktor Kolyada and Javier Soria PDF
- Trans. Amer. Math. Soc. 361 (2009), 5555-5574 Request permission
Abstract:
We study the Lorentz spaces $L^{p,s}(R,\mu )$ in the range $1<p<s\le \infty$, for which the standard functional \[ ||f||_{p,s}=\left (\int _0^\infty (t^{1/p}f^*(t))^s\frac {dt}{t}\right )^{1/s} \] 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: \[ ||f||_{(p,s)}=\inf \bigg \{\sum _{k}||f_k||_{p,s}\bigg \}, \] 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 \[ ||f||_{p,s}’= \sup \left \{ \int _R fg d\mu : ||g||_{p’,s’}=1\right \}\] agree for all values of $p,s>1$.References
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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
- 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. - © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 361 (2009), 5555-5574
- MSC (2000): Primary 46E30, 46B25
- DOI: https://doi.org/10.1090/S0002-9947-09-04739-4
- MathSciNet review: 2515823