Remote Access Transactions of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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$.

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

  • 1. C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press, Boston, 1988. MR 928802 (89e:46001)
  • 2. M. J. Carro, J. A. Raposo, and J. Soria, Recent Developments in the Theory of Lorentz Spaces and Weighted Inequalities, Mem. Amer. Math. Soc. 187, Providence, RI, 2007. MR 2308059 (2008b:42034)
  • 3. M. J. Carro and J. Soria, Weighted Lorentz spaces and the Hardy operator, J. Funct. Anal. 112 (1993), 480-494. MR 1213148 (94f:42025)
  • 4. E. DiBenedetto, Real Analysis, Birkhäuser, Boston, 2002. MR 1897317 (2003d:00001)
  • 5. I. Halperin, Function spaces, Canad. J. Math. 5 (1953), 273-288. MR 0056195 (15:38h)
  • 6. H. P. Heinig and L. Maligranda, Chebyshev inequality in function spaces, Real Anal. Exchange 17 (1991-92), 211-247. MR 1147365 (92k:26045)
  • 7. R. A. Hunt, On $ L(p, q)$, Enseignement Math. 12 (1966), 249-276. MR 0223874 (36:6921)
  • 8. V. I. Kolyada, Rearrangement of functions and embedding of anisotropic spaces of Sobolev type, East J. Approx. 4 (1998), no. 2, 111-199. MR 1638343 (99g:46043a)
  • 9. V. I. Kolyada, Inequalities of Gagliardo-Nirenberg type and estimates for the moduli of continuity, Russian Math. Surveys 60 (2005), 1147-1164. MR 2215758 (2007b:26026)
  • 10. G. G. Lorentz, Some new functional spaces, Ann. of Math. 51 (1950), 37-55. MR 0033449 (11:442d)
  • 11. G. G. Lorentz, On the theory of spaces $ \Lambda$, Pacific J. Math. 1 (1951), 411-429. MR 0044740 (13:470c)
  • 12. G. G. Lorentz, Bernstein polynomials, Univ. of Toronto Press, Toronto, 1953. MR 0057370 (15:217a)
  • 13. E. Sawyer, Boundedness of classical operators on classical Lorentz spaces, Studia Math. 96 (1990), 145-158. MR 1052631 (91d:26026)
  • 14. E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, 1971. MR 0304972 (46:4102)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 46E30, 46B25

Retrieve articles in all journals with MSC (2000): 46E30, 46B25

Additional Information

Sorina Barza
Affiliation: Department of Mathematics, Karlstad University, SE-65188 Karlstad, Sweden

Viktor Kolyada
Affiliation: Department of Mathematics, Karlstad University, SE-65188 Karlstad, Sweden

Javier Soria
Affiliation: Department of Applied Mathematics and Analysis, University of Barcelona, E-08007 Barcelona, Spain

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.

American Mathematical Society