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Zbl 1030.46012
Takahashi, Yasuji; Hashimoto, Kazuo; Kato, Mikio
On sharp uniform convexity, smoothness, and strong type, cotype inequalities. (English)
[J] J. Nonlinear Convex Anal. 3, No.2, 267-281 (2002). ISSN 1345-4773; ISSN 1880-5221/e

In the first part the authors show that certain modifications of Clarkson's inequality are equivalent to the definition of $p$-uniform smoothness and $q$-uniform convexity of a Banach space $X$. They also show duality for these modifications. \par Then the concepts of $p$-uniform smoothness and $q$-uniform convexity are shown to be equivalent to certain variants of Rademacher type $p$ and cotype $q$ inequalities, dubbed strong type $p$ and strong cotype $q$. The authors study duality of strong type and cotype and how these properties pass from $X$ to $L_p(X)$. \par All this continues and extends previous work by the authors {\it M. Kato, L.-E. Persson,} and {\it Y. Takahashi} [Collect. Math. 51, 327-346 (2000; Zbl 0983.46014)] and {\it M. Kato} and {\it Y. Takahashi} [Math. Nachr. 186, 187-195 (1997; Zbl 0901.46013)].
[Jörg Wenzel (Pretoria)]

MSC 2000:: *46B20 Geometry and structure of normed spaces
46B07 Local theory of Banach spaces
46E40 Spaces of vector-valued functions
46B04 Isometric theory of Banach spaces

Keywords: $p$-uniformly smooth space; $q$-uniformly convex space; type and cotype; strong type and cotype; Clarkson-Boas type inequality; Lebsgue-Bochner space

Citations: Zbl 0983.46014; Zbl 0901.46013

Cited in: Zbl 1087.46007 Zbl 1087.46008

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