Order convexity and concavity of Lorentz spaces ${\mit \Lambda }_{p,w}$, $0< p< \infty $
Volume 160 / 2004
Studia Mathematica 160 (2004), 267-286
MSC: Primary 46E30, 46B20; Secondary 46B42, 46B25.
DOI: 10.4064/sm160-3-5
Abstract
We study order convexity and concavity of quasi-Banach Lorentz spaces ${\mit \Lambda }_{p,w}$, where $0< p< \infty $ and $w$ is a locally integrable positive weight function. We show first that ${\mit \Lambda }_{p,w}$ contains an order isomorphic copy of $l^p$. We then present complete criteria for lattice convexity and concavity as well as for upper and lower estimates for ${\mit \Lambda }_{p,w}$. We conclude with a characterization of the type and cotype of ${\mit \Lambda }_{p,w}$ in the case when ${\mit \Lambda }_{p,w}$ is a normable space.
