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Clarkson and Random Clarkson Inequalities for Lr(X)

✍ Scribed by Yasuji Takahashi; Mikio Kato

Publisher: John Wiley and Sons
Year: 1997
Tongue: English
Weight: 367 KB
Volume: 188
Category: Article
ISSN: 0025-584X
DOI: 10.1002/mana.19971880118

No coin nor oath required. For personal study only.

✦ Synopsis

We first show how ( p , p ' ) Clarkson inequality for a Banach space X is inherited by Lebesgue -Bochner spaces L, (X), which extends C LARKSON'S procedure deriving his inequalities for L, from their scalar versions. Fairly many previous and new results on Clarkson's inequalities, and also those on Rademacher type and cotype at the same time (by a recent result of the authors), are obtained as immediate consequences. Secondly we show that if the (p, p') Clarkson inequality holds in X, then random Clarkson inequalities hold in Lp(X) for any 1 5 T 5 00; the converse is true if r = p'. As corollaries the original Clarkson and random Clarkson inequalities for L, are both directly derived from the parallelogram law for scalars.

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