标题： 关于Köthe对偶的Orlicz-Lorentz空间 显示英文标题

标题： On Köthe duals of Orlicz-Lorentz spaces

摘要： 在本文中，我们研究了Orlicz-Lorentz空间的Köthe对偶空间$\mathcal{M}_{\varphi,w}$的一些性质。提供了Orlicz-Lorentz空间$\mathcal{M}_{\varphi,w}$的序连续子空间的显式描述。此外，这些空间的可分性由增长条件$\Delta_2$来表征。因此，当无穷远处的N函数$\varphi$满足适当的$\Delta_2$条件时，Köthe对偶空间$\mathcal{M}_{\varphi,w}$具有Radon-Nikodým性质。通过Orlicz函数之间的标准顺序，表征了$\mathcal{M}_{\varphi,w}$空间之间的比较。作为这些结果的应用，我们提供了配备Luxemburg范数的Orlicz-Lorentz空间的M-嵌入序连续子空间的充分条件，并证明了配备Orlicz范数的Orlicz-Lorentz空间上唯一保范延拓的存在性。 显示英文摘要

摘要： In this article, we study a number of properties of the K\"othe duals $\mathcal{M}_{\varphi,w}$ of Orlicz-Lorentz spaces. An explicit description of the order-continuous subspace of $\mathcal{M}_{\varphi,w}$ is provided. Moreover, the separability of these spaces is characterized by the growth condition $\Delta_2$. Consequently, the K\"othe dual space $\mathcal{M}_{\varphi,w}$ has the Radon-Nikod\'ym property if and only if the N-function at infinity $\varphi$ satisfies the appropriate $\Delta_2$-condition. The comparison between $\mathcal{M}_{\varphi,w}$ spaces is characterized via standard orders between Orlicz functions. As applications of these results, we provide sufficient conditions for M-embedded order-continuous subspaces of Orlicz-Lorentz spaces equipped with the Luxemburg norm and prove the existence of a unique norm-preserving extension on Orlicz-Lorentz spaces equipped with the Orlicz norm.