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Complex convexity and monotonicity in quasi-Banach lattices

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Abstract

In this paper we study the monotonicity and convexity properties in quasi-Banach lattices. We establish relationship between uniform monotonicity, uniform ℂ-convexity, H-and PL-convexity. We show that if the quasi-Banach lattice E has α-convexity constant one for some 0 < α < ∞, then the following are equivalent: (i) E is uniformly PL-convex; (ii) E is uniformly monotone; and (iii) E is uniformly ℂ-convex. In particular, it is shown that if E has α-convexity constant one for some 0 < α < ∞ and if E is uniformly ℂ-convex of power type then it is uniformly H-convex of power type. The relations between concavity, convexity and monotonicity are also shown so that the Maurey-Pisier type theorem in a quasi-Banach lattice is proved.

Finally we study the lifting property of uniform PL-convexity: if E is a quasi-Köthe function space with α-convexity constant one and X is a continuously quasi-normed space, then it is shown that the quasi-normed Köthe-Bochner function space E(X) is uniformly PL-convex if and only if both E and X are uniformly PL-convex.

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Authors and Affiliations

  1. Department of Mathematics, POSTECH, San 31, Hyoja-dong, Nam-gu, Pohang-shi, Kyungbuk, Republic of Korea

    Han Ju Lee

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  1. Han Ju Lee
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The author acknowledges the financial support of the Korean Research Foundation made in the program year of 2002 (KRF-2002-070-C00005) and BK21 Project.

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Lee, H.J. Complex convexity and monotonicity in quasi-Banach lattices. Isr. J. Math. 159, 57–91 (2007). https://doi.org/10.1007/s11856-007-0038-2

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