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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Uniqueness of dilation invariant norms


Authors: E. Moreno and A. R. Villena
Translated by:
Journal: Proc. Amer. Math. Soc. 132 (2004), 2067-2073
MSC (2000): Primary 46E30, 46H40
Published electronically: January 29, 2004
MathSciNet review: 2053979
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Abstract: Let $\delta_a$ be a nontrivial dilation. We show that every complete norm $\Vert\cdot\Vert$ on $L^1(\mathbb{R} ^N)$ that makes $\delta_a$ from $(L^1(\mathbb{R} ^N),\Vert\cdot\Vert)$ into itself continuous is equivalent to $\Vert\cdot\Vert _1$. $\delta_a$ also determines the norm of both $C_0(\mathbb{R} ^N)$ and $L^p(\mathbb{R} ^N)$ with $1<p<\infty$ in a weaker sense. Furthermore, we show that even all the dilations do not determine the norm on $L^\infty(\mathbb{R} ^N)$.


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Additional Information

E. Moreno
Affiliation: Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain

A. R. Villena
Affiliation: Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain
Email: avillena@ugr.es

DOI: http://dx.doi.org/10.1090/S0002-9939-04-07327-7
PII: S 0002-9939(04)07327-7
Received by editor(s): November 19, 2002
Received by editor(s) in revised form: April 1, 2003
Published electronically: January 29, 2004
Additional Notes: The second author was supported by MCYT Grant BFM2003-01681.
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2004 American Mathematical Society