Strongly exposed points in Orlicz-Lorentz spaces equipped with the Orlicz norm

TL;DR

This work addresses the geometric structure of Orlicz-Lorentz spaces $\Lambda_{\varphi, \omega}^{o}$ endowed with the Orlicz norm by characterizing strongly exposed points in their unit ball. It develops a framework based on duality with the Köthe dual $\mathcal{M}_{\psi, \omega}$, level-interval machinery, and rearrangement methods to describe supporting functionals and the sets $K(x)$ that govern norm representations. The authors establish necessary and sufficient conditions for the strongly exposed property, including precise norm formulas when $K(x)$ is nonempty ( $\|x\|_{\varphi, \omega}^{o}=\frac{1}{k}(1+\rho_{\varphi, \omega}(kx))$ for $k\in K(x)$ ) and the case $K(x)=\emptyset$ ( $\|x\|_{\varphi, \omega}^{o}=B\int_{0}^{\infty} x^{*}\omega$ ), along with exact criteria on $\varphi$ (e.g., $\varphi\in\Delta_{2}$ and $\varphi\in\nabla_{2}$ and strict convexity) and measure-theoretic conditions on $kx^{*}$. These results advance the understanding of dentability and separation properties in symmetric Banach function spaces and provide concrete tools for identifying strongly exposed points in Orlicz-Lorentz contexts.

Abstract

The criterion for a point in the unit ball to be a strongly exposed point is given. The necessity and sufficiency conditions for Orlicz-Lorentz spaces to possess strongly exposed property are given. Besides, some useful methods are obtained to handle issues related to decreasing rearrangement.

Theorems & Definitions (26)

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